UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
POSGRADO EN CIENCIAS FÍSICAS
INSTITUTO DE FÍSICA
Divisibility classes of qubit maps and singular Gaussian
channels
TESIS
QUE PARA OPTAR POR EL GRADO DE:
DOCTOR EN CIENCIAS (FÍSICA)
PRESENTA:
DAVID DÁVALOS GONZÁLEZ
Director de Tesis:
Dr. Carlos Francisco Pineda Zorrilla
Instituto de Fı́sica
Codirector de Tesis:
Dr. Mario Ziman
Instituto de Fı́sica, Academia Eslovaca de Ciencias
Miembros del comité tutor:
Dr. Carlos Francisco Pineda Zorrilla
Instituto de Fı́sica
Dr. Luis Benet Fernández
Instituto de Ciencias Fı́sicas
Thomas Henry Seligman Schurch
Instituto de Ciencias Fı́sicas
Ciudad Universitaria, Ciudad de México, Enero 2020
 
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UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
POSGRADO EN CIENCIAS FÍSICAS
Divisibility classes of qubit maps and
singular Gaussian channels
TESIS
Que para obtener el grado de: Doctor en Ciencias (Fı́sica)
Presenta: David Dávalos González
Director de tesis: Dr. Carlos Francisco Pineda Zorrilla
Codirector: Dr. Mario Ziman
Miembros del Comité Tutoral:
Dr. Carlos Pineda, Dr. Luis Benet, y Dr. Thomas H. Seligman
México D. F., 2019

UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
POSGRADO EN CIENCIAS FÍSICAS
Divisibility classes of qubit maps and
singular Gaussian channels
THESIS
To obtain the degree: Doctor en Ciencias (Fı́sica)
Presents: David Dávalos González
Director: Dr. Carlos Francisco Pineda Zorrilla
Co-director: Dr. Mario Ziman
Members of the Tutorial Committee:
Dr. Carlos Pineda, Dr. Luis Benet, and Dr. Thomas H. Seligman
México D. F., 2019

Truth is ever to be found in simplicity, and not in the multiplicity and confusion
of things.
Isaac Newton

Gracias
La idea que nació cuando cursaba la secundarı́a, la de convertirme algún dı́a en
cientı́fico, no habrı́a sido posible de no haber nacido en el seno de una familia
estable, funcional y de sólidos valores. Por eso les agradezco infinitamente a mis
queridos Padres. A mi Madre, Sara, por dedicarme tanto de su tiempo y energı́a,
por ser una Madre muy amorosa, llena de valores y por ser la persona mas paciente
del mundo. Le agradezco a mi Padre, Juan Manuel, por siempre estar atento a que
fuera una persona de principios y un buen ciudadano, por ser un padre amoroso
y por darme su confianza. Le agradezco que siempre se haya preocupado por
tener una computadora en casa y por ser un entusiasta de la tecnologı́a, eso aportó
fundamentalmente a quien soy hoy. A mis hermanas y hermanos por preocuparse
por mi y por regalarme tantas veces su tiempo. A mi compañera de vida, a mi
esposa Lorena, gracias por tenerme tanta paciencia, por creer en mi y por quer-
erme tanto. A mi tutor y amigo, Carlos, le agradezco su paciencia, sus valiosas
enseñanzas, su gran apoyo y su amistad. A Thomas Seligman y Luis Benet por
siempre apoyarme. Le agradezco a mis amigos Luis Juarez, Arturo Carranza,
Thomas Gorin, Mario Ziman, Diego Wisniacki, Ignacio Garcı́a Mata, Juan Diego
Urbina, Peter Rapčan, Tomáš Rybár, Edgar Aguilar, David Amaro, Alvaro Dı́az,
Miguel Cardona, Chayo Camarena, Antonio Rosado, Sergio Sánchez, Nephtalı́
Garrido, Sergio Pallaleo, Samuel Rosalio. A mis gatitos Lola y Dalı́ por hacerme
feliz el poco tiempo que estuvieron en este mundo. A mi querida gatita Lulú por
hacer de mi hogar siempre un lugar feliz. Les agradezco a todos los seres queridos
que hicieron de esta parte de mi trayectoria algo memorable.
Agradezco el apoyo brindado por los proyectos PAPIIT número IG100518 y CONA-
CYT CB-285754.

SYNOPSIS
We present two projects concerning the main part of my PhD work. In the first
one we study quantum channels, which are the most general operations mapping
quantum states into quantum states, from the point of view of their divisibility
properties. We introduced tools to test if a given quantum channel can be imple-
mented by a process described by a Lindblad master equation. This in turn defines
channels that can be divided in such a way that they form a one-parameter semi-
group, thus introducing the most restricted studied divisibility type of this work.
Using our results, together the study of other types of divisibility that can be found
in the literature, we characterized the space of qubit quantum channels. We found
interesting results connecting the concept of entanglement-breaking channel and
infinitesimal divisibility. Additionally we proved that infinitely divisible channels
are equivalent to the ones that are implementable by one-parameter semigroups,
opening this question for more general channel spaces. In the second project
we study the functional forms of one-mode Gaussian quantum channels in the
position state representation, beyond Gaussian functional forms. We perform a
black-box characterization using complete positivity and trace preserving condi-
tions, and report the existence of two subsets that do not have a functional Gaus-
sian form. The study covers as particular limit the case of singular channels, thus
connecting our results with the known classification scheme based on canonical
forms. Our full characterization of Gaussian channels without Gaussian func-
tional form is completed by showing how Gaussian states are transformed under
these operations, and by deriving the conditions for the existence of master equa-
tions for the non-singular cases.
Keywords: divisibility, qubit channels, open quantum systems.

RESUMEN
En esta tesis se presentan dos proyectos realizados durante mis estudios de doctor-
ado. En el primero se estudian los canales cuánticos, que son las operaciones más
generales que transforman estados cuánticos en estados cuánticos, desde el punto
de vista de sus propiedades de divisibilidad. Introducimos herramientas para pro-
bar si un canal cuántico dado puede ser implementado por un proceso descrito
por una ecuación maestra de Lindblad. Ésto a su vez define a los canales que
pueden ser divididos de tal manera que ellos forman semigrupos de un parámetro,
introduciendo entonces el tipo más restringido de divisibilidad estudiado de este
trabajo. Usando nuestros resultados, junto con el estudio de otros tipos de divis-
ibilidad que pueden ser encontrados en la literatura, caracterizamos el espacio de
canales cuánticos de un qubit. Encontramos resultados interesantes que conectan
el concepto de canales que rompen el entrelazamiento (del sistema con cualquier
sistema auxiliar) y el de divisibilidad infinitesimal. Además probamos que el
conjunto de canales infinitamente divisibles es equivalente al de los canales im-
plementables por semigrupos de un parámetro. Ésto abre la pregunta sobre si esto
sucede para espacios de canales más generales. En el segundo proyecto estudi-
amos las formas funcionales de canales Gaussianos de un solo modo, más allá
de la forma funcional Gaussiana. Se hace una caracterización de caja negra uti-
lizando las condiciones de completa positividad y preservación de la traza, y se
reporta la existencia de dos subconjuntos que no poseen forma funcional Gaus-
siana. El estudio cubre en particular el lı́mite de los canales singulares, conectando
entonces nuestros resultados con la la clasificación basada en formas canónicas.
Nuestra caracterización de canales Gaussianos sin forma funcional Gaussiana es
completada mostrando como los estados Gaussianos se transforman bajo esas op-
eraciones, ası́ como al derivar las condiciones para la existencia de ecuaciones
maestras para los casos no singulares.

Contents
1 Introduction 1
2 Open quantum systems and quantum channels 5
2.1 Introduction to the scheme of open quantum systems . . . . . . . 5
2.2 Quantum channels . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Quantum channels of continuous variable systems . . . . . . . . . 23
3 Representations of quantum channels 27
3.1 Kraus representation . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Choi-Jamiołkowski representation . . . . . . . . . . . . . . . . . 29
3.3 Operational representations . . . . . . . . . . . . . . . . . . . . . 30
3.4 Qubit channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Representation of Gaussian quantum channels . . . . . . . . . . . 40
4 Divisibility of quantum channels and dynamical maps 47
4.1 Divisibility of quantum maps . . . . . . . . . . . . . . . . . . . . 47
4.2 Characterization of L-divisibility . . . . . . . . . . . . . . . . . . 56
4.3 Divisibility of unital qubit channels . . . . . . . . . . . . . . . . . 57
4.4 Non-unital qubit channels . . . . . . . . . . . . . . . . . . . . . . 68
4.5 Divisibility transitions and examples with dynamical processes . . 72
5 Singular Gaussian quantum channels 79
5.1 Allowed singular forms . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Existence of master equations . . . . . . . . . . . . . . . . . . . 84
6 Summary and conclusions 89
7 Appendices 93
A Exact dynamics with Lindblad master equation 95
ix
x CONTENTS
B On Lorentz normal forms of Choi-Jamiolkowski state 97
C Articles 99
C.1 Article: Divisibility of qubit channels and dynamical maps . . . . 99
C.2 Article: Gaussian quantum channels beyond the Gaussian func-
tional form: full characterization of the one-mode case . . . . . . 114
C.3 Article: Positivity and Complete positivity of differentiable quan-
tum processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.4 Article: Positivity and Complete positivity of differentiable quan-
tum processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Bibliography 148
Chapter 1
Introduction
In questions of science, the authority of a thousand is not worth the humble
reasoning of a single individual.
Galileo Galilei
The advent of quantum technologies opens questions aiming for deeper under-
standing of the fundamental physics beyond the idealized case of isolated quantum
systems. Also the well established Born-Markov approximation used to describe
open quantum systems (e.g. relaxation process such as spontaneous decay and
decoherence) is of limited use and a more general framework of open system dy-
namics is required. Recent efforts in this area have given rise to relatively novel
research subjects - non-markovianity and divisibility.
A central object of study in quantum information theory and open quantum sys-
tems are quantum channels, also called quantum operations. They describe, for
instance, the noisy communication between Alice and Bob or the changes that an
open quantum system undergoes at some fixed time. They can also be seen as the
basic building blocks of time-dependent quantum processes (also called quantum
dynamical maps). Conversely, families of quantum channels arise naturally given
a quantum dynamical map.
Given a quantum channel, for instance an spin flip or the approximation of the
universal NOT gate, one can wonder about how it can be implemented. The lat-
ter in the sense of, being quantum channels discrete operations, can we find a
continuous time-dependent processes that at some time it implements the given
channel?, or, does process such that we “just wait for a relaxation of the physical
system” implements such channel? It turns that this question is related with the
1
2 Chapter 1. Introduction
one of finding simpler operations such that their concatenation equals the given
quantum channel [WC08]. Such operations are simpler in the sense that they are
closer to the subset of unitary operations, or even “smaller” in the sense that they
are closer to the identity channel.
This thesis encompasses the results of two works developed during my PhD.
The first and the most extended one was devoted to study the divisibility proper-
ties of quantum channels (discrete evolutions of quantum systems), for the par-
ticular case of qubits. We revise the divisibility types introduced in the seminal
paper by Wolf et al. [WC08] and derived several useful relations to decide each
type of divisibility. In particular, we characterize channels that can be divided in
such a way that they belong to one-parameter semigroups (dynamics described by
Lindblad master equations), and extended the analysis of [WECC08] for channels
with negative eigenvalues. We did this using the results by Evans et al. [EL77]
and Culver [Cul66].
Beyond the mentioned characterization tools, the principal aim of the work was
to understand the forms of non-markovianity standing behind the observed quan-
tum channels. The non-markovianity character describe the back-action of the
system’s environment on the system’s future time evolution. Such phenomena is
identified as emergence of memory effects [ARHP14, VSL+11, PGD+16]. On
the other side, divisibility questions the possibility of splitting a given quantum
channel into a concatenation of other quantum channels. In this work we will in-
vestigate the relation between these two notions. Thus, we related features of of
continuous time evolutions of quantum systems, and the concept of divisibility of
quantum maps, which are discrete evolutions. A very first example of this is the
well known identification of one-parameter semigroups with Linbladian dynam-
ics [Lin76].
The second project is devoted to representation theory of continuous-variable
quantum systems, which is a central topic of study given its role in the descrip-
tion of physical systems like the electromagnetic field [CLP07], solids and nano-
mechanical systems [AKM14] and atomic ensembles [HSP10]. In this theory the
simplest states, both from a theoretical an experimental point of view, are the so-
called Gaussian states. An operation that transforms such family of states into
itself is called a Gaussian quantum channel (GQC). Even though Gaussian states
and channels form small subsets among general states and channels, they have
proven to be useful in a variate of tasks such as quantum communication [GVAW+03],
quantum computation [LB99] and the study of quantum entanglement in sim-
ple [BvL05] and complicated scenarios [LRW+18]. In this project we study the
possible functional forms that one-mode Gaussian quantum channels can have
3
in the position state representation, and characterize the particular case of sin-
gular channels. Although they are already characterized by their action on the
first and second moments of Gaussian states [Hol07, WPGP+12], we connect our
framework to such known results. Additionally we give an insight of the possible
functional forms of, for instance, Gaussian unitaries.
The thesis is organized as follows: In chapter 2 we discuss the most widely
adopted scheme to study open quantum systems, introducing the formalism of
bipartite systems and useful tools for it. Later on we present the general setting
for system plus reservoir dynamics and its formal solution. As a paradigmatic ex-
ample of open system dynamics, we present briefly the microscopic derivation of
the Lindblad master equation using the well known Born-Markov approximation,
and discuss the properties of the generator of the dynamics. Subsequently we in-
troduce the formalism of quantum channels, being the most general operations
over quantum systems (excluding post-selection), by introducing some useful
mathematical definitions and contrasting with its classical analog. Additionally
we discuss briefly the concept of local operations and classical communications
(LOCC), also known as filtering operations. Finally we give a very brief intro-
duction to continuous variable systems, giving special attention to Gaussian states
and channels.
In chapter 3 we discuss the different available representations for quantum chan-
nels and their relation with the concept of complete positivity. In particular we in-
troduce the well known Kraus representation and discuss the Choi-Jamiołkowski
theorem which in turn defines a very useful representation to study quantum
channels and their divisibility properties. Later on we introduce various matrix
representations of quantum channels, paying special attention to hermitian and
traceless bases types (without taking into account the component proportional to
identity). Furthermore we introduce useful decompositions of qubit channels into
unitary conjugations and one-way stochastic local operations, and classical com-
munication, both being analogous to the well known singular value decomposi-
tion. Finally we give an introduction to representations of Gaussian channels and
a detailed derivation of the position-state representations for Gaussian channels
without Gaussian functional form.
In chapter 4 we give the definition of divisible quantum channel, as well the def-
inition of various subclasses of divisible channels concerning additional proper-
ties. In particular we discuss the concepts of infinitesimal and infinitely divisible
channels and some relations and inclusions between them. Among infinitesimal
divisible channels we identify two subclasses, being the set of infinitesimal divis-
ible channels in complete positive and positive (but not complete positive) maps.
4 Chapter 1. Introduction
Later on we introduce the concept of L-divisible channels, defining the set of
channels which are members of one-parameter semigroups. We show that the set
of infinitely divisible channels is the same of the L-divisible Pauli channels.
In chapter 5 we study one-mode Gaussian quantum channels in continuous-variable
systems by performing a black-box characterization using complete positivity and
trace preserving conditions, and report the existence of two subsets that do not
have a functional Gaussian form. Our study covers as particular limit the case
of singular channels, thus connecting our results with their known classification
scheme based on canonical forms. Our full characterization of Gaussian channels
without Gaussian functional form is completed by showing how Gaussian states
are transformed under these operations, and by deriving the conditions for the
existence of master equations for the non-singular cases.
In chapter 6 we give a summary of the two projects introduced in this work and
conclusions.
Finally, in the appendix A we prove that the exact reduced dynamics of an open
quantum system never follow a Lindblad master equation unless they are unitary,
given a bounded global Hamiltonian. In appendix B we give an example that
shows that the set of Lorentz normal forms introduced in the literature, is incom-
plete. In the appendix C we attach copies of the articles produced during my PhD.
Chapter 2
Open quantum systems and
quantum channels
When we talk mathematics, we may be discussing a secondary language built on
the primary language of the nervous system.
John Von Neumann
In this chapter we introduce the usual scheme to study open quantum systems, the
widely known Born-Markov approximation and the concept of CP-divisibility.
Later on and based on the idea of (classical) stochastic map, we discuss the ax-
iomatic formulation of quantum channels and its connection with the usual con-
struction of open quantum systems. Finally, for continuous variable systems, we
discuss the paradigmatic example of Gaussian channels.
2.1 Introduction to the scheme of open quantum systems
The most widely used scheme to study open quantum systems is based on the idea
of study a closed system composed by the central system and its environment, see
fig. 2.1 for an schematic explanation. Thus, concepts as bipartite Hilbert spaces,
density matrix and partial trace are useful tools to study open systems. In what
follows we give a brief review of them.
Bipartite Hilbert space. Consider a bipartite closed quantum system described
by a Hilbert space with the structure H = HS ⊗HE, where HS is the Hilbert
5
6 Chapter 2. Open quantum systems and quantum channels
space of the open system and HE is the Hilbert space of the environment. If
{|φ S
i 〉}
dim(HS)
i=1 and {|φ E
i 〉}
dim(HE )
i=1 are basis for the spaces HS and HE, respec-
tively, a basis for H is simply {|φ S
i 〉⊗ |φ E
j 〉}
dim(HS),dim(HE)
i=1, j=1 . It is typical that for
finite dimensional systems one has that dim(HE)≫ dim(HS) as the environment
is usually “bigger” than the central system.
To describe the states of open quantum systems it is necessary to model the igno-
rance that the observer has with respect to the open system. Since the experimen-
talist cannot access the degrees of freedom of the environment, they are simply
ignored. To do this we need the two following concepts.
Density matrix. Let a quantum system that has probability pi to be in the state
|φi〉, and let the operator A an observable over such system. Using the average
formula 〈A〉= ∑i pi〈φi|A|φi〉 it is straightforward to show that 〈A〉= tr(Aρ) with
ρ = ∑
i
pi|φi〉〈φi|, (2.1)
and ∑i pi = 1. ρ is called density operator or density matrix. Note that ρ is a
positive-semidefinite operator given that pi ≥ 0, and the states |φi〉 don’t need
to be orthogonal. Also note that since ρ is hermitian, together with positive-
semidefiniteness, implies that we can always write any density matrix as in eq. (2.1)
with the states |φi〉 being orthogonal. Thus, every operator ρ acting on a Hilbert
space H , fulfilling ρ ≥ 0, ρ = ρ† and tr(ρ) = 1 is a density matrix. The set of
density matrices will be denoted along this work as S (H ).
Comparing the notion of density matrices with the notion of state vectors in the
Hilbert space |ψ〉 ∈ H , density matrices describe physical systems where the
observer has an incomplete knowledge of the system’s state. Thus, while state
vectors are naturally equipped with intrinsic or quantum probabilities, density
operators are additionally equipped with classical probabilities. The density ma-
trices enjoying the form ρ = |ψ〉〈ψ|, or equivalently ρ2 = ρ , i.e. projectors, are
pure states. It is clear that in this case the system is prepared in the state |ψ〉 with
probability one.
A useful quantity to characterize quantum states is the purity, defined as
P(ρ) = tr
(
ρ2
)
. (2.2)
It ranges from dim(H )−1 to 1; 1 is obtained for pure states and dim(H )−1 for
the complete mixture ✶/dim(H ).
2.1. Introduction to the scheme of open quantum systems 7
Additionally the set S is convex, i.e. any convex combination of density matrices
is another density matrix, in the same way as classical distributions do. In fact,
mixed states (P(ρ) < 1) can be written always as convex combinations of pure
states, see eq. (2.1). Furthermore the set S (H ) is a subset of the bigger set of
trace-class operators, T (H ), defined as the ones containing operators with finite
trace norm. The latter is defined as |∆|tr = tr
√
A†A. This set is in turn a subset
of the set of bounded operators B(H ), containing operators with finite operator
norm, defined as |A|op = sup|ψ〉 |A|ψ〉|, where |A|ψ〉| =
√
(〈Aψ|Aψ〉), i.e. the
standard Hilbert space norm, with normalized vectors |ψ〉.
It is worth to note that for the finite dimensional, bounded operators always have
finite trace norm and vice versa, thus T (H ) = B(H ). But the identification
of such sets is relevant for infinite dimensional systems, where counter-examples
of the non-equivalence of such sets exist [HZ12]. Additionally B(H ) is the
dual space of T (H ) under the Hilbert-Schmidt product, defined as 〈A,B〉 =
tr(A†B) [Hol01].
Now, to ignore the degrees of freedom of the unaccessible part of the system,
we have to perform an operation in a very analogous way as computing marginal
distributions in classical probability theory. For density operators this introduces
the concept of partial trace.
Partial trace. Let ρ ∈ S (HA ⊗HB) and HA,B the Hilbert spaces of systems
A and B. Thus, ρ describes a state of a bipartite system composed by A and B. If
we want to know the state of the system A alone, one performs a partial trace over
B defined as
ρA = trB(ρAB) =
dB
∑
i=1
(
✶⊗〈φ B
i |
)
ρAB
(
✶⊗|φ B
i 〉
)
,
where {|φ B
i 〉}dB
i=1 is a complete orthonormal basis on HB. The resulting operator
ρA is a density matrix describing the state of the system A alone. It is trivial to
show that it is a density operator. A similar formula holds for ρB. An alternative
definition is trB (A⊗B) = A tr(B) plus linearity.
In general for composite systems, in a pure state, knowing the reduced states (for
instance for bipartite systems, ρA and ρB) is in general not enough to know the
whole state of a system. This captures the non-local nature of quantum corre-
lations, demanding simultaneous measurements on both parts of the system. In
such case we say that the subsystems A and B are entangled. To see this, consider
the example of the Bell state |Ω〉= 1/
√
2(|00〉+ |11〉), where {|0〉, |1〉} is an or-
thogonal basis of a qubit system. It is trivial to show that |Ω〉 cannot be written
8 Chapter 2. Open quantum systems and quantum channels
as |φ〉 ⊗ |ψ〉, a factorizable state, prohibiting the observer to know the state of
the whole system only by non-simultaneous measurements on A and B (described
by reduced density matrices). In fact it is easy to show that ρA,B = ✶/2 are the
reduced density matrices, appearing also when the total state is ρAB = ✶/4. For
composite systems in mixed states the situation is quite different. In this case si-
multaneous measurements are needed to access classical correlations. To see this
consider the state
ρAB = ∑
i
piρ
i
A ⊗ρ i
B, (2.3)
being a convex combination of factorizable mixed states. This state is a mixed
separable state [HHHH09], i.e. subsystems A and B are not entangled. Notice
now that performing only local non-simultaneous measurements, the accessible
reduced states are ρ ′
A,B = ∑ piρ
i
A,B. This state also arise when the total system is
in the factorizable state ρ ′
A ⊗ρ ′
B. Therefore local simultaneous measurements are
needed.
2.1.1 System plus reservoir dynamics
The most widely used scheme to study open quantum systems is to consider a
bipartite system, where the central system S, is interacting with its environment,
E. The full system S+E undergoes a closed system evolution, i.e. Hamiltonian
dynamics, see fig. 2.1. The total Hamiltonian H, describing the whole system,
has the following general structure
H = HS +HE +V, (2.4)
where HS,E are the free Hamiltonians of the central system and the environment,
respectively, and V is the interaction Hamiltonian among them. Now let ρSE(0)
be the state of the total system at the time t = 0. Thus, the state of the system S at
the time t is simply:
ρS(t) = trE
(
U(t)ρSE(0)U
†(t)
)
, (2.5)
where U(t) = e−iHt (taking h̄ = 1) and trE is the partial trace over the environ-
mental degrees of freedom. Note that for a general initial state ρSE(0), where
one allows classical and quantum correlations, ρSE(t) depends on general on ini-
tial information about the environment and its correlations with the central sys-
tem S. Thus, to compute the dynamics of the central system such that we end
up to universal reduced dynamics, i.e. the same for every initial state and inde-
pendent of the initial information in the environment, we take a factorized initial
2.1. Introduction to the scheme of open quantum systems 9
S+E
|ψ〉
ρS = trE |ψ〉〈ψ|
Figure 2.1: Diagram of the scheme to study open quantum systems. The let-
ters S and E state for the open (or central) system and environment parts
of the total closed system, S+E. The latter is described (typically) by
a pure state |ψ〉 ∈ HS ⊗HE and the central system is described by the
reduced state computed using the partial trace over the environmental
degrees of freedom, see main text.
state ρSE(0) = ρS(0)⊗ ρE [BP07, RH12]. We don’t write explicitly the time-
dependence of the environmental state since one is not usually interested on its
evolution. With the choice of a factorizable total initial state and using equa-
tion eq. (2.5), we have the following expression for the evolution of the central
system,
ρS(t) = trE
[
U(t)(ρS(0)⊗ρE)U†(t)
]
. (2.6)
Therefore we have that the dynamics over S only depend on the total Hamiltonian
H and the environmental initial state ρE , whereas ρS(t) depends only on its initial
condition.
Hence the equation eq. (2.6) defines a dynamical map, Et , parametrized by t. .
Thus, we have
Et [ρ(0)] = trE
[
U(t)(ρS(0)⊗ρE)U†(t)
]
. (2.7)
Such map possesses all the information concerning the dynamics of the system S,
thus knowing Et one can know entirely the evolution of the system S. The map
10 Chapter 2. Open quantum systems and quantum channels
ρS(0)⊗ρE ρSE(t)
ρS(0) ρS(t)
trE(·)
U(t)·U†(t)
trE(·)
Et
Figure 2.2: Scheme of the equivalences between the concept of dynamical
map and the theory of open quantum systems.
Et can be obtained numerically or experimentally (depending on the context) by
measuring only the system S by quantum process tomography [NC11]. In fig. 2.2
we present a schematic description of the two equivalent schemes under which
the system S evolves, and their connection throughout trE.
Eq. (2.7) can be reduced, by writing ρE = ∑ j pE
j |φ E
j 〉〈φ E
j |, in the following way,
Et [ρS(0)] = ∑
i, j
K(t)i, jρS(0)K(t)†
i, j, (2.8)
where the operators K(t)i j =
√
pE
j 〈φ E
i |U(t)|φ E
j 〉 are called Kraus operators and
act upon the system S alone [RH12]. The expression of eq. (2.8) is called sum
represention, also called Kraus representation of the map Et , this will be retaken
on chapter 3.
Now let us discuss the differential equation for the density matrix of an open quan-
tum system. The total state of the system evolves according to the Von Neumann
equation [BP07],
dρSE
dt
=−i[H,ρSE], (2.9)
which is the analog of the Liouville equation describing the evolution of a classical
distribution in the phase space.
Taking the partial trace on both sides of eq. (2.9) one arrives to the following:
dρS
dt
=−i tr[H,ρSE]
= Lt [ρS], (2.10)
where Lt is the generator of the master equation of the system S. Integrating time
in both sides from τ = 0 to τ = t, we arrive to the equivalent integral equation:
ρS(t) = ρS(0)+
∫ t
0
dτLτ [ρS(t)]. (2.11)
2.1. Introduction to the scheme of open quantum systems 11
To compute the formal solution of this equation, we use the method of succes-
sive approximations. This consists on substituting the whole expression for ρS(t)
defined by the right hand side of eq. (2.11). A first iteration leads to
ρS(t) = ρS(0)+
∫ t
0
dτ1Lτ1
[ρS(0)]+
∫ t
0
dτ1
∫ t
0
dτ2Lτ1
[Lτ2
[ρS(t)]]. (2.12)
Repeating this procedure infinite times, i.e. substituting ρS(t) defined by the right
hand side of the last equation in its second integrand several times, we arrive to
a power series solution for ρS(t) (powers of Lt). This leads to the well known
Dyson series for Lt . Compactly,
ρ(t) =~Texp
(
∫ t
0
dsLs
)
ρ(0) (2.13)
with~T the time-ordering operator, defined as
~T[H(τ1)H(τ2)] = θ(τ1 − τ2)H(τ1)H(τ2)+θ(τ2 − τ1)H(τ2)H(τ1),
with θ(x) the Heaviside step function. Eq. (2.13) constitutes the formal solution
to the Von Neumann equation with generator Lt , and we can easily identify Et =
~Texp
(
∫ t
0 dsLs
)
.
2.1.2 Born-Markov approach: microscopic derivation
In general the form of the generator Lt , given a global Hamiltonian, can be quite
involved [BP07], but in the limit of weak coupling and short memory we can
perform the very widely known Born-Markov approximation. A brief discussion
is presented in this subsection.
The Born-Markov approximation leads to the Lindblad master equation. We will
briefly overview its usual textbook derivation. The first step is to use the inter-
action picture, hence the total Hamiltonian becomes HI(t) = eiH0tHe−iH0t , where
H0 = HS +HE is the free Hamiltonian. Assuming that the dimension of HE is big
compared with the dimension of HS, the weak coupling limit leads to negligible
changes in the environmental state. Thus, at time t we can approximate
ρSE(t)≈ ρS(t)⊗ρE .
In other words, the state of the total system is left always approximately uncor-
related, while the state of the environment is never updated. Therefore the envi-
ronment forgets any information about the central system, while the state of the
12 Chapter 2. Open quantum systems and quantum channels
latter undergoes a non-trivial evolution. Additionally to simplify the derivation we
choose ρE a stationary state of HE, i.e. [HE,ρE] = 0 [RH12]. ρE is typically cho-
sen as a thermal state of the environmental Hamiltonian, ρE ∝ exp(−βHE), with
β = 1/(kBT ), kB the Boltzmann constant and T the environment temperature.
Now, in the interaction picture the Von Neumann equation becomes
dρS
dt
=−i trE[VI(t),ρS], (2.14)
where VI(t) = eiH0tVe−iH0t and the state ρS(t) are now written in the interaction
picture. Inserting ρS(t) from its integral equation eq. (2.11) in the differential
equation (2.14) and assuming trE[VI(t),ρS ⊗ρE] = 0 [BP07], we obtain
dρS
dt
=−
∫ t
0
dτ trE[VI(t), [VI(τ),ρS(τ)⊗ρE]]. (2.15)
If we assume that the dynamics of the state of the central system does not depend
on its past, we can change ρS(τ) to ρS(t), this is called the Markovian approxi-
mation. Additionally doing the variable change τ ′ = t − τ , we arrive to
dρS
dt
=−
∫ t
0
dτ ′ trE[VI(t), [VI(t − τ ′),ρS(t)⊗ρE]], (2.16)
this equation is known as Redfield equation [Red65] and it is local in time [BP07].
Assuming that the time scale on which the central system varies appreciably is
much larger than the time on which the correlations of the environment decay
(say τE), the integrand decays to zero rapidly for τ ′ ≫ τE. Then we can safely
replace t by ∞ in the integrand limits, obtaining
dρS
dt
=−
∫ ∞
0
dτ ′ trE[VI(t), [VI(t − τ ′),ρS(t)⊗ρE]]. (2.17)
Up to this point, eq. (2.17) has in general fast oscillating terms coming from the
explicit dependence on VI(t), this in turn can bring a generator that leads to a
quantum process that violates complete positivity [ARHP14, RH12]. In order to
get rid of such fast oscillations, one uses the aforementioned assumption that the
environment is initialized in a stationary state, and perform the so called secular
approximation [ARHP14]. A detailed derivation is outside of the scope of this
thesis, but it can be consulted on references [BP07, RH12]. After performing the
Markov, Born and secular approximations and changing back to the Schrödinger
2.1. Introduction to the scheme of open quantum systems 13
picture, the resulting master equation can be written in the following forms
dρS
dt
= i[ρS, H̃S]+
d2
S−1
∑
i, j=1
Gi j
(
FiρSF
†
j −
1
2
{F
†
j Fi,ρS}
)
, (2.18)
= i[ρS, H̃S]+
d2
S−1
∑
j=1
γ j
(
A jρSA
†
j −
1
2
{A
†
jA j,ρS}
)
, (2.19)
= L[ρS]. (2.20)
Fj ( j = 0, · · · ,d2
S − 1) are operators acting on the central system that addition-
ally form an orthonormal basis under Hilbert-Schmidt inner product, such that
F0 = ✶/
√
dS and trFj = 0 ∀ j > 0 (this will be revised in subsection 3.3.1); the
matrix G is called dissipator matrix. In the second inequality we have used the
singular value decomposition of matrix G, thus operators Ai are linear combina-
tions of Fi. The scalars γ j > 0 are called relaxation rates and the operator H̃S is
the shifted free Hamiltonian of the central system. The first term on both equa-
tions, the commutator, is called Hamiltonian part, while the second, the super-
operator defined with the summations, is called dissipator. Note that if γ j = 0
∀ j (uncoupled limit), one recovers the Hamiltonian dynamics over the system S.
The operator L is called Lindblad operator or Lindbladian and eq. (2.20) is called
Lindblad master equation. We will use along the work the notation L for Lindblad
operators.
Note that L is independent of time, hence the formal solution of the master equa-
tion eq. (2.20) equation is simply the exponentiation of L [see eq. (2.13)], i.e.
ρS(t) = eLtρS(0). (2.21)
Therefore the dynamics is homogeneous in time and, together with the fact that
Et = exp(Lt), we have Et+s = EtEs, i.e. the quantum process Et resulting from a
Lindblad master equation forms a one-parameter semigroup. In fact, Lindblad has
proven the converse, including the case of infinite dimension [Lin76]:
Theorem 1 (One-parameter quantum semigroups). Let Et with E0 = id and t ≥ 0
a quantum process, it is a one-parameter quantum semigroup if and only if it has
a generator with the form presented in eq. (2.20).
It is worth to point out that starting from global dynamics governed by a bounded
Hamiltonian, the reduced dynamics are never of Lindblad form. This can be stated
as the following,
14 Chapter 2. Open quantum systems and quantum channels
Theorem 2 (Exact dynamics with Lindblad master equation). Let Et = etL a quan-
tum process generated by a Lindblad operator L. The equation
Et [ρ] = trE
[
e−iHt (ρ ⊗ρE)eiHt
]
,
where H has finite dimension, holds if and only if Et is an unitary conjugation for
every t.
A proof made jointly with Sergey Filippov is given in the appendix A. It was made
using an specific matrix representation for operators that will be introduced in the
next chapter. But a more general proof can be found in [Exn85].
Let us point out that this is not the case for global unbounded Hamiltonians, they
can lead to Lindblad master equations for the reduced dynamics. This is shown
below together other illustrative examples.
Examples. To illustrate Lindblad dynamics we present several examples. The
first one, depolarizing dynamics, is constructed via a continuous and monotonic
contraction of the Bloch sphere. The second one corresponds to a system for
which the exact reduced dynamics have Lindblad generator.
Example 1 (Dephasing dynamics). Let ρ(0) =
(
ρ00 ρ01
ρ∗
01 ρ11
)
be the initial state,
written in a basis called decoherence basis, of a system that undergoes depolar-
izing dynamics. This is, only coherence terms (in this basis) are modified in the
following way:
Et : ρ(0) 7→
(
ρ00 ρ01e−γt
ρ∗
01e−γt ρ11
)
=: ρ(t),
with γ > 0. It is trivial to check that Et is a one-parameter semigroup with E0 = id.
For t → ∞, we get ρ(0) → diag(ρ00,ρ11). For this process it is easy to prove,
by taking 0 < t ≪ 1, that its generator is L[ρ] = γ/2(σzρσz −ρ), which has
Lindblad form. It has null Hamiltonian part and only one operator A0 = σz and
one relaxation ration, γ/2.
Example 2 (Dynamics from global unbounded Hamiltonian). Consider a bipar-
tite system composed by a qubit interacting with a particle in a line, with global
Hamiltonian H = σz⊗ x̂, where x̂ is position operator. Notice that H is unbounded
since the configuration space of the particle is the entire real line. Initializing the
environment in the state |ψ〉 with
〈x|ψ〉=
√
γ
π
1
x+ iγ
,
2.2. Quantum channels 15
it can be shown that the exact reduced dynamics for the qubit, without any ap-
proximation, is L[ρ] = γ/2(σzρσz −ρ) [AHFB15]. The same generator as in the
first example.
2.2 Quantum channels
In this section we give a brief introduction to classical stochastic processes, this
motivates the definition of quantum channel. We first give an overview of stochas-
tic processes; based on this we review the construction steps of quantum channels
and discuss several of their properties. Additionally we introduce the simplest
example of local operations and classical communication. Later on one we dis-
cuss the definition of CP-divisible processes based on the definition of classical
Markovianity. Finally we give a brief revision of Gaussian quantum states and
channels.
2.2.1 A classical analog
The classical analog of quantum channels are the widely known stochastic ma-
trices or stochastic maps which propagate classical probability distributions. To
introduce them consider, for sake of simplicity, a finite dimensional stochastic
system whose state xt (at time t) is described by the probability distribution (or
probability vector) ~p(t), i.e. xt ∼ ~p(t) [with ∑i pi(t) = 1 and pi(t)≥ 0]. Note that
probability vectors form a convex space in the very same way that density matri-
ces do. The distribution ~p(t) is the classical analogous object to density matrices.
They serve as the tool to model the accessible information of the observer about
the state of the classical stochastic system.
Consider now the most general linear transformation on probability vectors that
takes, for instance ~p(0) to ~p(t) and let us write it explicitly as a matrix multi-
plication, ~p(t) = Λ(t,0)~p(0). We have to impose further constrictions over Λ(t,0)
in order to preserve the normalization of ~p(t) and the non-negativity of its el-
ements. Since pi(t) = ∑ j
(
Λ(t,0)
)
i j
~p j(0), simple algebra leads us to note that
∑i
(
Λ(t,0)
)
i j
= 1 ∀ j and
(
Λ(t,0)
)
i j
≥ 0. Matrices that fulfill these conditions are
widely known as stochastic matrices, and form a convex set following the con-
vexity of the space of probability distributions.
A remarkable property of stochastic maps is that they are contractive with respect
to the Kolmogorov distance.
16 Chapter 2. Open quantum systems and quantum channels
Theorem 3 (Contractivity of stochastic maps). The matrix Λ is a stochastic matrix
if and only if
DK (Λ~p,Λ~q)≤ DK (~p,~q) , (2.22)
where DK (~p,~q) = ∑k |pk −qk| is the Kolmogorov distance.
It is worth to note Kolmogorov distance is a measure of distinguishability be-
tween classical distributions. A detailed proof of this theorem can be found in
Ref. [ARHP14].
A particular and interesting class of stochastic matrices are bistochastic matri-
ces. They are defined as the transformations that leave invariant the probabil-
ity distribution with maximum entropy, given by ~m = (1/N, . . . ,1/N)T
, where N
is the number that the system can have. Therefore a bistochastic matrix fulfills
~m = Λ(0,t)~m. Doing simple algebra lead us to note that bistochastic matrices ad-
ditionally fulfill ∑ j
(
Λ(t,0)
)
i j
= 1 ∀i. This implies that they are also stochastic
matrix acting from the right, i.e. mapping row probability vectors. This is also the
origin of the name bistochastic.
In the previous section we have introduced the concept of Markovianity in the
context of open quantum systems, the so called Markovian approximation. It
consisted on assuming that the system ’forgets the information about its previous
states’. This concept comes from the theory of classical stochastic processes. Let
us introduce the following definition [BP07, ARHP14],
Definition 1 (Classical Markovian process). Let xt be the state of a stochastic
system where t ∈ [0,τ], and χ = {t0, . . . tn} any ordered set of times such that
0 < t0 < t1 < · · ·< tn < τ , the process is Markovian if
P(xtn , tn|xtn−1
, tn−1; . . . ;xt0 , t0) = P(xtn , tn|xtn−1
, tn−1) ∀n > 0, (2.23)
where P(·|·) denotes conditional probability.
According to this definition, the conditional probability of the system to be at the
state xtn at the time tn, given the history of events {xtn−1
, tn−1; . . . ;xt0 , t0}, depends
only on the previous state. This definition captures the memoryless character of
Markovian processes.
Consider now a stochastic process and {Λ(t,0)}t∈χ its set of stochastic matrices
given some ordered set of times χ . If the process is Markovian then the matrices
Λ(tm,tn) are stochastic matrices for any χ , where tm > tn ∈ χ . The converse is not
true [ARHP14, BP07]. This condition implies that the map Λ(t,0) is divisible in
2.2. Quantum channels 17
the sense that it can always be written as
Λ(t1,t0) = Λ(t1,s)Λ(s,t0) ∀ t1 > s > t0, (2.24)
with Λ(t1,s), Λ(s,t0) and Λ(t1,t0) stochastic matrices, the latter two by definition. In-
termediate maps can be constructed as Λ(t1,s) = Λ(t1,t0)Λ
−1
(s,t0)
if Λ−1
(s,t0)
exists. Note
that theorem 3 implies that Markovian stochastic processes do not increase the
Kolmogorov distance.
2.2.2 Construction of quantum channels
The concept of quantum channel, also known as quantum operation, captures the
idea of stochastic map in the quantum setting. Thus, being the density matrices
the analogous objects to probability vectors, we seek for linear operations that
transform density matrices into density matrices. The operations that do such job
are defined as follows:
Definition 2 (Positive and trace preserving linear operations (PTP)). A linear op-
eration E : T (H )→ T (H ) is positive and trace preserving if, for all ∆ ∈ H ,
we have the following
• E [∆]≥ 0 ∀∆ ≥ 0,
• tr(E [∆]) = tr(∆).
A remarkable property of linear positive maps is that they are contractive respect
to the trace norm [ARHP14]. This is leads to a decrease of the distinguishability
of quantum states, similar to the classical case.
Theorem 4 (Contractivity of positive maps). A linear map E is PTP if and only
if |E [∆]|tr ≤ |∆|tr ∀∆† = ∆ ∈ B(H ).
A simple proof for the finite dimensional case can be found in Ref. [ARHP14].
Now, given that any hermitian operator can be written as
∆ = (tr∆)Hp, for tr∆ 6= 0,
∆ = tr∆+ (ρ1 −ρ2) , for tr∆ = 0,
where Hp = pρ1 − (1− p)ρ2 a Helstrom matrix and p ∈ [0,1], by theorem 4 the
generalized trace distance defined as Dp(ρ1,ρ2) = |pρ1 − (1− p)ρ2|tr decreases
after the application of a positive map E , i.e.
Dp (E [ρ1],E [ρ2])≤ Dp (ρ1,ρ2) .
18 Chapter 2. Open quantum systems and quantum channels
It is worth to point out that this is directly related to the two-state discrimination
problem where we have, for instance, probability p of erroneously identify ρ1
with ρ2 [NC11, ARHP14]. In this setting the probability of failing with such
identification is
Perr =
1−Dp(ρ1,ρ2)
2
.
Therefore if the distance is zero, the probability of correctly identify ρ1 is the
same as choosing randomly between ρ1 and ρ2, but if it is 1, we identify ρ1 from
ρ2 with certainty. For p = 1/2 we recover the standard unbiased trace distance.
It is well known that any quantum system can be entangled with another, for
instance a central system can be entangled with its environment. Thus, in the
context of quantum operations we must handle this fact carefully. Let us define
the following:
Definition 3 (k-positive operations). A linear map E is k-positive if
idk ⊗E [∆̃]≥ 0 ∀∆̃ ≥ 0 ∈ B(Hk ⊗H ),
with k a positive integer, being the dimension of Hk and idk the identity map in
that space.
Therefore a positive map is k-positive if the expended map idk ⊗E is positive, the
trace preserving of k-positive maps follows immediately from the trace preserving
of E . Such maps transform properly density matrices of the extended system
(with ancilla of dimension k) into density matrices, apart from the fact that they
transform properly the density matrices of the system, hence handling quantum
entanglement correctly for this ancilla.
Since the dimension of any other quantum system is arbitrary, being for example
the rest of the universe, one must have that quantum maps must transform quan-
tum states for every positive integer k. Therefore one defines complete positive
and trace preserving linear maps as the following,
Definition 4 (Complete positive and trace preserving operations (CPTP)). A trace
preserving linear operation E : T (H )→ T (H ) is complete positive if
idk ⊗E [∆̃]≥ 0 ∀∆̃ ≥ 0 ∈ B(Hk ⊗H ),∀k ∈ Z
+
0 ,
where Z
+ is the set of the positive integers.
It will be shown later in chapter 3, that deciding complete positivity is straightfor-
ward using the so called Choi matrix.
2.2. Quantum channels 19
It is trivial to check that unitary operations, U [ρ] =UρU†, are CPTP maps as ex-
pected. Additionally they leave invariant the maximally mixed state, ✶/dim(H ).
In fact, unitary operations belong to a wider class of CPTP maps called unital
quantum maps, similar to its classical counterpart. The set of unital channels
is defined simply as the one containing CPTP maps E that additionally fulfill
E [✶] = ✶.
Additionally notice that due to the trace preserving property the adjoint operator
of E is always unital. The adjoint is defined in the usual way,
〈A,E [B]〉= 〈E ∗[A],B〉, (2.25)
where the inner product is the Hilbert-Schmidt product and A ∈ T (H ) and
B ∈ B(H ) [Hol01, HZ12]. Now, ∀∆ ∈ T (H ) we write the trace preserving
condition as tr∆ = trE [∆] = 〈✶,E [∆]〉= 〈E ∗[✶],∆〉, therefore E ∗[✶] = ✶.
Let us now illustrate the connection of the concept of quantum channel with the
scheme of open quantum systems introduced above. Consider the following the-
orem [Sti06]:
Theorem 5 (Stinespring dilation theorem). Let E a CPTP map, there exist an
environmental Hilbert space HE and ρE ∈ S (HE) such that
E [ρ] = trE
[
U (ρ ⊗ρE)U†
]
,
with the unitary matrix U : H ⊗HE → H ⊗HE.
The unitary U and the state ρE are not unique [HZ12]. Stinespring theorem is
an important result given that one can always understand a CPTP operation as a
Hamiltonian evolution in a bigger space, such that we recover the given operation
at some fixed time and by performing a partial trace over the environmental de-
grees of freedom. Later in this chapter we will discuss an important implication
of this theorem for Markovian processes.
Along the work we will also denote the set of CPTP linear maps simply as C.
A remarkable property of C is its convexity. To show this consider the following
convex combination of CPTP maps: E = pE1+(1− p)E2, acting upon the density
matrix ρ0. By linearity we have E [ρ0] = pE1[ρ0] + (1− p)E2[ρ0]. Defining the
density matrices ρi = Ei[ρ0] ∈ S (H ), it follows from the convexity of S (H )
that E is another CPTP map. Therefore the set C is convex.
Unitary maps are extremal channels of C i.e. they cannot be written as convex
combinations of other channels, but they can be used to construct other maps,
20 Chapter 2. Open quantum systems and quantum channels
id
U1
U2
U3
Figure 2.3: The figure shows an schematic slice of CPTP maps, one can see
the identity map and other extremal channels. The straight lines are
convex combinations of those channels, the curve contains channels in
the boundary that cannot be written as convex combinations of unitary
channels.
see fig.2.3. For instance consider a simple convex combination of unitary maps
E [ρ] = ∑i piUρU†, with ∑i pi = 1 and pi ≥ 0. This channel is a more general
example of a unital channel, in fact it turns that every unital qubit channel has
such form. This can be shown easily using the Ruskai’s decomposition that will
be introduced in the next chapter. Convex combinations of unitary channels can
be implemented in the laboratory, for instance choosing unitaries randomly by
tossing a die.
Regarding the algebraic properties of the set C, it enjoys the structure of a semi-
group. It is closed under the composition operation, i.e. E1E2 ∈ C, ∀E1,E2 ∈ C,
and is associative, (E1E2)E3 = E1 (E2E3). Additionally it contains an identity ele-
ment. C does not contain the inverse elements, this captures the irreversible char-
acter of general quantum operations, being only the unitaries the ones their inverse
elements in C. Furthermore, C contains another remarkable convex structure,
Definition 5 (Entanglement-breaking channels). A map E ∈ C is entanglement-
breaking if it breaks the entanglement of the system with any ancilla, i.e. ∀k ∈ Z
+
and ∀σ ∈ S (Hk ⊗H ), the state (idk ⊗E ) [σ ] is separable.
This set is convex given that convex combinations of separable states is separa-
ble [HHHH09].
Quantum channels can be seen as the basic building of time-dependent quantum
processes, also called quantum dynamical maps.
Definition 6 (Quantum dynamical maps). A continuous family of channels {Et ∈
C : t≥ 0,E0 = id} is called quantum dynamical map.
2.2. Quantum channels 21
id
U1
U2
U3Et
Figure 2.4: Scheme of an smooth dynamical map inside a slice of the set C.
Given some interval I , if the family is smooth respect to t ∈ I and invertible, it
admits a master equation
ρ̇(t) = At [ρ(t)] with At = ĖtE
−1
t .
An schematic description is shown in fig. 2.4. Note that the standard scheme open
quantum systems, introduced at the beginning of this chapter, leads to quantum
dynamical maps.
2.2.3 Non-Linear CPTP operations
Notice that the set C does not contain everything that can be performed on a quan-
tum system; it contains only linear operations. Therefore C do not contain post-
seletion procedures, i.e. updating the state once a measurement is done and the
result is known. For instance, let ρ the state of some system and {Mi} a collection
of measurement operators over it, where the index i refers to the measurement
outcome. The probability of measuring i is p(i) = tr
(
MiρM
†
i
)
, while the opera-
tion performed over the state is
ρ 7→ MiρM
†
i
tr
(
MiρM
†
i
) .
This operation is explicitly non-linear but it is trivially complete positive and trace
preserving. Note that if the action of the measurement apparatus is performed but
the experimentalist do not read the outcome, or it is simply forgotten, the resulting
map belongs to C [NC11]. This is shown by noting that the operation MiρM
†
i is
applied with probability p(i), then the performed operation is ∑i p(i)MiρM
†
i and
it is linear and CPTP by construction. Complete positivity follows immediately
22 Chapter 2. Open quantum systems and quantum channels
from the complete positivity of ρ 7→ MiρM
†
i and the trace preserving property
from the weighted summation.
A more general set of operations including measurements, postselection and ex-
change of classical information will be introduced in the next subsection.
2.2.4 Local operations and classical communication
Several types of quantum operations can be found and studied, in particular in
Ref. [HZ12] there is a classification mainly based on its locality. A paradigmatic
and widely studied type are the so called local operations and classical com-
munication [HHHH09]. A surprising feature of this operations is that they can
increase the entanglement of entangled states of a system (at the cost of throw-
ing some members of the ensemble), but cannot create them from non-entangled
ones [VDD01, HHHH09].
In this work we are particularly interested in one-way stochastic local operations
and classical communication channels (1wSLOCC). Consider a bipartite system
where one part is controlled by Alice and the other by Bob. Alice performs an
operation which includes measurements with postselection, and then she commu-
nicates its outcome to Bob. Then Bob performs a local operation that can be again
a measurement with postselection, finishing the protocol. The stochasticity comes
from the fact that this operation, for each particular set of measurement outcomes,
has a certain probability generally less than 1 of occurrence. And the one-way
comes from the fact that no feedback is given to Alice and no more operations
and classical communications are performed. This operations can be written in
the following way:
ρ 7→ ρ ′ =
(X ⊗Y )ρ (X ⊗Y )†
tr
[
(X ⊗Y )ρ (X ⊗Y )†
] . (2.26)
Additionally we will consider detX 6= 0 and detY 6= 0, this is the usual choice
as projective measurements destroy entanglement [VDD01]. This operations are
complete positive and trace preserving, but non-linear unless X and Y are uni-
taries. Additional notice that given ρ and ρ ′, the matrices X and Y can always be
chosen such that detX = detY = 1 (for the invertible case). Therefore for two-
level systems it is enough to consider X ,Y ∈ SL(2,C) [Tun85], where the latter
is the special linear group of 2× 2 matrices with complex entries. Furthermore
notice that the operation
ρ 7→ (X ⊗Y )ρ (X ⊗Y )†
(2.27)
2.3. Quantum channels of continuous variable systems 23
preserves the determinant, i.e. detρ = detρ ′. In the next chapter we will ex-
ploit this to show that there is a correspondence between 1wSLOCC and Lorentz
transformations. We use this to introduce a decomposition analogous to the sin-
gular value decomposition, but using the Lorentz metric instead of the Euclidean,
enjoying an useful physical meaning.
2.3 Quantum channels of continuous variable systems
Many of the definitions and tools introduced in the previous sections are also rele-
vant for the infinite dimensional case. Although we can always choose countable
basis for the Hilbert space as long it is separable [HZ12], it is often of interest
to consider non-countable bases, typically phase-space variables. This introduces
the theory of continuous variable systems. It is a central topic of study given
that they appear naturally in the description of many physical systems. A few
examples are the electromagnetic field [CLP07], solids and nano-mechanical sys-
tems [AKM14] and atomic ensembles [HSP10]. In particular, in this section we
introduce and discuss a set of continuous variable channels called Gaussian quan-
tum channels.
2.3.1 Gaussian quantum states
To introduce the definition of Gaussian quantum channel, consider first the sim-
plest state type of quantum states in continuous variable, both from a theoretical
an experimental point of view, the so-called Gaussian states. The operations that
transform such family of states into itself are called Gaussian quantum channels
(GQC). Even though Gaussian states and channels are small subsets of all possi-
ble states/channels, they have proven to be useful in a very wide variate of tasks
such as quantum communication [GVAW+03], quantum computation [LB99] and
the study of quantum entanglement in simple [BvL05] and complicated scenar-
ios [LRW+18].
Gaussian states are defined as those having Gaussian Wigner function. In partic-
ular, for one-mode the Wigner function is
W (~u) =
1
2π
√
detσ
e−
1
2(~u−~d)
T
σ−1(~u−~d), (2.28)
where ~u = (q, p)T
[EW07]. The mean vector ~d and the covariance matrix σ are
24 Chapter 2. Open quantum systems and quantum channels
the first and second moments, respectively. They are given by
σ =
(
〈q̂2〉−〈q̂〉2 1
2
〈q̂p̂+ p̂q̂〉−〈q̂〉〈 p̂〉
1
2
〈q̂p̂+ p̂q̂〉−〈q̂〉〈 p̂〉 〈 p̂2〉−〈 p̂〉2
)
,
~d = (〈q̂〉,〈p̂〉)T .
The observables q̂ and p̂ are the standard canonical conjugate position and mo-
mentum variables. As for any other Gaussian variable, Gaussian quantum states
are characterized completely by first and second probabilistic moments. Therefore
a Gaussian state S can be denoted as S = S
(
σ , ~d
)
.
2.3.2 Gaussian quantum channels
To start with, we recall the following definition [WPGP+12]:
Definition 7 (Gaussian quantum channels). A quantum channel is Gaussian (GQC)
if it transforms Gaussian states into Gaussian states.
This definition is strictly equivalent to the statement that any GQC, say A , can be
written as
A [ρ] = trE
[
U (ρ ⊗ρE)U†
]
(2.29)
where U is a unitary transformation, acting on a combined global state obtained
from enlarging the system with an environment E, that is generated by a quadratic
bosonic Hamiltonian (i.e. U is a Gaussian unitary) [WPGP+12]. The environ-
mental initial state ρE is a Gaussian state and the trace is taken over the environ-
mental degrees of freedom.
Following definition 7, a GQC is fully characterized by its action over Gaussian
states, and this action is in turn defined by affine transformations [WPGP+12].
Specifically, A = A (T,N,~τ) is given by a tuple (T,N,~τ) where T and N are
2×2 real matrices with N = NT [WPGP+12] acting on Gaussian states according
to
A (T,N,~τ)
[
S
(
σ , ~d
)]
= S
(
TσTT +N,T~d +~τ
)
.
In the particular case of closed systems we have N = 0 and T is a symplectic
matrix. The particular form and properties of Gaussian quantum channels in the
continuous variable representations, as well their connection the mentioned affine
transformations, will be given in chapter 3.
2.3. Quantum channels of continuous variable systems 25
In this work we explore GQCs without Gaussian functional form in the position
state representation. In particular we study channels that can arise when singu-
larities on the coefficients of Gaussian forms GF occur (they will be denoted by
δGQC). Such channels can lead immediately to singular Gaussian operations.
Thus, we characterize which forms in δGQC lead to valid quantum channels,
and under which conditions singular operations lead to valid singular Gaussian
quantum channels (SGQC).
Let us note that although channels with Gaussian form trivially transform Gaus-
sian states into Gaussian states, the definition goes beyond GF. We will use the
typical difference and sum coordinates, x = q2 − q1 and r = (q1 + q2)/2, respec-
tively. Defining ρ(x,r) =
〈
r− x
2
∣
∣ ρ̂
∣
∣r+ x
2
〉
, a quantum channel in this represen-
tation is defined such that
ρ f (x f ,r f ) =
∫
R2
dxidriJ(x f ,xi;r f ,ri)ρi (xi,ri) , (2.30)
where ρ̂i and ρ̂ f are the initial and final states, respectively, and J(x f ,xi;r f ,ri) is
the representation of the quantum channel in the aforementioned variables. An
example of a channel without GF can be constructed from the general form of
Gaussian quantum channel with GF [MP12]:
JG(x f ,xi;r f ,ri) =
b3
2π
exp
[
ı
(
b1x f r f +b2x f ri +b3xir f
+b4xiri + c1x f + c2xi
)
−a1x2
f −a2x f xi −a3x2
i
]
, (2.31)
where all coefficients are real and no quadratic terms in ri, f are allowed. Choosing
an = αnε−1 + ãn
and
bn = βnε−1/2 + b̃n,
with ε > 0, αn,βn, ãn, b̃n ∈ R ∀n and b̃3 = 0. Taking the limit ε → 0 and using
the formula
δ (x) = lim
ε→0
1
2
√
πε
e
−x2
4ε , (2.32)
we arrive to
lim
ε→0
JG(x f ,xi;r f ,ri) = N δ (αx f −βxi)e
Σ′(x f ,xi;r f ,ri), (2.33)
where α , β ∈R and Σ′(x f ,xi;r f ,ri) is a quadratic form that now admits quadratic
terms in ri, f , arising from the completion of the square of the exponent of eq. (2.31)
26 Chapter 2. Open quantum systems and quantum channels
to take the limit of eq. (2.32). This is the first example of a δGQC. This channel
is still a GQC according to the definition. A physical, but complicated realization
occurs in the system of one Brownian quantum particle with harmonic potential
and linearly coupled to the bath. In such system, channels with the functional
form of eq. (2.33) are realized at isolated points in time, see equations 6.71-75 of
Ref. [GSI88].
Since the form of eq. (2.33) admits quadratic terms in ri, f in the exponent, it
suggest that a form with two deltas exist and can be defined using the same limit,
see eq. (2.32). In fact, the identity map is a particular case; it is realized setting
J(x f ,xi;r f ,ri) = δ (x f − xi)δ (r f − ri). In any case, to avoid working with such
limits, it is convenient to perform a black-box characterization of general forms
involving Dirac’s deltas, which will be done in the next chapter. This will lead to
explicit relations between position state representation and affine representations
of Gaussian quantum channels without Gaussian functional form.
Chapter 3
Representations of quantum
channels
Simplicity is the ultimate sophistication.
Leonardo da Vinci
In this chapter we introduce several and useful representations of quantum chan-
nels for the finite dimensional case. We start with the Kraus representation, al-
ready mentioned in the previous chapter, but additionally we will show that quan-
tum channels always have this form. Later on we introduce Choi’s theorem (and
the so called Choi-Jamiołkowski representation) which is cornerstone tool to study
many properties of quantum channels. We also discuss operational representations
by introducing two types of basis. These representations are useful to prove sev-
eral results in this work. Next, we apply the introduced tools to the qubit case.
Additionally we discuss two decomposition of qubit channels, leading to two nor-
mal forms that are essential to study divisibility properties of quantum channels.
3.1 Kraus representation
In the previous chapter we have shown that starting from the usual scheme of
open quantum systems, we arrive to the Kraus representation, see eq. (2.8). Later
on, using the Stinespring dilation theorem, see Theorem 5, we show that CPTP
maps can always fit in the scheme of open quantum systems for some global
27
28 Chapter 3. Representations of quantum channels
unitary evolution. Since the latter scheme always has a Kraus representation, one
concludes that CPTP maps always have a Kraus representation. It turns out that
the converse also holds [KBDW83].
Theorem 6 (Kraus). A linear operation E : T (H ) → T (H ) belongs to C if
and only if there exist a set of bounded operators {Ki} such that
E [∆] = ∑
i
Ki∆K†
i ∀∆ ∈ T (H ),
with ∑i K
†
i Ki = ✶.
Proof. The ’only if’ part is already commented in the main text and follows
the logic: every E ∈ C has a dilation such that it has the familiar form of the
open quantum systems dynamics, i.e. there exists U and ρE such that E [ρ] =
trE
[
U (ρ ⊗ρE)U†
]
. We already showed that writing ρE in terms of its eigenbasis,
the latter expression leads to the Kraus representation, see eq. (2.8). To prove the
’if’ part, we only have to construct the extended map to test its complete positivity.
Let k > 0 ∈Z and τk = (idk ⊗E ) [∆̃k], where ∆̃k ∈B(Hk ⊗H ) and ∆̃k ≥ 0, using
Kraus decomposition and evaluating 〈φ |τk|φ〉 with |φ〉 ∈ Hk ⊗H , one arrives to
〈φ |τk|φ〉= ∑
i
〈φ |(✶k ⊗Ki) ∆̃k
(
✶⊗K
†
i
)
|φ〉
= ∑
i
〈φi|∆̃k|φi〉
≥ 0.
The latter follows immediately from the positive-semidefinitiveness of ∆̃k, i.e.
〈φi|∆̃k|φi〉 ≥ 0. The condition ∑i K
†
i Ki = ✶ comes from the trace-preserving of E
and the cyclic property of the trace,
trE [∆] = ∑
i
tr
[
Ki∆K
†
i
]
= ∑
i
tr
[
K
†
i Ki∆
]
= tr
[(
∑
i
K
†
i Ki
)
∆
]
= tr∆,
3.2. Choi-Jamiołkowski representation 29
Therefore ∑i K
†
i Ki = ✶. It is worth to note that Kraus operators are not unique.
Defining a new set of operators, Ak = ∑l uklKl , it is easy to show that ∑i Ki∆K
†
i =
∑k Ak∆A
†
k if and only if ukl are the components of an unitary matrix. Therefore
different Kraus representations are related by unitary conjugations.
3.2 Choi-Jamiołkowski representation
The Choi-Jamiołkowski representation arises as part of a very useful theorem in
quantum information theory, the so called Choi’s theorem [Cho75, HZ12].
Theorem 7 (Choi). Let E : Cn×n → C
m×m be a linear map. The following state-
ments are equivalent:
i) E is n−positive.
ii) The matrix
CE =
n
∑
i, j=1
|ϕi〉〈ϕ j|⊗E [|ϕi〉〈ϕ j|] ∈ C
n×m ⊗C
n×m
is positive-semidefinite with {|ϕi〉}n
i=1 an orthonormal basis in C
n.
iii) E is completely positive.
Proof. The proof of iii) → i) is trivial, if E is completely positive then it is
n−positive. The implication i) → ii) can be proved easily by noticing that nor-
malizing CE → CE /n =: τE , where τE can be obtained as the application τE =
(idn ⊗E ) [ω], where ω = |Ω〉〈Ω| with |Ω〉 = 1/
√
n∑
n
i |ϕi〉⊗ |ϕi〉 a Bell state be-
tween two copies of Cn. Therefore, by the n−positivity of E it follows that τE is
positive-semidefinite.
What remains to prove is ii) → iii). To do this observe that the space C
n×m is
isomorphic to the direct sum of n copies of Cm, i.e. Cn×m ∼= C
m
1 ⊕C
m
2 ⊕·· ·⊕C
m
n ,
and define the projector into the kth copy as Pk = 〈ϕk| ⊗✶, such that PkCE Pl =
E [|ϕk〉〈ϕl|]. Now, given that CE is positive-semidefinite, it can be written as CE =
∑
nm
i |Ψi〉〈Ψi|, where |Ψi〉 ∈ C
n×m are generally unnormalized vectors. Thus, we
have that E [|ϕk〉〈ϕl|] = ∑i Pk|Ψi〉〈Ψi|Pl , where Pk|Ψi〉 ∈ C
m
k . Defining the opera-
tors {Ki : Cn →C
m}i via the equation Pk|Ψi〉= Ki|ϕk〉, where choosing for exam-
ple |ϕk〉 as the canonical basis, the columns of Ki contain the n projections of |Ψi〉
into the copies of Cm. Finally we arrive to E [|ϕk〉〈ϕl|] = ∑i Ki|ϕk〉〈ϕl|K†
i ∀k, l =
1, . . . ,n. In conclusion, since {|ϕk〉〈ϕl|}k,l is a complete basis of Cn×n, by linearity
and by theorem 6, the map E is completely positive.
30 Chapter 3. Representations of quantum channels
The matrix CE is commonly known as Choi matrix and τE as Choi-Jamiołkowski
state. Both define the Choi-Jamiołkowski representation, in this work labeled as
τE since it is normalized.
Choi’s theorem provides a simple test of complete positivity, which I find beau-
tiful. For instance, if we want to know if a given PTP map E is a valid quantum
map, we just have to consider two copies of our system in only one state, the Bell
state, then apply E to one of the copies and check if the result, τE = (idn ⊗E ) [ω],
is a density matrix.
The Choi-Jamiołkowski representation enjoys other useful properties, if E pre-
serves the trace, the matrix
τE =
1
n



E [|ϕ1〉〈ϕ1|] . . . E [|ϕ1〉〈ϕn|]
...
. . .
...
E [|ϕn〉〈ϕ1|] . . . E [|ϕn〉〈ϕn|]



(3.1)
has blocks of trace 1/n and 0, since trE [|ϕi〉〈ϕ j|] = δi j. This property additionally
means that not every density matrix in C
n×m ⊗C
n×m has a corresponding CPTP
map.
The matrix rank of τE coincides with the so called Kraus rank, i.e. the number
of linearly independent Kraus operators required to write the channel. This can
be shown easily noticing that computing τE from the Kraus sum, one arrives to
the equality |Ψi〉= 1/
√
n✶⊗Ki|Ω〉, therefore the linear independence of {|Ψi〉}i
follows immediately from the linear independence of {Ki}i. Therefore the max-
imum Kraus rank is mn and the minimum 1. Channels with Kraus rank equal to
1 are trivially unitary channels given that E [∆] = K∆K† with K†K = ✶. Channels
with the maximum rank are called full Kraus rank channels.
Another interesting property is that if τE is separable (i.e. not entangled) , then E
is entanglement-breaking, see definition 5. For qubit channels it is enough to test
that the concurrence is zero [RFZB12].
3.3 Operational representations
It has been shown that the Choi-Jamiołkowski representation is useful to test sev-
eral properties of quantum channels. In this section we will introduce other repre-
sentations, this time with operational meanings. They are basically operator basis
that give matrix and vector forms to channels and density matrices, respectively.
3.3. Operational representations 31
The vectorization of density matrices can be achieved simply “making them flat”,
this is,



ρ11 . . . ρ1d
...
. . .
...
ρd1 . . . ρdd



7→





ρ11
ρ12
...
ρdd





=:~ρ.
Using this mapping, the matrix form of operators acting on T (H ) is build using
the simple rule [GTW09]
AρB 7→
(
A⊗BT
)
~ρ. (3.2)
For instance applying this rule to a commutator, [H,ρ] 7→
(
H ⊗✶−✶⊗HT
)
~ρ .
This representation is useful to prove various results involving operators acting on
the space of density matrices, see for instance the appendix A. Additionally it is
simple to prove that the Hilbert-Schmidt inner product is mapped to 〈γ,ρ〉 7→~γ†~ρ .
One can use other operator basis accordingly to our purposes. In general we have
the following, consider {Ai}i an orthonormal operator basis in the space T (H ),
the components of the density matrix are
αi = 〈Ai,ρ〉= tr
[
A
†
i ρ
]
,
so
ρ = ∑
i
αiAi.
Correspondingly, the components of operators acting on B(H ), for instance E ,
are simply
Êi j = 〈Ai,E [A j]〉= tr
[
A
†
i E [A j]
]
.
Using this equation it is easy to prove that the representation of the adjoint opera-
tor of E , see eq. (2.25), is simply Ê ∗ = Ê †.
3.3.1 Hermitian and traceless basis
Two types of basis are specially useful in this work, the first one are the hermitian
basis. This is, every orthonormal basis {Ai}i that fulfills Ai = A
†
i , ∀i. To show the
utility of this kind of basis, let us introduce the following definition,
Definition 8 (Hermiticity preserving operators). A linear operator E : B(H )→
B(H ) preserves hermiticity if
E [∆]† = E [∆†], ∀∆ ∈ B(H ).
32 Chapter 3. Representations of quantum channels
Using the Kraus representation is trivial to prove that linear CPTP maps preserve
hermiticity, using complete positivity. Furthermore, hermiticity preserving maps
enjoy an hermitian Choi-Jamiołkowski representation, i.e. τE = τ†
E
[Wol11].
Using an hermitian basis it is straightforward to prove the following,
Proposition 1 (Representation with real entries). Let E be a linear and hermiticity
preserving map. Its matrix representation using an hermitian basis {Ai} has real
entries.
Proof. Let Êi j = tr [AiE [A j]], where the line over denotes complex conjugation.
Distributing the latter inside the argument of the trace and using the hermiticity of
Ai, we get Êi j = tr
[
AiE [A j]
†
]
, finally stressing that E [A j]
† = E [A†
j ] = E [A j], we
arrive to Êi j = Êi j.
This simple property will be used later to prove the equivalence of the problem of
finding channels that can be written as E = exp(L), with L a Lindblad operator.
The second useful type of basis are the so called traceless bases. They are defined
as follows. Let {Fi}d2−1
i=0 be an orthogonal basis, where we have indicated the
dimension of the space T (H ) as d2 with d = dim(H ), it is traceless if F0 =
✶/
√
n and trFi = 0 ∀i > 0. The traceless property comes from the fact that only
one element has non-zero trace, it is easy to prove that it must be proportional to
the identity, given that one can write the identity matrix using such basis.
This basis is useful to prove that generators of quantum dynamical maps, Lt , de-
fined with E(t+ε,t)[ρ] = ρ + εLt [ρ]+O(ε2), have the following specific structure.
Theorem 8 (Specific form of generators of dynamical maps). Let L : T (H )→
T (H ) be a linear operator fulfilling L[∆]† = L[∆†] and tr [L[∆]] = 0 (or equiva-
lently L∗[✶] = 0), then it has the following form,
L[ρ] = i[ρ,H]+
d2−1
∑
i, j=1
Gi j
(
FiρF
†
j −
1
2
{F
†
j Fi,ρ}
)
, (3.3)
where d = dim(H ), H ∈C
d×d and G∈C(d2−1)×(d2−1) are hermitian, and {Fi}d2−1
i=0
is an orthonormal traceless basis of B(H ).
Notice that Lindblad operators enjoy such form with the additional condition that
G ≥ 0, see eq. (2.20). A proof of this is given in Ref. [EL77] for the infinite
dimensional case using technicalities beyond this work. Here we will prove it for
3.3. Operational representations 33
the finite dimensional case, using the notation of an incomplete proof given in
Ref. [WECC08].
Proof. Since L preserves hermiticity, it has an hermitian Choi-Jamiołkowski ma-
trix, τL ∈ C
d2×d2
. We can write such matrix always as
τL = τφ −|Ψ〉〈Ω|− |Ω〉〈Ψ|, (3.4)
where |Ψ〉 = −ω⊥τL|Ω〉− (λ/2) |Ω〉, λ = 〈Ω|τL|Ω〉, ω⊥τLω⊥ = ω⊥τφ ω⊥ = τφ
and ω⊥ = ✶−ω . Observe that choosing the traceless operator basis {Fi}d2−1
i=0 ,
it is simple to prove that the matrix τφ can be understood also as the Choi-
Jamiołkowski matrix of the following operator:
φ [ρ] =
d2−1
∑
i, j=1
Gi jFiρF
†
j , (3.5)
with G hermitian, i.e. τφ = (idd2 ⊗φ) [ω]. This can be shown noticing that the
summation ∑
d2−1
i, j=1 goes over only traceless operators, therefore the projections
into the one-dimensional space of |Ω〉 of the Choi matrix of φ are null,
ωτφ =
d2−1
∑
i, j=1
Gi j
1
d
tr(Fi)ω
(
✶⊗F
†
j
)
= 0 ,
and similarly for
τφ ω = 0 .
For the second and third terms of Eq. 3.4, it is easy to show that the corresponding
operator is simply ρ 7→ −κρ −ρκ†, where we identify |Ψ〉= (✶⊗κ) |Ω〉.
Up to now we have shown that hermiticity preserving generators have the form
L [ρ] = φ [ρ]−κρ −ρκ†. (3.6)
Using the condition L∗[✶] = 0, we have that
κ +κ† = φ ∗[✶],
i.e. the hermitian part of κ is given by 1
2 ∑
d2−1
i, j=1 Gi jF
†
j Fi. Simply writing the anti-
hermitian part as iH we end up with
κ = iH +
1
2
d2−1
∑
i, j=1
Gi jF
†
j Fi.
Substituting this expression and eq. (3.5), in eq. (3.6), we arrive to the desired
form, see 4.5.
34 Chapter 3. Representations of quantum channels
Notice that the operator φ is completely positive if and only if G≥ 0 [HZ12], thus,
G ≥ 0 ⇔ τφ ≥ 0. In such case L has exactly a Lindblad form. This condition will
be introduced later as conditional complete positivity [EL77, WECC08].
The following is a central and useful result for our work.
Proposition 2 (Conditional complete positivity). An hermiticity preserving linear
operator L : T (H )→ T (H ) fulfilling tr [L∗[✶]] = 0, has Lindblad form if and
only if
ω⊥τLω⊥ ≥ 0.
Additionally choosing an arbitrary basis of the Hilbert space to write operators
{Fi}d2
i=1, it is easy to prove that G and ω⊥τLω⊥ are related by an unitary conjuga-
tion [CDG19].
3.4 Qubit channels
We shall devote some time to the most simple but non-trivial quantum system,
the qubit. This case turns to be rich enough to use and test the tools provided
by the literature and the ones developed here, in the context of divisibility. We
recall a particular representation and a couple of decomposition theorems for qubit
channels.
3.4.1 Pauli representation and Ruskai’s decomposition
In the case of qubit channels we can have at the same time an hermitian, traceless
and unitary basis, it is the simple Pauli basis 1√
2
{1,σx,σy,σz}. This induces a
simple 4×4 representation with real entries given by
Ê =
(
1 ~0T
~t ∆
)
, (3.7)
where ∆ is a 3×3 matrix with real entries and~t a column vector. This describes
the action of the channel in the Bloch sphere picture in which the points ~r are
identified with density matrices ρ~r =
1
2
(✶+~r ·~σ) [RSW02]. Therefore the action
of the channel is described by E (ρ~r) = ρ∆~r+~t .
3.4. Qubit channels 35
In order to study qubit channels with simpler expressions, we will consider a
decomposition in unitaries such that
E = U1DU2. (3.8)
This can be achieved using Ruskai’s decomposition [RSW02], which can be per-
formed by decomposing ∆ in rotation matrices, i.e. ∆ = R1DR2, where D =
diag(λ1,λ2,λ3) is diagonal and the rotations R1,2 ∈ SO(3) (of the Bloch sphere)
correspond to the unitary channels U1,2. Notice that as D is not required to be
positive-semidefinite, Ruskai’s decomposition must not be confused with the sin-
gular value decomposition. The latter allows decompositions that include total re-
flections. Such operations do not correspond to unitaries over a qubit, in fact they
are not CPTP. An example of this is the universal NOT gate defined by ρ 7→ ✶−ρ ,
it is PTP but not CPTP. The resulting form from Ruskai’s decomposition is stated
in the following theorem,
Theorem 9 (Special orthogonal normal form). For any qubit channel E , there
exist two unitary conjugations , U1 and U2, such that E = U1DU2, where D has
the following form in the Pauli basis,
D̂ =
(
1 ~0T
~γ D
)
, (3.9)
and is called special orthogonal normal form of E .
Here, RT
1 ∆RT
2 = D and ~γ = RT
1
~t. The latter describes the shift of the center of the
Bloch sphere under the action of D . The parameters ~λ determine the length of
semi-axes of the Bloch ellipsoid, being the deformation of Bloch sphere under the
action of E . In particular detD̂ = det Ê = λ1λ2λ3.
To develop geometric intuition in the space determined by the possible values of
the three parameters of~λ , consider the Choi-Jamiołkowski representation of the
special orthogonal normal form of an arbitrary channel in the basis that diago-
nalises D,
τD =
1
4




γ3 +λ3 +1 γ1 − iγ2 0 λ1 +λ2
γ1 + iγ2 −γ3 −λ3 +1 λ1 −λ2 0
0 λ1 −λ2 γ3 −λ3 +1 γ1 − iγ2
λ1 +λ2 0 γ1 + iγ2 −γ3 +λ3 +1




. (3.10)
Complete positivity is determined by the non-negativity of its eigenvalues, given
that it is hermitian, but it turns that for the general case they have complicated
expressions. To overcome this problem we use the fact that if D is a channel, then
36 Chapter 3. Representations of quantum channels
its unital part, defined by taking ~γ =~0, is a channel too [Wol11]. Therefore the
set of the possible values of~λ for the general case is contained in the set arising
from the unital case. The complete positivity conditions for the latter are
1+λi ± (λ j +λk)≥ 0, (3.11)
with i, j and k all different, this implies that the possible set of lambdas lives inside
the tetrahedron with corners (1,1,1), (1,−1,−1), (−1,1,−1) and (−1,−1,1),
see fig. 3.1. For unital channels, all points in the tetrahedron are allowed. The
corner ~λ = (1,1,1) corresponds to the identity channel, ~λ = (1,−1,−1) to σx
~λ = (−1,1,−1) to σy and~λ = (−1,−1,1) to σz (Kraus rank 1 operations). Points
in the edges correspond to Kraus rank 2 operations, points in the faces to Kraus
rank 3 operations and in the interior of the tetrahedron to Kraus rank 4 operations.
In particular, this tetrahedron defines the set of Pauli channels, which are defined
to have diagonal special orthogonal normal form.
Definition 9 (Pauli channels). A qubit channel E is a Pauli channel if
E [ρ] =
3
∑
i=0
piσiρσi, (3.12)
with σ0 := ✶, pi ≥ 0 and ∑
3
i=0 pi = 1.
For non-unital channels more restrictive conditions arise, an example will be given
later.
3.4.2 1wSLOCC and singular value decomposition using the Lorentz
metric
There is another parametrization for qubit channels called Lorentz normal decom-
position [VDD01] which is specially useful to characterize infinitesimal divisi-
bility CInf. To introduce it, let us resort to chapter 2 where we discussed local
operations and classical communication. For the two-qubit case, the operations
that Alice and Bob apply for their reduced states are
ρA 7→ XρAX†
ρB 7→ Y ρBY †, (3.13)
where we have shown that it is enough to consider X ,Y ∈ SL(2,C) for X and Y
invertible, see chapter 2 Now we are going to show that such operations can be
3.4. Qubit channels 37
λ1
λ3
λ2
id
ρ 7→ σxρσx
ρ 7→ σzρσz
Figure 3.1: Set of the possible values of~λ . This set has the shape of a tetra-
hedron where the corners are the Pauli unitaries (✶, σx and σz are indi-
cated in the figure, while σy lies behind). The rest of the body contains
convex combinations of Pauli unitaries. Unital qubit channels can be
obtained by concatenating Pauli channels with unitary conjugations, see
theorem 9.
38 Chapter 3. Representations of quantum channels
understood as proper orthochronous Lorentz transformations in the Pauli repre-
sentation.
Consider an arbitrary hermitian operator ∆ and its representation in the Pauli basis,
∆ = (✶ tr∆+~r∆ ·~σ)/2
with det∆ = (tr∆)2 − |~r∆|2. Now observe that det∆ can be understood as the
squared Lorentz norm of the four-vector r∆ = (tr∆,~r∆)
T
, lying in the Minkowski
vector space, denoted as (R4,η), where η = diag(1,−1,−1,−1) is the Lorentz
metric. Therefore we have
det∆ = |r∆|2Lorentz = 〈r∆,ηr∆〉, (3.14)
with 〈·〉 the standard inner product. Then tr∆ is a time-like component and ~r∆
space-like components.
Given that operations shown in eq. (3.13) preserve the determinant, they are isome-
tries in the Minkowski space. That is, they preserve the norm shown in eq. (3.14)
for any vector r∆ (with ∆ hermitian). Additionally, due to linearity of eq. (3.13),
these operations belong to SO(3,1) (the Lorentz group). In fact, due to the
positivity of quantum operations, they do not change the sign of the trace (the
time-like component); therefore the transformations are orthochronous. Also no-
tice that SL(2,C) contains the identity transformation, therefore the set of one-
way stochastic local operations and classical communication is identified with the
proper orthochronous Lorentz group, SO+(1,3) [Wol11, Tun85]. However, since
−X and X give the same result, see eq. (3.13), and both belong to SL(2,C), one
says that the latter is a double cover of SO+(1,3). This map is also called spinor
map.
Given this map, it is expected that the operations mentioned in eq. (3.13) are
explicitly Lorentz matrices, when writing them in the Pauli basis. Also notice
that unitary conjugations are particular cases of them [RSW02], therefore one can
think of a different decomposition using the Lorentz metric instead of the three
dimensional Euclidean metric, used in Ruskai’s decomposition.
The Lorentz normal form was introduced first for two-qubit states by writing them
as
τ =
1
4
∑
i j
Ri jσi ⊗σ j, (3.15)
where we have used the notation of Choi-Jamiołkowski states for convenience.
This decomposition is derived from the theorem 3 of Ref. [VDD01], which essen-
tially states that the matrix R can be decomposed as
R = L1ΣLT
2 . (3.16)
3.4. Qubit channels 39
Here L1,2 are proper orthochronous Lorentz transformations and Σ is either Σ =
diag(s0,s1,s2,s3) with s0 ≥ s1 ≥ s2 ≥ |s3|, or
Σ =




a 0 0 b
0 d 0 0
0 0 −d 0
c 0 0 −b+ c+a




. (3.17)
Note that Σ corresponds to an unnormalized state, with trace trΣ = a. Thus, the
normalization constant is α = a−1.
To introduce the Lorentz normal decomposition of qubit channels, let us first in-
troduce the following. Let E a qubit channel and Ê its matrix representation
using the Pauli basis. The latter is related with the matrix R, which defines its
Choi-Jamiołkowski state, see eq. (3.15),
Ê ΦT = R, (3.18)
where ΦT = diag(1,1,−1,1). This can be shown by defining a generic Pauli chan-
nel, computing its Choi matrix and extracting R using the Hilbert-Schmidt inner
product with the basis {σi⊗σ j}i, j. Now, defining the decomposition for channels
throughout decomposing the Choi-Jamiołkowski state, we can easily compute the
corresponding Lorentz transformations using equations (3.18) and (3.16):
RΦT = αL1ΣLT
2 ΦT
Ê = αL1ΣLT
2 ΦT (3.19)
= L1 (αΣΦT)ΦTLT
2 ΦT (3.20)
= L1
ˆ̃
E L̃T
2 , (3.21)
where ˆ̃
E = αΣΦT is the Lorentz normal form of Ê . Also notice that L̃T
2 =
ΦTLT
2 ΦT is proper and orthochronous, given that its determinant is positive and
ΦT is proper. Therefore the possible Lorentz normal forms for channels are
ˆ̃
E = diag(s0,s1,−s2,s3) with s0 ≥ s1 ≥ s2 ≥ |s3|, or
Σ =




a 0 0 b
0 d 0 0
0 0 d 0
c 0 0 −b+ c+a




. (3.22)
Using this, the authors of Ref. [VV02] introduced a theorem (theorem 8 of the
reference) defining the Lorentz normal form for channels by forcing b = 0, in
order to have normal forms proportional to trace-preserving operations. The latter
40 Chapter 3. Representations of quantum channels
is equivalent to say that the decomposition of Choi-Jamiołkowski states leads to
states that are also Choi-Jamiołkowski. We didn’t find a good argument to justify
such assumption, and found a counterexample that shows that Lorentz normal
forms with b 6= 0 exist (see appendix B). Therefore in general we can find a Σ
with form of eq. (3.17) with b 6= 0. The consequence of this is that the theorem
8 of Ref. [VV02] is incomplete, but given that form of eq. (3.17) is Kraus rank
deficient (it has rank three for b 6= c and two for b = c), the full Kraus rank case
is still useful. Thus, we propose a restricted version of their theorem:
Theorem 10 (Restricted Lorentz normal form for qubit quantum channels).
For any full Kraus rank qubit channel E there exist rank-one completely positive
maps T1,T2 such that T = T1E T2 is proportional to
(
1 ~0T
~0 Λ
)
, (3.23)
where Λ = diag(s1,s2,s3) with 1 ≥ s1 ≥ s2 ≥ |s3|.
The channel T is called the Lorentz normal form of the channel E . For unital
qubit channels D coincides with Λ.
3.5 Representation of Gaussian quantum channels
In this section we start from two ansätze, that put together with the Gaussian
functional form considered in Ref. [MP12], lead to the complete set of functional
forms in position state representation of one-mode Gaussian channels.
We will show that only two possible forms of δGQC hold according to trace pre-
serving (TP) and hermiticity preserving (HP) conditions. The one corresponding
to eq. (2.33) is one of these, as expected. Later on we will impose complete posi-
tivity in order to have valid GQC, i.e. complete positive and trace preserving (C)
Gaussian operations.
Following definition 7, those channels can be characterized by how they act over
Gaussian states. It is well known that the action of GQCs on Gaussian states is
described by affine transformations [WPGP+12]. Let A be a GQC defined by
a tuple such that A = A (T,N,~τ), where T and N are 2× 2 real matrices with
N = NT [WPGP+12]. The transformation acts on Gaussian states according to
A (T,N,~τ)
[
S
(
σ , ~d
)]
= S
(
TσTT +N,T~d +~τ
)
.
3.5. Representation of Gaussian quantum channels 41
In the particular case of closed systems, where the system is governed by a Gaus-
sian unitary, we have that N = 0 and T is a symplectic matrix.
3.5.1 Possible functional forms of δGQC operations
Let us introduce the ansätze for the possible forms of GQC in the position rep-
resentation, to perform the black-box characterization. Following eq. (2.29) and
taking the continuous variable representation of difference and sum coordinates,
the trace becomes an integral over position variables of the environment. Then
we end up with a Fourier transform of a multivariate Gaussian. Since the Fourier
transform of a Gaussian is again a Gaussian (unless there are singularities in the
coefficients, as in the example of eq. (2.33)), the result of the Fourier transform
for one mode can have the following structures: a Gaussian form [eq. (2.31)], a
Gaussian form multiplied with one-dimensional delta or a Gaussian form multi-
plied by a two-dimensional delta. No more deltas are allowed given that there are
only two integration variables when applying the channel, see eq. (2.30). Thus, in
order to start with the black-box characterization, we shall propose the following
general Gaussian operations with one and two deltas, respectively
JI(x f ,r f ;xi,ri) = NIδ (~α
T~v f +~β T~vi)e
Σ(x f ,xi;r f ,ri), (3.24)
JII(x f ,r f ;xi,ri) = NIIδ (A~v f −B~vi)e
Σ(x f ,xi;r f ,ri), (3.25)
with ~vi, j = (ri, j,xi, j), and NI,II are normalization constants. Coefficient arrays
A, B, ~α , and ~β have real entries since initial and final coordinates must be real.
Finally, the exponent reads:
Σ(x f ,xi;r f ,ri) = ı
(
b1x f r f +b2x f ri +b3xir f +b4xiri + c1x f + c2xi
)
−a1x2
f −a2x f xi −a3x2
i − e1r2
f − e2r f ri − e3r2
i −d1r f −d2ri.
They provide, together with eq. (2.31) all possible ansätze for GQC.
3.5.2 Hermiticity and trace preserving conditions
Before studying CPTP conditions it is useful to simplify expressions of equations
(3.24) and (3.25). To do this we use the fact that linear CPTP operations preserve
hermiticity and trace. For channels of continuous variable systems in the position
42 Chapter 3. Representations of quantum channels
state representation, J(q f ,q
′
f ;qi,q
′
i), HP condition is derived as follows,
ρ f (q
′
f ,q f )
∗ =
∫
R2
dqidq′iJ(q
′
f ,q f ;qi,q
′
i)
∗ρi(qi,q
′
i)
∗
=
∫
R2
dqidq′iJ(q
′
f ,q f ;q′i,qi)
∗ρi(qi,q
′
i)
= ρ f (q f ,q
′
f ), (3.26)
where the last equality holds if
J(q f ,q
′
f ;qi,q
′
i) = J(q′f ,q f ;q′i,qi)
∗.
Using sum and difference coordinates, HP becomes
J(−x f ,r f ;−xi,ri) = J(x f ,r f ;xi,ri)
∗. (3.27)
Following this equation and comparing exponents of the both sides of the last
equations, it is easy to note that the coefficients an, bn, cn, en and dn must be real.
Concerning the delta factors, in eq. (3.27) we end up with expressions like
δ (α1x f +α2xi +β1r f +β2ri) = δ (−α1x f −α2xi +β1r f +β2ri)
for both cases. Therefore the equality holds for eq. (3.24) only for two possible
combinations of variables: i) δ (αx f − βxi) and ii) δ (αr f − β ri). For the case
of eq. (3.25), equality holds only for iii) δ (γr f −ηri)δ (αx f −βxi). Let us now
analyze the trace preserving condition (TP), since the trace of ρ f in sum and dif-
ference coordinates is
trρ f =
∫
R
dr′f ρ f (x f = 0,r′f )
=
∫
R
dr′f dridxiJ(x f = 0,r′f ;xi,ri)ρi(xi,ri)
=
∫
R
driρi(xi = 0,ri).
To fulfill the last equality, the following must be accomplished
∫
R
dr′f J(x f = 0,r′f ;xi,ri) = δ (xi). (3.28)
This condition immediately discards ii) from the above combinations of deltas,
thus we end up with cases i) and iii). For case i) TP reads:
NI
∫
dr f δ (−βxi)e
Σ =
NI
|β |
√
π
e1
δ (xi)e
(
e2
2
4e1
−e3
)
r2
i
, (3.29)
3.5. Representation of Gaussian quantum channels 43
thus, the relation between the coefficients assumes the form
e2
2
4e1
− e3 = 0,d1 = 0,d2 = 0, (3.30)
and the normalization constant NI = |β |
√
e1
π with β 6= 0 and e1 > 0. For case iii)
the trace-preserving condition reads
NII
∫
dr f δ (γr f −ηri)δ (−βxi)e
Σ =
NII
|βγ|δ (xi)e
−
(
e1(
η
γ )
2+e2
η
γ +e3
)
r2
i −
(
d1
η
γ +d2
)
ri .
Thus, the following relation between en and dn coefficients must be fulfilled:
e1
(
η
γ
)2
+ e2
η
γ
+ e3 = 0, d1
η
γ
+d2 = 0, (3.31)
with γ,β 6= 0 and NII = |βγ|. In the particular case of η = 0, eq. (3.31) is reduced
to e3 = d2 = 0. As expected from the analysis of limits above, we showed that
δGQC’s admit quadratic terms in ri, j.
3.5.3 Complete positivity conditions
Up to this point we have hermitian and trace preserving Gaussian operations; to
derive the remaining CPTP conditions, it is useful to write its Wigner’s function
and Wigner’s characteristic function. The representation of the Wigner’s charac-
teristic function reads
χ(~k) = tr
[
ρD(~k)
]
= exp
[
−1
2
~kT
(
ΩσΩT
)
~k− ı(Ω〈x̂〉)T~k
]
(3.32)
and its relation with Wigner’s function:
W (x) =
∫
R2
d~xe−ı~xTΩ~kχ
(
~k
)
(3.33)
=
∫
R
eıpxdx
〈
r− x
2
∣
∣
∣ ρ̂
∣
∣
∣r+
x
2
〉
, (3.34)
where ~k = (k1,k2)
T
, ~x = (r, p)T
and h̄ = 1 (we are using natural units). Using
the previous equations to construct Wigner and Wigner’s characteristic functions
of the initial and final states, and substituting them in the equation 2.30, it is
44 Chapter 3. Representations of quantum channels
straightforward to get the propagator in the Wigner’s characteristic function rep-
resentation:
J̃
(
~k f ,~ki
)
=
∫
R6
dΓK(~l)J(~v f ,~vi), (3.35)
where the transformation kernel reads
K(~l) =
1
(2π)3
e[ı(k
f
2 r f −k
f
1 p f −ki
2ri+ki
1 pi−pixi+p f x f )],
with dΓ = d p f d pidx f dxidr f dri and~l = (p f , pi,x f ,xi,r f ,ri)
T
. By elementary in-
tegration of eq. (3.35) one can show that for both cases
J̃I,III
(
~k f ,~ki
)
= δ
(
ki
1 −
α
β
k
f
1
)
δ
(
ki
2 −~φ T
I,III
~k f
)
ePI,III(~k f ), (3.36)
where PI,III(~k f ) = ∑
2
i, j=1 P
(I,III)
i j k
f
i k
f
j +∑
2
i=1 P
(I,III)
0i k
f
i with P
(I,III)
i j = P
(I,III)
ji . For case
i) we obtain
P
(I)
11 =−
(
(
α
β
)2(
a3 +
b2
3
4e1
)
+
α
β
(
a2 +
1
2
b1b3
e1
)
+a1 +
b2
1
4e1
)
,
P
(I)
12 =−
(
α
β
b3
2e1
+
b1
2e1
)
,
P
(I)
22 =− 1
4e1
. (3.37)
For case iii) we have
P
(III)
11 =−
(
(
α
β
)2
a3 +
α
β
a2 +a1
)
,
P
(III)
12 = P
(III)
22 = 0. (3.38)
And for both cases we have P
(I,III)
01 = ı
(
α
β c2 + c1
)
and P
(I,III)
02 = 0. Vectors ~φ are
given by
~φI =
(
α
β
(
b4 −
b3e2
2e1
)
− b1e2
2e1
+b2,−
e2
2e1
)T
,
~φIII =
(
α
β
η
γ
b3 +
α
β
b4 +
η
γ
b1 +b2,
η
γ
)T
. (3.39)
3.5. Representation of Gaussian quantum channels 45
We are now in position to write explicitly the conditions for complete positivity.
Having a Gaussian operation characterized by (T,N,~τ), the CP condition can be
expressed in terms of the matrix
C = N+ ıΩ− ıTΩTT, (3.40)
where Ω =
(
0 1
−1 0
)
is the symplectic matrix. An operation A (T,N,~τ) is
CP if and only if C ≥ 0 [Lin00, WPGP+12]. Applying the propagator on a test
characteristic function, eq. (3.32), it is easy compute the corresponding tuples.
For both cases we get:
NI,III = 2
(
−P22 P12
P12 −P11
)
,
~τI,III =
(
0, ıP
(I,III)
01
)T
, (3.41)
while for case i) matrix T is given by
TI =
(
e2
2e1
0
~φI,1 −α
β
)
, (3.42)
where ~φI,1 denotes the first component of vector ~φI, see eq. (3.39). The complete
positive condition is given by the inequalities raised from the eigenvalues of ma-
trix eq. (3.40):
±
√
α2e2
2 +4αβe2e1 +4β 2e2
1
(
4P
(I)
12
2
+
(
P
(I)
11 −P
(I)
22
)2
+1
)
2βe1
≥ P
(I)
11 +P
(I)
22 .
(3.43)
For case iii) matrix T is
TIII =
(
−η
γ 0
~φIII,1 −α
β
)
, (3.44)
and complete positivity conditions read:
±
√
(βγ −αη)2 +β 2γ2P
(III)
11
2
βγ
−P
(III)
11 ≥ 0. (3.45)
Note that in both cases the complete positivity conditions do not depend on ~φ .

Chapter 4
Divisibility of quantum channels
and dynamical maps
Wine is sunlight, held together by water.
Galileo Galilei
In this chapter we introduce the formal definition of divisibility of quantum chan-
nels, inspired by questioning how can we implement a given quantum channel
via the concatenation of simpler channels. Later on we define further types of
divisibility by adding extra conditions, such as channels being infinitesimal di-
visible and channels belonging to one-parameter semigroups. These types are
physically relevant since both lead to Markovian dynamical maps [ARHP14]. We
additionally prove three theorems, which are the central contributions of this part
of the work. Finally, a complete characterization of channels belonging to one-
parameter semigroups that is given.
4.1 Divisibility of quantum maps
A quantum channel E is said to be divisible if it can be expressed as the concate-
nation of two non-trivial channels,
Definition 10 (Divisibility). A linear map E ∈ C is divisible if there exists a de-
composition,
E = E2E1, (4.1)
such that E1 and E2 are non-unitary channels, or E is unitary.
47
48 Chapter 4. Divisibility of quantum channels and dynamical maps
Notice that this definition ensures that unitary channels are divisible, and that non-
unitary channels must be divisible in non-unitary channels. This prevents one to
consider simple changes of basis as a “division” of a given quantum operation.
This type of divisibility, which is the most general and less restrictive one, defines
a set that will be denoted by Cdiv. The set of indivisible channels is the comple-
ment of Cdiv in C, therefore it will be denoted as Cdiv. Notice that this definition
is different to the one given in Ref. [WC08] where unitary channels are excluded
to be divisible.
The concept of indivisible channels resembles the concept of prime numbers, uni-
tary channels play the role of unity (which are not indivisible/prime), i.e. a com-
position of indivisible and a unitary channel results in an indivisible channel.
We now introduce three results from Ref. [WC08] that shall be used later. We
only give the proof for the second for the sake of brevity.
Theorem 11 (Full Kraus rank channels). Let E : T (H )→T (H ) be a quantum
channel. If it has full Kraus Kraus rank, i.e. d2 with d = dim(H ), then it is
divisible.
An example of full Kraus rank channel is the total depolarizing channel ρ 7→
✶/dim(H ), which maps every state into the maximal mixed one.
Theorem 12 (Indivisible channels). Consider the set Cd of channels acting on the
space of density matrices of d×d, i.e. E : T (H )→ T (H ) with d = dim(H ).
The channel with minimal determinant, E0 ∈ Cd, is indivisible.
Proof. To prove this we use the fact that channels with negative determinant ex-
ist [WC08] (two examples are given below), and the property of monotonicity of
the determinant.
Let E ∈ C with detE < 0 and E = E2E1 an arbitrary division of E with E1,E2 ∈ C.
The monotonicity of the determinant implies the following,
|det(E2E1) |= |detE2||detE1| ≤ |detE1|.
Assuming, without loss of generality that detE1 < 0 and detE2 > 0, we have that
detE1 detE2 ≤ detE1.
Multiplying both sides by −1 we arrive to
|detE2||detE1| ≥ |detE1|.
4.1. Divisibility of quantum maps 49
Therefore, by monotonicity of the determinant, we have
|detE2||detE1|= |detE1|,
which implies that detE2 = 1, i.e. E2 is an unitary conjugation [WC08] and E has
minimum determinant. By definition 10 E is indivisible.
Two examples for the qubit case are the approximate NOT and the approximate
transposition maps:
ρ 7→ tr(ρ)✶+ρT
3
(approximate transposition),
ρ 7→ tr(ρ)✶−ρ
3
(approximate NOT gate), (4.2)
both have minimal determinant corresponding to −1/27, which can be computed
from their matrix representation.
Theorem 13 (Unital Kraus rank three channels). A unital qubit channel is indi-
visible if and only if it has Kraus rank equal to three.
This is a restricted version of theorem 23 of Ref. [WECC08], where authors
proved the theorem for any qubit channel instead of only unital ones. Since their
proof rely on the validity of the Lorentz normal decomposition for channels, we
have written here a restricted version, where Lorentz normal form is equivalent to
the special orthogonal normal form (see theorem 10 and its discussion).
These results can be used immediately to identify the divisibility character of uni-
tal qubit channels, see fig. 4.5. The faces of the tetrahedron (without edges) corre-
spond to indivisible channels, in particular the center of every face corresponds to
channels with minimal determinant. The body (full Kraus rank channels) contain
divisible channels.
4.1.1 Subclasses of divisible maps
Divisibility of quantum dynamical maps
We motivate the extra conditions to define new types of divisibility on the con-
cept of Markovian process. In subsection 2.2.1 we have introduced the definition
of Markovian process and its consequences at the level of propagators of one-
point probabilities, see eq. (2.24). Based on this, we introduce the concept of
50 Chapter 4. Divisibility of quantum channels and dynamical maps
CP-divisibility of quantum dynamical maps, which is often used as definition of
Markovianity in the quantum realm [ARHP14].
Definition 11 (CP-divisible quantum dynamical maps). Consider a quantum dy-
namical map E(t,0) : T (H ) → T (H ) with t ∈ R
+. It is CP-divisible in the
interval [0, t]⊂ R
+ if for every decomposition of the form
E(t,0) = E(t,s)E(s,0),
E(t,s) is a quantum channel for every s ∈ (0, t).
A remarkable theorem on CP-divisible maps is the following [Kos72b, Kos72a,
Gor76, Lin76, ARHP14],
Theorem 14 (Gorini-Kossakowski-Susarshan-Lindblad). An operator Lt is the
generator of a CP-divisible process if and only if it can be written in the following
form:
Lt [ρ] =−i[H(t),ρ]+∑
i, j
Gi j(t)
(
Fi(t)ρF
†
j (t)−
1
2
{F
†
j (t)Fi(t),ρ}
)
, (4.3)
where G is hermitian and positive semidefinite, H(t),Fk(t)∈C
d×d are time-dependent
operators acting on H , with H(t) hermitian for every t ∈R
+, and d = dim(H ).
In Ref. [RH12] a proof is given starting from the Kraus representation of quantum
dynamical maps and the definition of CP-divisibility. Here we will give a simpler
proof resorting to theorem 8.
Proof. Notice that for each time t we can define the “instant” map E(t+ε,t)[ρ] =
ρ +εLt [ρ]+O(ε2), with ε > 0, therefore the hermiticity preserving of Lt follows
from the hermiticity preserving of E(t,0). Also note that we can always choose the
same traceless basis, {Fi}d2−1
i=0 , to write eq. (4.3), such that the time dependence
is dropped only in G(t) ∈ C
d2×d2
and H(t). By theorem 8, Lt has the form stated
in eq. (4.3), the only thing that remains to prove is that G(t) ≥ 0 for every t.
To do this we construct the Choi-Jamiołkowski matrix of the instant map, τt,ε =
ω+ε (idd2 ⊗Lt) [ω]+O(ε2). We remind the reader that ω = |Ω〉〈Ω|, where |Ω〉 is
the Bell state between two copies of Cd . Now we test positive-semidefinitiveness
of τt,ε ,
〈ϕ|τt,ε |ϕ〉= 〈ϕ|Ω〉〈Ω|ϕ〉+ ε〈ϕ|(idd2 ⊗Lt) [|Ω〉〈Ω|]|ϕ〉+O(ε2)≥ 0,
4.1. Divisibility of quantum maps 51
∀|ϕ〉 ∈ C
d2
. The inequality always holds for any 〈ϕ|Ω〉 6= 0 and ε > 0. For
〈ϕ|Ω〉 = 0 we have that for ε > 0 the inequality 〈ϕ|(idd2 ⊗Lt) [|Ω〉〈Ω|]|ϕ〉 ≥ 0
must be accomplished, i.e. ω⊥τLω⊥ ≥ 0 (conditional complete positivity). There-
fore by proposition 2, one has that G(t)≥ 0.
Analogously to CP-divisible processes, if we relax the condition of the interme-
diate maps to be PTP (and not necessarily CPTP), we arrive to the following
definition:
Definition 12 (P-divisible quantum dynamical maps). Consider a quantum dy-
namical map E(t,0) : T (H )→T (H ) with t ∈R
+. It is P-divisible in the interval
[0, t]⊂ R
+ if for every decomposition of the form
E(t,0) = E(t,s)E(s,0),
E(t,s) belongs to PTP for every s ∈ (0, t).
Unfortunately, to the best of our knowledge, there doesn’t exist a statement similar
to theorem 14, nor a simple test of P-divisibility. But for certain types of genera-
tors of dynamical maps, conditions for P-divisibility were derived in Ref. [CDG19].
Divisibility of quantum channels
Let us discuss these two types of divisibility but now from a statical point of
view. First notice that instant operations E(t+ε,t) are arbitrarily close to the identity
map as ε → 0+, for both P-divisible and CP-divisible processes. In other words,
they are infinitesimal. Consider now the idea of quantum channels divisible in
infinitesimal parts, i.e. what is given this time is a quantum channel instead of a
dynamical map. This idea motivates the following definition [WC08],
Definition 13 (Infinitesimal divisible channels in CPTP). Let LCP be the set con-
taining operations E ∈ C with the property that ∀ε > 0 there exist a finite number
of channels Ei ∈ C such that |Ei− id|< ε and E = ∏i Ei, see fig. 4.1. It is said that
a channel is infinitesimal divisible if it belongs to the closure of LCP. This set is
denoted as CCP.
The necessity of the closure can be motivated using the following example. Con-
sider the qubit channel defined as follows:
E∞ :
(
ρ00 ρ01
ρ∗
01 ρ11
)
7→
(
ρ00 0
0 ρ11
)
. (4.4)
52 Chapter 4. Divisibility of quantum channels and dynamical maps
E ∈ C
CP
E =
E 1 E 2 E Nε
id
E
Ei
Figure 4.1: Diagrammatic decomposition of channels belonging to LCP
whose closure is CCP, see definition 13. We show the circuit repre-
senting the decomposition of E into channels (left) arbitrarily close to
the identity map (right).
This channel is singular, i.e. does not belong to LCP. Now observe that using
the dynamical process, Et , given in example 1, one can get arbitrarily close to
E∞ when t → ∞, i.e. E∞ = limt→∞ Et . Note that Et ∈ LCP for every t ∈ R
+, see
theorem 14, therefore E∞ is an accumulation point of LCP. Thus, the closure is
taken to define infinitesimal divisible channels, to include channels such as E∞.
Up to this point we have shown that CP-divisible processes are infinitesimal di-
visible, i.e. CP-divisible processes parametrize families of channels belonging to
CCP. In Ref. [WECC08], authors have shown that channels in CCP can always be
implemented with CP-divisible processes. This can be roughly shown as follows.
Since C is connected, we can understand infinitesimal channels as the ending point
of an arbitrarily small curve parametrized by t, i.e. channels Ei in definition 13
can be written approximately as Ei ≈ id+Li ≈ exp(Li). We have shown that Li
has Lindblad form, see theorem 14. Therefore we have that if E ∈ CCP, it can be
written as
E = ∏
i
eLi .
Therefore E can be implemented using a CP-divisible dynamical processes. Bounds
of the convergence ratio using channels of the form exp(Li) instead of general in-
finitesimal channels, are computed in Ref. [WECC08].
Analogous to infinitesimal divisible channels in C and its relation with CP-divisible
processes, one can also define the following set involving PTP maps.
Definition 14 (Infinitesimal divisible channels in PTP). Let LP be the set con-
4.1. Divisibility of quantum maps 53
E ∈ C
P
E =
E 1 E 2 E Nε
id
E
Ei
CPTP
PTP
Figure 4.2: Diagramatic decomposition of channels belonging to LP which
closure is CP, see definition 14. At the left we show the circuit rep-
resenting the decomposition of E into channels arbitrarily close to the
identity map, see figure at the right. In contrast to figure 4.1, note that
infinitesimal channels can be outside the set of CPTP maps, but inside
PTP.
taining operations E ∈ C with the property that ∀ε > 0 there exist a finite number
of channels Ei ∈ PTP such that |Ei − id|< ε and E = ∏i Ei, see fig. 4.2. It is said
that a channel is infinitesimal divisible in PTP if it belongs to the closure of LP.
This set is denoted as CP.
Infinitesimal divisibility in PTP maps is interesting since this kind of maps can
arise in settings where the system is initially correlated with its surroundings, or
if the operation is correlated with the initial state [CTZ08].
Infinitesimal divisible (either in CPTP and PTP) channels have non-negative de-
terminant due to its continuity [WECC08]. To see this note that channels arbitrar-
ily close to the identity map have positive determinant; and by its multiplicative
property, the channel resulting from the concatenation of infinitesimal channels
has non-negative determinant.
Proposition 3 (Determinant of infinitesimal divisible channels). If a quantum
map E belongs either to CP or CCP, then detE ≥ 0.
It turns out that a non-negative determinant is a sufficient condition for a channel
to be infinitesimal divisible in PTP, see theorem 25 of Ref. [WECC08].
Other interesting type of divisibility that in turn forms a subset of CCP is the fol-
lowing [WECC08, Den89].
54 Chapter 4. Divisibility of quantum channels and dynamical maps
E ∈ C
∞
E =
En En En
n times
Figure 4.3: Diagrammatic decomposition of channels belonging to C∞, see
definition 15. This set contains channels for which every n-root exist
and is a valid quantum channel, denoted in the circuit as En.
Definition 15 (Infinitely divisible channels). A quantum channel E is infinitely
divisible if ∀n ∈ Z
+ ∃En ∈ C such that E = (En)
n
. This set is denoted as C∞,
see fig. 4.3.
This set contains channels for which every n-root exists and is a valid quantum
channel. Denisov has shown in [Den89] that infinitely divisible channels can be
written as E = E0 exp(L), with L a Lindblad generator, and an E0 idempotent
operator that fulfills E0LE0 = E0L. In this work we will prove that every infinitely
divisible Pauli channel has the simple form exp(L).
Let us now introduce the most restricted type of divisibility studied in this work,
Definition 16 (Channels belonging to one-parameter semigroups (L-divisibility)).
Let LL be the set containing non-singular operations E ∈ C, such that there exist
at least one logarithm, denoted as L = logE , such that
L[ρ] = i[ρ,H]+∑
i, j
Gi j
(
FiρF
†
j −
1
2
{F
†
j Fi,ρ}
)
, (4.5)
where H and G are hermitian with G ≥ 0, and {Fi}i are bounded operators acting
on T (H ). It is said that a channel is L-divisible if it belongs to the closure of
LL. This set is denoted as CL.
Analogous to the relation of CP-divisible dynamical maps and its relations with
CCP, time-independent Markovian processes form families of L-divisible chan-
nels. The converse is true by definition. One of the principal objectives of this
work is to construct a test to check whether a given channel belongs to CL or not.
4.1. Divisibility of quantum maps 55
C
C
L
C
div
C
P
C
CP
C
∞
C
P
∩ Cdiv
Unitary
channels
Figure 4.4: Scheme illustrating the different sets of quantum channels for
a given dimension. In particular, the inclusion relations presented in
eq. (4.6) are depicted.
4.1.2 Relation between channel divisibility classes
Let us summarize the introduced divisibility sets and the relations between them.
Since channels belonging to CCP can be implemented with time-dependent Lind-
blad master equations, and time-independent ones are a particular case of time de-
pendent ones, we have CL ⊂CCP. Now, since infinitely divisible channels have the
form E0 exp(L), channels with form exp(L) are a particular case of C∞, therefore
CL ⊆ C∞. Also, given that CPTP maps are also PTP, then CCP ⊂ CP. Finally, ev-
ery set except CP is subset of Cdiv, given that an infinitesimal divisible channels in
PTP is not necessarily divisible in CPTP channels. In summary we have [WC08],
C∞ ⊂ CCP ⊂ Cdiv
⊆
CL ⊂ CCP ⊂ CP
. (4.6)
The intersection of CP and Cdiv is not empty since CCP ⊂ Cdiv and CCP ⊂ CP, later
on we will investigate if CP ⊆ Cdiv or not. A scheme of the inclusions is given in
fig. 4.4.
56 Chapter 4. Divisibility of quantum channels and dynamical maps
4.2 Characterization of L-divisibility
Deciding L-divisibility is equivalent to proving the existence of a hermiticity pre-
serving generator, which additionally fulfills the ccp condition, see proposition 2.
To prove hermiticity preserving we recall that every HP operator has a real ma-
trix representation when choosing an hermitian basis, see subsection 3.3.1. Since
quantum channels preserve hermiticity, the problem is reduced to find a real loga-
rithm log Ê given a real matrix Ê , where the hat means that E is written using an
hermitian basis. This problem was already solved by Culver [Cul66] who charac-
terized completely the existence of real logarithms of real matrices. In this work
we restrict the analysis to diagonalizable channels. The results can be summarized
as follows.
Theorem 15 (Existence of hermiticity preserving generator). A non-singular
matrix with real entries Ê has a real generator (i.e. a log Ê with real entries)
if and only if the spectrum fulfills the following conditions: i) negative eigenval-
ues are even-fold degenerate; ii) complex eigenvalues come in complex conjugate
pairs.
We now discuss the multiplicity of the solutions of log Ê and its parametriza-
tion, as finding an appropriate one is essential to test for the ccp condition. If Ê
has positive degenerate, negative, or complex eigenvalues, its real logarithms are
not unique, and are spanned by real logarithm branches [Cul66]. The latter are
defined using the real quaternion, which coincides with iσy, using the fact that
✶ = exp(iσy2πk), with k ∈ Z. In case of having negative eigenvalues, it turns
out that real logarithms always have a continuous parametrization, in addition to
real branches due to the freedom of the Jordan normal form transformation matri-
ces [Cul66].
To compute the logarithm given a real representation of E , i.e. Ê , we calculate
its Jordan normal form, J, such that Ê = wJw−1 = w̃Jw̃−1, where w = w̃K and
K belongs to a continuum of matrices that commute with J [Cul66]. In the case
of diagonalizable matrices, if there are no degeneracies, K commutes with log(J).
In the case of having degeneracies, matrix K is responsible of the continuous
parametrization of the logarithm. We compute explicitly the logarithms for the
case of Pauli channels in section 4.3.4.
4.3. Divisibility of unital qubit channels 57
4.3 Divisibility of unital qubit channels
We will apply various of the results from the literature [WECC08] to decide if a
given unital qubit channel belongs to CL, CCP and/or CP. The non-unital case will
be discussed later.
Before starting with the characterization let us point out the following. From
the definition of divisibility, the concatenation of a given channel with unitary
conjugations (which are infinitesimal divisible) do not change its divisibility char-
acter, except for L-divisibility. In addition to this, since unitary conjugations are
infinitesimal divisible, they do not change the infinitesimal divisible character ei-
ther. We can summarize this in the following,
Proposition 4 (Divisibility of special orthogonal normal forms). Let E a qubit
quantum channel and D its special orthogonal normal form, E belongs to CX if
and only if D does, where X = {“Div”, “P”, “CP”}.
This proposition is in fact a consequence of theorem 17 of Ref. [WECC08]. No-
tice that this result does not apply for CL since conjugating with unitaries breaks
the implementability by means of time-independent Lindblad master equations.
Thus, if a channel belongs to CL, unitary conjugations can bring it to CInf \CL and
vice versa.
Therefore, by proposition 4 and the theorem 9, to study CP and CCP of unital qubit
channels, it is enough to study Pauli channels.
4.3.1 Channels belonging to C
div
Divisibility in CPTP of unital qubit channels is completely characterized by means
of theorem 13. Therefore the only indivisible channels lie in the faces of the
tetrahedron (without the edges), see fig. 4.5.
4.3.2 Channels belonging to C
P
Recalling that all unital qubit channels belonging to CP have non-negative deter-
minant [WC08], and using special orthogonal normal forms, see theorem 9, the
condition in terms of its parameters is given by
λ1λ2λ3 ≥ 0. (4.7)
58 Chapter 4. Divisibility of quantum channels and dynamical maps
This set is the intersection of the tetrahedron with the octants where the product
of all λ s is positive. In fact, it consists of four triangular bipyramids starting
in each vertex of the tetrahedron and meeting in its center, see fig. 4.5. Let us
study the intersection of this set with the set of unital entanglement-breaking (EB)
channels [ZB05], see definition 5. In the case of unital qubit channels, the set
entanglement-breaking channels is an octahedron that lie inside the tetrahedron
of unital qubit channels, see fig. 4.6. The inequalities that define such octahedron
are the following,
λi ± (λ j +λk)≤ 1, (4.8)
with i, j and k all different [ZB05], together eq. (3.11). It follows that unital
qubit channels that are not achieved by P-divisible dynamical maps are necessarily
entanglement-breaking (see fig. 4.6 and fig. 4.9). In fact this holds for general
qubit channels, see section 4.4.
4.3.3 Channels belonging to C
CP
To characterize CP-divisible channels it is useful to consider the Lorentz normal
form for channels, see theorem 10. A remarkable property of the Lorentz normal
decomposition is that it preserves the infinitesimal divisible character of E , see
Corollary 13 of [WC08]. To use it, we resort to theorem 24 of Ref. [WC08]. Due
to the mentioned drawback of Lorentz normal forms, see appendix B, we must
modify such to theorem to a restricted class of channels.
Theorem 16 (Restricted characterization of channels belonging to CCP). A qubit
channel E with diagonal Lorentz normal form belongs to CCP if and only if
i) the rank of the form is smaller than three or
ii) s2
min ≥ s1s2s3 > 0, where smin is the smallest of s1, s2 and s3, see theorem 10.
For non-unital Kraus deficient channels, the pertinent theorems are based on non-
diagonal Lorentz normal forms [VV02, WC08]. According to our appendix B
such results should be reviewed and are out of the scope of this work.
Notice that the Lorentz normal form coincides with the special orthogonal nor-
mal form for unital qubit channels. Therefore, by theorem 16, unital channels
belonging to CCP are non-singular with
0 < λ1λ2λ3 ≤ λ 2
min , (4.9)
or singular with a matrix rank less than three. They determine a body that is
symmetric with respect to permutation of Pauli unitary channels (i.e. in λ j), hence,
4.3. Divisibility of unital qubit channels 59
λ1
λ2
λ3
Figure 4.5: Tetrahedron of Pauli channels, see fig. 3.1. The bipyramid in blue
correspond to channels with λi > 0 ∀i, i.e. channels of the positive octant
belonging to CP. The whole set CP includes other three bipyramids
corresponding to the other vertexes of tetrahedron. This implies that CP
enjoys the symmetries of the tetrahedron, see eq. (4.7). The faces of the
bipyramids matching the corners of the tetrahedron are subsets of the
faces of the tetrahedron, i.e. contain Kraus rank three channels. Such
channels are both CP and Cdiv, showing that the intersection shown in
fig. 4.4 is not empty.
60 Chapter 4. Divisibility of quantum channels and dynamical maps
λ1
λ2
λ3
Figure 4.6: Tetrahedron of Pauli channels with the octahedron of entangle-
ment breaking channels shown in red, see eq. (4.8). The blue pyramid
inside the octahedron is the intersection of the bipyramid shown in 4.5,
with the octahedron. The complement of the intersections of the four
bipyramids forms the set of divisible but not infinitesimal divisible chan-
nels in PTP. Thus, a central feature of the figure is that the set Cdiv\CP
is always entanglement-breaking, but the converse is not true.
4.3. Divisibility of unital qubit channels 61
λ1
λ2
λ3
Figure 4.7: Tetrahedron of Pauli channels with part of the set of CP-divisible,
see eq. (4.9), but not L-divisible channels (CCP\CL) shown in purple.
The whole set CCP is obtained applying the symmetry transformations
of the tetrahedron to the purple volume.
the set of CCP of Pauli channels possesses the symmetries of the tetrahedron. The
set CCP\CL is plotted in fig. 4.7.
4.3.4 L-divisible unital qubit channels
We restrict our analysis of L-divisibility for two particular sets of unital channels,
Pauli channels and a family with complex eigenvalues that will be introduced later.
Pauli channels with non-degenerate positive eigenvalues
First let us now derive the conditions for L-divisibility of Pauli channels with pos-
itive eigenvalues λ1,λ2,λ3 (λ0 = 1). The logarithm of D , induced by the principal
logarithm of its eigenvalues is
L = Kdiag(0, logλ1, logλ2, logλ3)K
−1 , (4.10)
62 Chapter 4. Divisibility of quantum channels and dynamical maps
which is real (hermiticity preserving). In case of no-degeneration the dependency
on K vanishes and L is unique. In such case the ccp conditions, see theorem 2,
read
logλi − logλ j − logλk ≥ 0 ⇒ λi
λ jλk
≥ 1 (4.11)
for all combinations of mutually different i, j,k. This set (channels belonging to
CL with positive eigenvalues) forms a three dimensional manifold, see fig. 4.8.
Pauli channels with degenerate positive eigenvalues
In case of degeneration, let us label the eigenvalues η , λ and λ . In this case,
the real solution for L is not unique and is parametrized by real branches in the
degenerate subspace and by the continuous parameters of K [Cul66]. Let us study
the principal branch with K = ✶. Eq. (4.11) is then reduced to
λ 2 ≤ η ≤ 1 . (4.12)
Therefore, if this inequalities are fulfilled, the generator has Lindblad form. If
not, then a priori other branches can fulfill ccp condition and consequently have
a Lindblad form. Thus, Eq. (4.12) provides a sufficient condition for the channel
to be in CL. We will prove it is also necessary.
The complete positivity condition requires η ,λ ≤ 1, thus, it remains to verify only
the condition λ 2 ≤ η . It holds trivially for the case λ ≤ η . If η ≤ λ , then this
condition coincides with the CP-divisibility condition from eq. (4.9). Since CL
implies CCP the proof is completed. In conclusion, the condition in eq. (4.11) is
a necessary and sufficient for a given Pauli channel with positive eigenvalues to
belong to CL.
Let us stress that the obtained subset of L-divisible channels does not possess the
tetrahedron symmetries. In fact, composing D with a σz rotation
Uz = diag(1,−1,−1,1)
results in the Pauli channel D ′ = diag(1,−λ1,−λ2,λ3). Clearly, if λ j are pos-
itive (D is L-divisible), then D ′ has non-positive eigenvalues. Moreover, if all
λ j are different, then D ′ does not have any real logarithm, therefore, it cannot
be L-divisible. In conclusion, the set of L-divisible unital qubit channel is not
symmetric with respect to tetrahedron symmetries.
4.3. Divisibility of unital qubit channels 63
Pauli channels with negative eigenvalues
In what follows we will investigate the case of negative eigenvalues. Theorem 15
implies that that eigenvalues have the form (modulo permutations) η ,−λ ,−λ ,
where η ,λ > 0. The corresponding Pauli channels are
Dx = diag(1,η ,−λ ,−λ ), Dy = diag(1,−λ ,η ,−λ ), Dz = diag(1,−λ ,−λ ,η),
thus forming three two-dimensional regions inside the tetrahedron. Take, for in-
stance, Dz that specifies a plane (inside the tetrahedron) containing I, σz and com-
pletely depolarizing channel N = diag(1,0,0,0). The real logarithms for this
case are given by
L = K




0 0 0 0
0 log(λ ) (2k+1)π 0
0 −(2k+1)π log(λ ) 0
0 0 0 log(η)




K−1, (4.13)
where k ∈ Z and K, as mentioned above, belongs to a continuum of matrices that
commute with Dz. Note that L is always non-diagonal. For this case (similarly for
Dx and Dy) the ccp condition reduces again to conditions specified in Eq. (4.12).
Using the same arguments one arrives to more general conclusion:
Theorem 17 (L-divisibility of Pauli channels). Let E be a non-singular Pauli
channel. It belongs to CL if and only if its non-trivial eigenvalues fulfill
λi
λ jλk
≥ 0 (4.14)
for i, j and k mutually different.
This is one of the central results of this work, and it implies that for testing L-
divisibility of Pauli channels, it is enough to consider the principal real logarithm
branch and K = ✶. The singular cases are included in the closure of channels
fulfilling eq. (4.14). The set of L-divisible Pauli channels is illustrated in fig. 4.8.
To get a detailed picture of the position and inclusions of the divisibility sets, we
illustrate in fig. 4.9 two slices of the tetrahedron where different types of divisi-
bility are visualized. Notice the non-convexity of the considered divisibility sets.
64 Chapter 4. Divisibility of quantum channels and dynamical maps
λ1
λ2
λ3
Figure 4.8: Tetrahedron of Pauli channels with the set of L-divisible channels
(or equivalently infinitely divisible, see Theorem 19) shown in green,
see equations (4.11) and (4.12). The solid set corresponds to channels
with positive eigenvalues, and the 2D sets correspond to the negative
eigenvalue case. The point where the four sets meet correspond to the
total depolarizing channel. Notice that this set does not have the sym-
metries of the tetrahedron.
4.3. Divisibility of unital qubit channels 65
λ1
λ2λ3
 CL
 CCP\CL
 CP\CCP
 Cdiv\CP
 Cdiv
 EB boundary
Figure 4.9: We show two slices of the unitary tetrahedron (figure in the
left) determined by ∑i λi = 0.4 (shown in the center) and ∑i λi = −0.4
(shown in the right). The non-convexity of the divisibility sets can be
seen, including the set of indivisible channels. The convexity of sets C
and entanglement breaking channels can also be noticed in the slices. A
central feature is that the set Cdiv \CP is always inside the octahedron of
entanglement breaking channels.
66 Chapter 4. Divisibility of quantum channels and dynamical maps
Family of unital channels with complex eigenvalues
To give an insight to the case of complex eigenvalues, consider the following
family of channels with real logarithm, written in the Pauli basis,
Ecomplex =




1 0 0 0
0 c 0 0
0 0 a −b
0 0 b a




. (4.15)
The latter has complex eigenvalues a± ib and a real one c > 0, together with the
trivial eigenvalue equal to 1. Its real logarithm is given by,
L = K log
(
Ecomplex
)
k
K−1
= K




0 0 0 0
0 log(c) 0 0
0 0 log(|z|) arg(z)+2πk
0 0 −arg(z)−2πk log(|z|)




K−1
with z = a+ ib. The non-diagonal block of the logarithm has the same structure of
Ecomplex, so K also commutes with log(Ecomplex)k, leading to a countable paramet-
ric space of hermitian preserving generators. The ccp condition, see proposition 2,
is reduced to
a2 +b2 ≤ c ≤ 1. (4.16)
Note that it does not depend on k and the second inequality is always fulfilled for
CPTP channels. The set containing them is shown in fig. 4.10.
4.3.5 Relation of L-divisibility with other divisibility classes
Consider a Pauli channel with 0 < λmin = λ1 ≤ λ2 ≤ λ3 < 1, thus the condition
λ1λ2 ≤ λ3 trivially holds. Since λ1λ2 ≤ λ1λ3 ≤ λ2λ3 ≤ λ2, it follows that λ1λ3 ≤
λ2, thus, two (out of three) L-divisibility conditions hold always for Pauli chan-
nels with positive eigenvalues. Moreover, one may observe that CP-divisibility
condition eq. (4.9) reduces to one of L-divisibility conditions λ2λ3 ≤ λ1. In con-
clusion, the conditions of CP-divisibility and L-divisibility for Pauli channels with
positive eigenvalues coincide, thus, in this case CCP implies CL.
Concatenating (positive-eigenvalues) L-divisible Pauli channels with Dx,y,z, one
can generate the whole set of CCP Pauli channels. In other words, Dx,y,z brings
the body (with vertex in id) shown in fig. 4.8 to the bodies shown in fig. 4.7 (with
vertexes x,y,z). Therefore we can formulate the following theorem:
4.3. Divisibility of unital qubit channels 67
λ1
λ2
λ3
Figure 4.10: Tetrahedron of Pauli channels, with qubit unital L-divisible
channels of the form Êcomplex (see main text). Note that the set does
not have the symmetries of the tetrahedron.
Theorem 18 (Infinitesimal divisible unital channels). Let E CP
unital be an arbitrary
infinitesimal divisible unital qubit channel. There exists at least one L-divisible
Pauli channel Ẽ , and two unitary conjugations U1 and U2, such that
E
CP
unital = U1Ẽ U2 .
Notice that if E CP
unital is invertible, Ẽ = eL.
Let us continue with another equivalence relation valid for Pauli channels. In
general, CL ⊂ C∞; however, for Pauli channels these two subsets coincide.
Theorem 19 (Infinitely divisible Pauli channels). The set of L-divisible Pauli
channels is equivalent to the set of infinitely divisible Pauli channels.
Proof. A channel is infinitely divisible if and only if it can be written as E0eL,
where E0 is an idempotent channel satisfying E0LE0 = E0L and L has Lindblad
form, see definition 15. The only idempotent qubit channels are contractions of
the Bloch sphere into single points, diagonalization channels Ediag transforming
Bloch sphere into a line connecting a pair of basis states, and the identity channel.
Among the single-point contractions, the only one that is a Pauli channel is the
68 Chapter 4. Divisibility of quantum channels and dynamical maps
contraction of the Bloch sphere into the complete mixture; let us call it N . Notice
that E = N eL = N for all L. The channel N belongs to the closure of CL,
because a sequence of channels eLn with L̂n = diag(0,−n,−n,−n) converges to
ˆN in the limit n → ∞. For the case of E0 being the identity channel we have
E = eL, thus, trivially such infinitely divisible channel E is in CL too. It remains
to analyze the case of diagonalization channels. First, let us note that the matrix of
eL̂ is necessarily of full rank, since detÊ 6= 0. It follows that the matrix Ê = ÊdiageL̂
has rank two as Êdiag is a rank two matrix, thus, it takes one of the following forms
Ê λ
x = diag(1,λ ,0,0), Ê λ
y = diag(1,0,λ ,0), Ê λ
z = diag(1,0,0,λ ). The infinitely
divisibility implies λ > 0 in order to keep the roots of λ real. In what follows we
will show that Êz belongs to (the closure of) CL. Let us define the channels Ê
λ ,ε
z =
diag(1,ε,ε,λ ) with ε > 0. The complete positivity and ccp conditions translate
into the inequalities ε ≤ 1+λ
2
and ε2 ≤ λ , respectively; therefore one can always
find an ε > 0 such that Ê
λ ,ε
z is a L-divisible channel. If we choose ε =
√
λ/n
with n ∈ Z
+, the channels Êz,n = diag
(
1,
√
λ/n,
√
λ/n,λ
)
form a sequence of
L-divisible channels converging to Ê λ
z when n → ∞. The analogous reasoning
implies that Ê λ
x , Ê λ
y ∈ CL too. Let us note that one parameter family Ez are convex
combinations of the complete diagonalization channel Ê 1
z = diag(1,0,0,1) and
the complete mixture contraction ˆN . This completes the proof.
Finally, let us remark that using the theorem 13 we conclude that the intersection
CP ∩Cdiv depicted in fig. 4.4 is not empty. To show this, notice that there are
channels with positive determinant inside the faces (i.e. CP but not Cdiv), for
example diag
(
1, 4
5
, 4
5
, 3
5
)
. Therefore we conclude that up to unitaries, CP ∩Cdiv
corresponds to the union of the four faces faces of the tetrahedron minus the faces
of the octahedron that intersect with the faces of the tetrahedron, see fig. 4.6. We
have to remove such intersection since it corresponds to channels with negative
determinant, and thus not in CP.
4.4 Non-unital qubit channels
Similar to unital channels, using theorem 4 we are able to characterize Cdiv, CP
and CCP by studying special orthogonal normal forms. Such channels are char-
acterized by ~λ and ~τ , see eq. (3.9). Thus, we can study if a channel is Cdiv by
computing the rank of its Choi matrix, see theorem 11. For this case algebraic
equations are in general fourth order polynomials. In fact, in Ref. [RPZ18] a con-
dition in terms of the eigenvalues and ~τ is given. For special cases, however, we
4.4. Non-unital qubit channels 69
can obtain compact expressions, see fig. 4.11. The characterization of CP is given
by again by the condition λ1λ2λ3 ≥ 0. CCP is tested, for full Kraus rank non-unital
channels, using theorem 16, the calculation of si’s is done using the algorithm pre-
sented in Ref. [VDD01]. For the characterization of CL we use theorem 15 and
evaluate numerically the cpp condition.
We can plot illustrative pictures even though the whole space of qubit channels has
12 parameters. This can be done using special orthogonal normal forms and fixing
~τ , exactly in the same way as the unital case. Recall that unitaries only modify
CL, leaving the shape of other sets unchanged. CPTP channels are represented
as a volume inside the tetrahedron presented in fig. 4.5, see fig. 4.11. In the
later figure we show a slice corresponding to ~τ = (1/2,0,0)T
. Indeed, it has
the same structure of the slices for the unital case, but deformed, see fig. 4.9. A
difference with respect to the unital case is that L-divisible channels with negative
eigenvalues (up to unitaries) are not completely inside CP-divisible channels. A
part of them are inside the CP channels.
A central feature of Figs. 4.9 and 4.11 is that the set Cdiv \CP is inside the convex
slice of the set of entanglement breaking channels (deformed octahedron). Indeed,
we can proof the following theorem.
Theorem 20 (Entanglement-breaking channels and divisibility). Consider a qubit
channel E . If det Ê < 0, then E is entanglement-breaking, i.e. all qubit channels
outside CP are entanglement breaking.
Before introducing the proof, let us first show that the proper orthochronous
Lorentz transformations present in the Lorentz normal decomposition for chan-
nels, see sec. 3.4.2, correspond to 1wSLOCC at the level of their Choi-Jamiołkowski
state. Consider a channel E and its Lorentz normal form Ẽ given by
Ẽ = αF2E F1, (4.17)
where
Fi : ρ 7→ XiρX
†
i , with Xi ∈ SL(2,C), i = 1,2,
and α is a constant that must be included for Ẽ to be trace preserving, We showed
already that SL(2,C) is a double cover of SO+(3,1), i.e. Fi’s correspond to the
proper orthochronous Lorentz transformations of the decomposition.
Now let us compute the Choi-Jamiołkowski state of Ẽ , τ̃ , using the Kraus decom-
70 Chapter 4. Divisibility of quantum channels and dynamical maps
λ1
λ2
λ3
Figure 4.11: (left) Set of non-unital unital channels up to unitaries, defined
by ~τ = (1/2,0,0), see eq. (3.9). This set lies inside the tetrahedron.
For this particular case the CP conditions reduce to the two inequali-
ties 2± 2λ1 ≥
√
1+4(λ2 ±λ3)2. A cut corresponding to ∑i λi = 0.3
is presented inside and in the right, see fig. 4.9 for the color coding.
The structure of divisibility sets presented here has basically the same
structure as for the unital case except for CL. A part of the channels
with negative eigenvalues belonging to CL lies outside CCP \CL, see
green lines. As for the unital case a central feature is that the chan-
nels in Cdiv \CP are entanglement breaking channels. Channels in the
boundary are not characterized due to the restricted character of Theo-
rem 10.
4.4. Non-unital qubit channels 71
position of E [Wol11],
τ̃ = α (id2 ⊗F2E F1) [ω]
= α ∑
i
(✶⊗X2)(✶⊗Ki)(✶⊗X1) |Ω〉〈Ω|
(
✶⊗X
†
1
)(
✶⊗K
†
i
)(
✶⊗X
†
2
)
= α ∑
i
(
XT
1 ⊗X2
)
(✶⊗Ki) |Ω〉〈Ω|)
(
✶⊗K
†
i
)(
X1 ⊗X
†
2
)
= α(XT
1 ⊗X2)τ(X
T
1 ⊗X2)
†, (4.18)
where {Ki}i are a choice of Kraus operators of E and
τ = ∑
i
(✶⊗Ki) |Ω〉〈Ω|
(
✶⊗K
†
i
)
its Choi-Jamiołkowski matrix. Here, |Ω〉 is the Bell state between two copies of
the system, in this case a qubit, for which the identity A⊗ ✶|Ω〉 = ✶⊗ AT|Ω〉
holds. It can be observed that eq. (4.18) has exactly the form of the normalized
1wSLOCC scheme, where α turns to be the normalization constant, see eq. (2.27),
i.e. tr τ̃ = 1. That’s why we have introduced it at the first place. Now let us proceed
with the proof of theorem 20,
Proof. Let E be a qubit channel with negative determinant and Ê its matrix rep-
resentation using the Pauli basis, see eq. (3.7). Recall that the matrix R defining
the Choi-Jamiołkowski state of E ,
τE =
1
4
3
∑
jk
R jkσ j ⊗σk,
and Ê are related by
R = Ê ΦT,
where ΦT = diag(1,1,−1,1). It follows immediately that R has positive determi-
nant,
detR =−det Ê > 0,
since detΦT =−1. Using the aforementioned Lorentz normal decomposition for
matrix R, we have
R = LT
1 R̃L2
where detL1,2 > 0 and det R̃ > 0. Stressing that transformations L1,2 correspond to
1wSLOCC (see eq. (4.18)), then R̃ parametrizes an unnormalized two-qubit state.
72 Chapter 4. Divisibility of quantum channels and dynamical maps
Let us first discuss the case when R̃ is diagonal. The channel corresponding to R̃
(in the Pauli basis) is
Ĝ = R̃ΦT/R̃00,
where R00 = tr R̃ = trτG . Since R̃ is diagonal, then G is a Pauli channel with
det Ĝ < 0. A Pauli channel has a negative determinant, if either all λ j are negative,
or exactly one of them is negative. In Ref. [ZB05] it has been shown that the set
of channels with λ j < 0 ∀ j are entanglement breaking channels. Now, using
the symmetries of the tetrahedron, one can generate all channels with negative
determinant by concatenating this set with the Pauli rotations. Therefore every
Pauli channel with negative determinant is entanglement breaking, thus, τG is
separable. Given that LOCC operations can not create entanglement [HHHH09],
we have that τE is separable, therefore E is entanglement breaking.
The case when R̃ is non-diagonal corresponds to Kraus deficient channels (the
matrix rank of 3.17 is at most 3). This case can be analyzed as follows. Since
the neighborhood of any Kraus deficient channel with negative determinant con-
tains full Kraus rank channels, by continuity of the determinant such channels
have negative determinant too. The last ones are entanglement breaking since full
Kraus rank channels have diagonal Lorentz normal form. Therefore, by continu-
ity of the concurrence [ZB05], Kraus deficient channel with negative determinant
are entanglement breaking.
4.5 Divisibility transitions and examples with dynamical
processes
The aim of this section is to use illustrative examples of quantum dynamical pro-
cesses to show transitions between divisibility types of the instantaneous channels.
From the slices shown above (see figures 4.9 and 4.11) it can be noticed that every
transition between the studied divisibility types is permitted. This is due to the
existence of common borders between all combinations of divisibility sets; we
can think of any continuous line inside the tetrahedron [FPMZ17] as describing
some quantum dynamical map.
We analyze two examples. The first is an implementation of the approximate NOT
gate, ANOT throughout a specific collision model [RFZB12]. The second is the
well known setting of a two-level atom interacting with a quantized mode of an
optical cavity [HR06]. We define a simple function that assigns a particular value
4.5. Divisibility transitions and examples with dynamical processes 73
to a channel Et according to divisibility hierarchy, i.e.
δ [E ] =







1 if E ∈ CL ,
2/3 if E ∈ CCP \CL ,
1/3 if E ∈ CP \CCP ,
0 if E ∈ C\CP .
(4.19)
A similar function can be defined to study the transition to/from the set of entanglement-
breaking channels, i.e.
χ[E ] =
{
1 if E is entanglement breaking ,
0 if E if not.
(4.20)
The quantum NOT gate is defined as NOT : ρ 7→ ✶−ρ , i.e. it maps pure qubit
states to its orthogonal state. Although this map transforms the Bloch sphere into
itself it is not a CPTP map, and the closest CPTP map is ANOT : ρ 7→ (✶−ρ)/3.
This is a rank-three qubit unital channel, thus, it is indivisible [WC08]. Moreover,
detANOT = −1/27 implies that this channel is not achievable by a P-divisible
dynamical map. It is worth noting that ANOT belongs to Cdiv.
A specific collision model was designed in Ref. [RFZB12] simulating stroboscop-
ically a quantum dynamical map that implements the quantum NOT gate ANOT in
finite time. The dynamical map is given by
Et(ρ) = cos2(t)ρ + sin2(t)ANOT(ρ)+
1
2
sin(2t)F (ρ) , (4.21)
where F (ρ) = i 1
3 ∑ j[σ j,ρ]. It achieves the desired gate ANOT at t = π/2.
Let us stress that this dynamical map is unital, i.e. Et(✶) = ✶ for all t, thus, its
special orthogonal normal form can be illustrated inside the tetrahedron of Pauli
channels, see fig. 4.12. In fig. 4.13 we plot δ [Et ], χ[Et ] and the value of the detEt .
We see the transitions CL → CP \CCP → Cdiv \CP → Cdiv and back. Notice that in
both plots the trajectory never goes through the CCP \CL region. This means that
when the parametrized channels, up to rotations, belong to CL, so do the original
ones. The transition between P-divisible and divisible channels, i.e. CP\CCP and
Cdiv\CP, occurs at the discontinuity in the yellow curve in fig. 4.12. Let us note
that this discontinuity only occurs in the space of ~λ ; it is a consequence of the
special orthogonal normal decomposition, see eq. (3.9). The complete channel is
continuous in the full convex space of qubit CPTP maps. The transition from CP \
Cdiv and back occurs at times π/3 and 2π/3. It can also be noted that the transition
to entanglement breaking channels occurs shortly before the channel enters in the
74 Chapter 4. Divisibility of quantum channels and dynamical maps
λ1
λ2 λ3
Figure 4.12: (top left) Tetrahedron of Pauli channels with the trajectory, up
to rotations, of the quantum dynamical map eq. (4.21) leading to the
ANOT gate, as a yellow curve. (right) Cut along the plane that con-
tains the trajectory; there one can see the different regions where the
channel passes. For this case, the characterization of the CL of the
channels induced gives the same conclusions as for the corresponding
Pauli channel, see eq. (3.9). The discontinuity in the trajectory is due
to the reduced representation of the dynamical map, see eq. (4.21); the
trajectory is continuous in the space of channels. See fig. 4.9 for the
color coding.
4.5. Divisibility transitions and examples with dynamical processes 75
0
π
3
π
2
2 π
3
π
0
1
3
2
3
1
δ
[E
t
],
d
et
E
t
,
χ
[E
t
]
t
C
P\CCP
C
CP\CL
C
L
C
div\CP
Figure 4.13: Evolution of divisibility, determinant, and entanglement break-
ing properties of the map induced by eq. (4.21), see eq. (4.19) and
eq. (4.20). Notice that the channel ANOT, implemented at t = π/2,
has minimum determinant. The horizontal gray dashed lines show the
image of the function δ , with the divisibility types in the right side.
It can be seen that the dynamical map explores the divisibility sets as
CL → CP \CCP → Cdiv \CP → Cdiv and back. The channels are entan-
glement breaking in the expected region.
76 Chapter 4. Divisibility of quantum channels and dynamical maps
Cdiv \CP region; likewise, the channel stops being entanglement breaking shortly
after it leaves the Cdiv \CP region, see theorem 20.
Consider now the dynamical map induced by a two-level atom interacting with
a mode of a boson field. This model serves as a workhorse to explore a great
variety of phenomena in quantum optics [GKL13]. Using the well known rotating
wave approximation one arrives to the Jaynes-Cummings model [JC63], whose
Hamiltonian is
H =
ωa
2
σz +ω f
(
a†a+
1
2
)
+g
(
σ−a† +σ+a
)
. (4.22)
By initializing the environment in a coherent state |α〉, one gets the familiar
collapse and revival setting. Considering a particular set of parameters shown
in fig. 4.14, we constructed the channels parametrized by time numerically, and
studied their divisibility and entanglement-breaking properties. In the same figure
we plot functions δ [Et ] and χ[Et ], together with the probability of finding the atom
in its excited state pe(t), to study and compare the divisibility properties with the
features of the collapses and revivals. The probability pe(t) is calculated choos-
ing the ground state of the free Hamiltonian ωa/2σz of the qubit, and it is given
by [KC09]:
pe(t) =
〈σz(t)〉+1
2
, (4.23)
where
〈σz(t)〉=−
∞
∑
n=0
Pn
(
∆2
4Ω2
n
+
(
1− ∆2
4Ω2
n
)
cos(2Ωnt)
)
,
with Pn = e−|α|2 |α|2n/n!, Ωn =
√
∆2/4+g2n and ∆ = ω f −ωa the detuning.
The divisibility indicator function δ exhibits an oscillating behavior, roughly at the
same frequency of pe(t), see inset in fig. 4.14. The figure shows fast periodic tran-
sitions between CP \CCP and CCP \CL occurring in the region of revivals. There
are also few transitions among CCP \CP and CL in the second revival. Respect
to the entanglement breaking and the function χ , there are no fast transitions in
the former, and during revivals, channels are not entanglement breaking. We also
observe that channels belonging to Cdiv \CP are entanglement breaking, which
agrees with theorem 20 for the non-unital case.
4.5. Divisibility transitions and examples with dynamical processes 77
3.85 3.9 3.95
0 2 4 6 8 10
0
1
3
2
3
1
δ
[E
t
],
p
e
(t
),
χ
[E
t
]
t
C
P\CCP
C
CP\CL
C
L
C
div\CP
Figure 4.14: Black and red curves show functions δ and χ of the chan-
nels induced by the Jaynes-Cummings model over a two-level system,
see eq. (4.22) with the environment initialized in a coherent state |α〉.
The blue curve shows the probability of finding the two-level atom in
its excited state, pe(t). The figure shows that the fast oscillations in δ
occur roughly at the same frequency as the ones of pe(t), see the inset.
Notice that there are fast transitions between CP \CCP and CCP \CL
occurring in the region of revivals, with a few transitions between
CCP \CP and CL in the second revival. The function χ shows that dur-
ing revivals channels are not entanglement breaking, but we find that
channels belonging to Cdiv \CP are always entanglement breaking, in
agreement with theorem 20. The particular chosen set of parameters
are α = 6, g = 10, ωa = 5, and ω f = 20.

Chapter 5
Singular Gaussian quantum
channels
Self-education is, I firmly believe, the only kind of education there is.
Isaac Asimov
In this chapter we derive the conditions for δGQC to be singular, see sec. 3.5.1.
In particular we will show that only the functional form involving one Dirac delta
can be singular, together with the Gaussian form. Additionally we derive, for
the non-singular cases, the conditions for the existence of master equations that
parametrize channels that have always the same functional form. We do this by
letting the channels parameters to depend on time.
5.1 Allowed singular forms
There are two classes of Gaussian singular channels. Since the inverse of a Gaus-
sian channel A (T,N,~τ) is A
(
T−1,−T−1NT−T ,−T−1~τ
)
, its existence rests on
the invertibility of T. Therefore, studying the rank of the latter we are able to ex-
plore singular forms. We are going to use the classification of one-mode channels
developed by Holevo [Hol07]. For singular channels there are two classes charac-
terized by its canonical form [Hol08], i.e. any channel can be obtained by apply-
ing Gaussian unitaries before and after the canonical form. The class called “A1”
corresponds to singular channels with Rank(T) = 0 and coincide with the family
of total depolarizing channels. The class “A2” is characterized by Rank(T) = 1.
79
80 Chapter 5. Singular Gaussian quantum channels
Both channels are entanglement-breaking [Hol08].
Before analyzing the functional forms constructed in this work, let us study chan-
nels with GF. The tuple of the affine transformation, corresponding to the prop-
agator JG, eq. (2.31), were introduced in Ref. [MP12] up to some typos. Our
calculation for this tuple, following eq. (3.35), is:
TG =
(
−b4
b3
1
b3
b1b4
b3
−b2 −b1
b3
)
,
NG =


2a3
b2
3
a2
b3
− 2a3b1
b2
3
a2
b3
− 2a3b1
b2
3
−2
(
−a3b2
1
b2
3
+ a2b1
b3
−a1
)

 ,
~τG =
(
− c2
b3
,
b1c2
b3
− c1
)T
. (5.1)
It is straightforward to check that for b2 = 0, TG is singular with Rank(TG) = 1,
i.e. it belongs to class A2. Due to the full support of Gaussian functions, it was
surprising that Gaussian channels with GF have singular limit. In this case the
singular behavior arises from the lack of a Fourier factor for x f ri, see eq. (2.31).
This is the only singular case for GF.
Now we analyze functional forms derived in sec. 3.5.1. The complete positivity
conditions of the form J̃III, presented in eq. (3.45), have no solution for α → 0
and/or γ → 0, thus, this form cannot lead to singular channels. This is not the case
for J̃I, eq. (3.36), which leads to singular operations belonging to class A2 for
αe2 = 0, (5.2)
and to class A1 for
e2 = α = b2 = 0. (5.3)
For the latter, the complete positivity conditions, see eq. (3.40), read:
e1 ≤ a1. (5.4)
By using an initial state characterized by σi and ~di we can compute the explicit
dependence of the final states on the initial parameters. The final states for chan-
nels of class A2 with the functional form involving one delta, see eq. (3.25), and
5.1. Allowed singular forms 81
with e2 = 0, are
(σ f )11
=
1
2e1
,
(σ f )22
=
(
α
β
)2(
b2
3
2e1
+2a3
)
+
α
β
(
2a2 +
b1b3
e1
)
+2a1 +
b2
1
2e1
+ s1,
(σ f )12
=−α
β
b3
2e1
− b1
2e1
,
~d f (s3) =
(
0,−α
β
c2 − c1 + s2
)T
, (5.5)
where
s1 =
(
b2
2 +2
α
β
b2b4 +
(
α
β
)2
b2
4
)
(σi)11 −2
(
α
β
b2 +
(
α
β
)2
b4
)
(σi)12
+
(
α
β
)2
(σi)22 ,
s2 =
(
α
β
b4 +b2
)
(di)1 −
α
β
(di)2. (5.6)
For the same functional form but now with α = 0, the final states are
(σ f )11
=
e2
2
4e2
1
(σi)11 +
1
2e1
,
(σ f )12
=
(
b2e2
2e1
− b1e2
2
4e2
1
)
(σi)11 −
b1
2e1
,
(σ f )22
= 2a1 +
(
b2 −
b1e2
2e1
)2
(σi)11 +
b2
1
2e1
, (5.7)
and
~d f =
(
e2
2e1
(
~di
)
1
,
(
b2 −
b1e2
2e1
)
(
~di
)
1
− c1
)T
. (5.8)
The explicit formulas of the final states for channels of class A2 with Gaussian
82 Chapter 5. Singular Gaussian quantum channels
form are
(σ f )11
(s1) =
2a3
b2
3
+ s1,
(σ f )12
(s1) =
a2
b3
− 2a3b1
b2
3
−b1s1,
(σ f )22
(s1) =
b1 (b3 (b1b3s1 −2a2)+2a3b1)
b2
3
+2a1,
~d f (s2) =
(
s2 −
c2
b3
,b1
(
c2
b3
− s2
)
− c1
)T
, (5.9)
where
s1 =
b2
4
b2
3
(σi)11 −
2b4
b2
3
(σi)12 +
1
b2
3
(σi)22 ,
s2 =
1
b3
(di)2 −
b4
b3
(di)1 . (5.10)
See fig. 5.1 for an schematic description of the final states. From such combina-
tions it is obvious that we cannot solve for the initial state parameters given a final
state as expected; this is because the parametric space dimension is reduced from
5 to at most 3. The channel belonging to A1 [see eq. (3.42) with e2 = α = b2 = 0
and eq. (5.4)] maps every initial state to a single one characterized by σ f = N and
~d f = (0,−c1)
T
, see fig. 5.2 for a schematic description.
According to our ansätze [see equations (3.24) and (3.25)], we conclude that one-
mode SGQC can only have the functional forms given in eq. (2.31) and eq. (3.24).
This is the central result of this chapter and can be stated as:
Theorem 21 (One-mode singular Gaussian channels). A one-mode Gaussian quan-
tum channel is singular if and only if it has one of the following functional forms
in the position space representation:
1. b3
2π exp
[
ı
(
b1x f r f +b3xir f +b4xiri + c1x f + c2xi
)
−a1x2
f −a2x f xi −a3x2
i
]
,
2. |β |
√
e1/πδ (αx f −βxi)exp
[
−a2x f xi −a1x2
f −a3x2
i
+ı
(
b2x f ri +b3r f xi +b1r f x f +b4rixi +c1x f +c2xi
)
−e1r2
f −e2r f ri − e2
2r2
i
4e1
]
,
with e2α = 0.
Corollary 1 (Singular classes). A one-mode singular Gaussian channel belongs
to class A1 if and only if its position representation has the following form:
√
e1/πδ (xi)exp
[
−a1x2
f + ı
(
b2x f ri +b1r f x f + c1x f
)
− e1r2
f
]
.
5.1. Allowed singular forms 83
Otherwise the channel belongs to class A2.
Since channels on each class are connected each other by unitary conjugations [Hol07],
a consequence of the theorem and the subsequent corollary is that the set of al-
lowed forms must remain invariant under unitary conjugations. To show this we
must know the possible functional forms of Gaussian unitaries. They are given by
the following lemma for one mode:
Lemma 1 (One-mode Gaussian unitaries). Gaussian unitaries can have only GF
or the one given by eq. (3.25).
Proof. Recalling that for a unitary GQC, T must be symplectic (TΩTT = Ω) and
N = 0. However, an inspection to eq. (3.37) lead us to note that N 6= 0 unless e1
diverges. Thus, Gaussian unitaries cannot have the form JI [see eq. (3.24)]. An
inspection of matrices T and N of GQC with GF [see eq. (5.1)] and the ones for JII
[see equations (3.38) and (3.44)] lead us to note the following two observations:
(i) in both cases we have N = 0 for an = 0 ∀n; (ii) the matrix T is symplectic for
GF when b2 = b3, and when αη = βγ for JII. In particular the identity map has
the last form. This completes the proof.
One can now compute the concatenations of the SGQCs with Gaussian unitaries.
This can be done straightforward using the well known formulas for Gaussian in-
tegrals and the Fourier transform of the Dirac delta. Given that the calculation is
elementary, and for sake of brevity, we present only the resulting forms of each
concatenation. To show this compactly we introduce the following abbreviations:
Singular channels belonging to class A2 with form JI and with α = 0, e2 = 0 and
α = e2 = 0, will be denoted as δ α
A2
, δ e2
A2
and δ α,e2
A2
, respectively; singular channels
belonging to the same class but with GF will be denoted as AA2
; channels belong-
ing to class A1 will be denoted as δA1
; finally Gaussian unitaries with GF will be
denoted as AU and the ones with form JII as δU . Writing the concatenation of
two channels in the position representation as
J(f)(x f ,r f ;xi,ri) =
∫
R2
dx′dr′J(1)
(
x f ,r f ;x′,r′
)
J(2)
(
x′,r′;xi,ri
)
, (5.11)
the resulting functional forms for J(f) are given in table 5.1. As expected, the table
shows that the integral has only the forms stated by our theorem. Additionally it
shows the cases when unitaries change the functional form of class A2, while for
class A1 J(f) has always the unique form enunciated by the corollary.
84 Chapter 5. Singular Gaussian quantum channels
J(1) J(2) J(f)
δ α
A2
AU AA2
AU δ α
A2
δ α
A2
δ α
A2
δU δ α
A2
δU δ α
A2
δ α
A2
δ e2
A2
AU δ e2
A2
AU δ e2
A2
AA2
δ e2
A2
δU δ e2
A2
δU δ e2
A2
δ e2
A2
AU ,δU δ
α,e2
A2
δ
α,e2
A2
δ
α,e2
A2
AU ,δU δ
α,e2
A2
δU ,AU δA1
δA1
δA1
δU ,AU δA1
Table 5.1: The first and second columns show the functional forms of J(1)
and J(2), respectively. The last column shows the resulting form of the
concatenation of them, see eq. (5.11). See main text for symbol coding.
5.2 Existence of master equations
In this section we show the conditions under which master equations, associated
with the channels derived in sec. 3.5.1, exist. To be more precise, we study if the
functional forms derived above parametrize channels belonging to one-parameter
differentiable families of GQCs. As a first step, we let the coefficients of forms
presented in equations (3.24) and (3.25) to depend on time. Later we derive the
conditions under which they bring any quantum state ρ(x,r; t) to ρ(x,r; t + ε)
(with ε > 0 and t ∈ [0,∞)) smoothly, while holding the specific functional form
of the channel, i.e.
ρ(x,r; t + ε) = ρ(x,r; t)+ εLt [ρ(x,r; t)]+O(ε2), (5.12)
where both ρ(x,r; t) and ρ(x,r; t + ε) are propagated from t = 0 with channels
either with the form JI or JII, and Lt is a bounded superoperator in the state
subspace. This is basically the problem of the existence of a master equation
∂tρ(x,r; t) = Lt [ρ(x,r; t)] , (5.13)
for such functional forms. Thus, the problem is reduced to prove the existence of
the linear generator Lt , also known as Liouvillian.
5.2. Existence of master equations 85
Class A2
r
p
(σ f )11 (s1 ,s2)
(σ f )22 (s1 ,s2)
(σ f )12 (s1 ,s2)
~d i
7→
~d f
(s3
)
(
σi , ~di
)
Figure 5.1: Schematic picture of the channels belonging to class A2. The
explicit dependence of the final state in terms of the combinations s1,
s2 and s3 are presented in the appendix. As well the formulas for si
depending on the form of the channel.
To do this we use an ansatz proposed in Ref. [KG97] to investigate the existence
and derive the master equation for GFs,
L = Lc(t)+(∂x,∂r)X(t)
(
∂x
∂r
)
+(x,r)Y(t)
(
∂x
∂r
)
+(x,r)Z(t)
(
x
r
)
(5.14)
where Lc(t) is a complex function and
X(t) =
(
Xxx(t) Xxr(t)
Xrx(t) Xrr(t)
)
(5.15)
is a complex matrix as well as Y(t) and Z(t), whose entries are defined in a similar
way as in eq. (5.15). Note that X(t) and Z(t) can always be chosen symmetric, i.e.
Xxr = Xrs and Zxr = Zrx. Thus, we must determine 11 time-dependent functions
from eq. (5.14). This ansatz is also appropriate to study the functional forms
introduced in this work, given that the left hand side of eq. (5.13) only involves
quadratic polynomials in x, r, ∂/∂x and ∂/∂ r, as in the GF case.
Notice that singular channels do not admit a master equation since its existence
implies that channels with the functional form involved can be found arbitrarily
86 Chapter 5. Singular Gaussian quantum channels
Class A1
r
p
1
2e1
2a1 +
b2
1
2e1
− b1
2e1
(0,−c1)
T
(
σi , ~di
)
Figure 5.2: Schematic picture of the class A1. Every channel of this class
maps every initial quantum state, in particular GSs characterized by
(
σi, ~di
)
, to a Gaussian state that depends only on the channel param-
eters. We indicate in the figure the values of the corresponding compo-
nents of the first and second moments of the final Gaussian state.
5.2. Existence of master equations 87
close from the identity channel. This is not possible for singular channels due to
the continuity of the determinant of the matrix T.
For the non-singular cases presented in equations (3.24) and (3.25), the condition
for the existence of a master equation is obtained as follows. (i) Substitute the
ansatz of eq. (5.14) in the right hand side of the eq. (5.13). (ii) Define ρ(x,r; t)
using eq. (2.30), given an initial condition ρ(x,r;0), for each functional form JI,II.
(iii) Take ρ f (x f ,r f ) → ρ(x,r; t) and ρi(xi,ri) → ρ(x,r;0). Finally, (iv) compare
both sides of eq. (5.13). Defining A(t) = α(t)/β (t) and B(t) = γ(t)/η(t), the
conclusion is that for both JI and JII, a master equations exist if
c(t) ∝ A(t) (5.16)
holds, where c(t) = c1(t)+A(t)c2(t). Additionally, for the form JI the solutions
for the matrices X(t), Y(t) and Z(t) are given by
Xxx = Xxr = Yrx = Zrr = 0,
Yxx =
Ȧ
A
,
Lc = Yrr =
ė1
e1
− ė2
e2
,
Xrr =
ė1
4e2
1
− ė2
2e1e2
,
Yxr = ı
(
λ1ė2
e1e2
+
λ2Ȧ
e2A
− λ1ė1
2e2
1
− λ̇2
e2
)
,
Zxx =
λ 2
1
2
(
ė2
e1e2
− ė1
2e2
1
)
+
λ1
e2
(
λ2Ȧ
A
− λ̇2
)
+2λ3
Ȧ
A
− λ̇3,
Zxr = ı
(
Ȧ
A
(
e1λ2
e2
− λ1
2
)
+
λ̇1
2
− λ̇2e1
e2
+
λ2
2
(
ė2
e2
− ė1
e1
))
,
(5.17)
where we have defined the following coefficients: λ1 = b1 +Ab3, λ2 = b2 +Ab4
and λ3 = a1 +Aa2 +A2a3.
88 Chapter 5. Singular Gaussian quantum channels
For the form JII the solutions are the following
Lc = Xxx = Xxr = Xrr = Zrr = 0,
Yrx = Yxr = 0,
Yxx =
Ȧ
A
, Yrr =
Ḃ
B
.
Zxx = a2(t)Ȧ(t)+
2a1(t)Ȧ(t)
A(t)
−A(t)2
−ȧ3(t)−A(t)ȧ2(t)− ȧ1(t),
Zxr = ı
(
1
2
λ̇ − λ
2
(
Ȧ
A
+
Ḃ
B
))
,
(5.18)
where λ = b1 +Ab3 +B(b2 +Ab4).
Chapter 6
Summary and conclusions
Living is worthwhile if one can contribute in some small way to this endless
chain of progress.
Paul A.M. Dirac
In this thesis we have introduced two works developed during my PhD. The first
one was devoted to study quantum channels from the point of view of their di-
visibility properties. We made use of several results from the literature, specially
from the seminal work by M. M. Wolf and J. I. Cirac [WECC08], and completed
and fixed some results of Ref. [WC08]. This led to the construction of a tool to
decide whether a quantum channel can be implemented using time-independent
Markovian master equations or not, for the finite dimensional case. We addition-
ally proved three theorems relating some of the studied divisibility types. Some
of the tools introduced in chapter 3 are results from other paper developed during
my PhD, where I am a secondary author, see Ref. [CDG19]. In the second work
we have studied one-mode Gaussian channels without Gaussian functional form
in the position state representation. We performed a characterization based on the
universal properties that quantum channels must fulfill; in particular we studied
the case of singular channels. We showed that the transition from unitarity to
non-unitarity can correspond directly to a change in the functional form of the
channel, in particular it turns out that functional form with one Dirac delta factor
do not parametrize unitary channels. Additionally in this project we derived the
conditions under which master equations for particular functional forms exist.
Let us summarize the results for the first project in more detail. We imple-
mented the known conditions to decide the compatibility of channels with time-
independent master equations (the so called L-divisibility) for the general diag-
89
90 Chapter 6. Summary and conclusions
onalizable case, and a discussion of the parametric space of Lindblad generators
was given. We additionally clarified one of the results of the paper [WECC08].
There, the authors arrived to erroneous conclusions for the case of channels with
negative eigenvalues. In our work we handled this case carefully. For unital qubit
channels it was shown that every infinitesimal divisible map can be written as a
concatenation of one L-divisible channel and two unitary conjugations. For the
particular case of Pauli channels case, we have shown that the sets of infinitely
divisible and L-divisible channels coincide. We made an interesting observation,
connecting the concept of divisibility with the quantum information concept of
entanglement-breaking channels: we found that divisible but not infinitesimal di-
visible qubit channels (in positive but not necessarily completely positive maps)
are necessarily entanglement-breaking. We also noted that the intersection of in-
divisible and P-divisible channels is not empty. This allows us to implement indi-
visible channels with infinitesimal positive and trance preserving maps. Finally,
we studied the possibility of dynamical transitions between different classes of
divisibility channels. We argued that all the transitions are, in principle, possible,
given that every divisibility set appears connected in our plots. We exploited two
simple models of dynamical maps to demonstrate that these transitions exist. They
clearly illustrate how the channels evolutions change from being implementable
by Markovian dynamical maps (infinitesimal divisible in complete positive maps
and/or L-divisible) to non-Markovian (divisible but not infinitesimal divisible or
infinitesimal divisible in positive but not complete positive maps), and vice versa.
For the second project we have critically reviewed the deceptively natural idea
that Gaussian quantum channels always admit a Gaussian functional form. To
this end, we went beyond the pioneering characterization of Gaussian channels
with Gaussian form presented in Ref. [MP12] in two new directions. First we
have shown that, starting from their most general definition (a quantum ma that
takes Gaussian states to Gaussian states), a more general parametrization of the
coordinate representation of the one-mode case exists, that admits non-Gaussian
functional forms. Second, we were able to provide a black-box characterization of
such new forms by imposing complete positivity (not considered in Ref. [MP12])
and trace preserving conditions. While our parametrization connects with the
analysis done by Holevo [Hol08] in the particular cases where besides having a
non-Gaussian form the channel is also singular, it also allows the study of Gaus-
sian unitaries, thus providing similar classification schemes. We completed the
classification of the studied types of channels by deriving the form of the Liou-
villian super operator that generates their time evolution in the form of a master
equation. Surprisingly, Gaussian quantum channels without Gaussian form can
be experimentally addressed by means of the celebrated Caldeira-Legget model
for the quantum damped harmonic oscillator [GSI88], where the new types of
91
channels described here naturally appear in the sub-ohmic regime.
We are interested in several directions to continue the investigation. From the
project of divisibility of quantum channels, an extension of this analysis to larger-
dimensional systems could give a deeper sight to the structure of quantum chan-
nels. In particular we are interested on proving if the equivalence of infinitely
divisible channels and L-divisible channels is present also in the general qubit
case. Additionally a plethora of interesting questions are related to design of effi-
cient verification procedures of the divisibility classes for channels and dynamical
maps. For instance, can we define an extension of the Lorentz normal decompo-
sition to systems composed of many qubits?, this would be useful to character-
ize infinitesimal divisibility of many particle systems; or Is the non-countable
parametrization of channels with negative eigenvalues relevant on deciding L-
divisibility?. Finally the area of channel divisibility contains several open struc-
tural questions, e.g. the existence of at most n-divisible channels. From the project
concerning one-mode Gaussian channels, a natural direction to follow is to extend
the analysis for other types of channels (or more modes) by following the classifi-
cation introduced by Holevo, see Ref. [Hol07]. The latter is based on the form of
a canonical form of one-mode Gaussian channels. Therefore a connection of this
classification with ours could be useful to assess quantum information features, in
particular for systems for which position state representation is advantageous.

Chapter 7
Appendices
93

Appendix A
Proof of theorem “Exact
dynamics with Lindblad master
equation”
The theorem announced in chapter 2 is,
Theorem 2 (Exact dynamics with Lindblad master equation) Let Et = etL a quan-
tum process generated by a Lindblad operator L. The equation
Et [ρS] = trE
[
e−iHt (ρS ⊗ρE)eiHt
]
,
where H has finite dimension, holds if and only if Et is a unitary conjugation for
every t.
Proof. To prove this theorem, we will compute ρS(t) to first order in t, see eq. (2.12).
Following the master equation of eq. (2.10) and taking t = ε ≪ 1, we have
ρS(ε)≈ ρS + trE
∫ ε
0
dt {i [ρS ⊗ρE,H]}
= ρS + trE {i [ρS ⊗ρE,H]}ε
= ρS +LExact[ρS]ε.
where LExact = trE {i [ρS ⊗ρE,H]}. Since Et is generated by a Lindblad master
equation, LExact must coincide with the Lindblad generator since the process is
homogeneous in time, i.e. LExact is time-independent. Writing the global Hamil-
tonian as
H = ∑
k,l=0
hklF
(S)
k ⊗F
(E)
l ,
95
96 Chapter A. Exact dynamics with Lindblad master equation
where hkl ∈ R, and
{
F
(S)
k
}
k
and
{
F
(E)
k
}
k
are orthogonal hermitian bases of
B
(
H (S)
)
and B
(
H (E)
)
, respectively, with H (S) and H (E) are the Hilbert
spaces of the central system S and the environment E. We have,
LExact[ρS] = i trE
{
∑
k,l
hkl[ρS ⊗ρE,F
(S)
k ⊗F
(E)
l ]
}
= i∑
k,l
hkl
{
ρSF
(S)
k tr[ρEF
(E)
l ]−F
(S)
k ρS tr[F
(E)
l ρE]
}
= i∑
k,l
hkl tr[F (E)
l ρE]
{
ρSF
(S)
k −F
(S)
k ρS
}
= i[ρS, H̃],
where H̃ = ∑k,l hkl tr[F (E)
l ρE]F
(S)
k is an hermitian operator. Therefore LExact is the
generator of Hamiltonian dynamics with Hamiltonian H̃, thus Et is unitary for all
t.
Appendix B
On Lorentz normal forms of
Choi-Jamiolkowski state
In this appendix we compute the Lorentz normal decomposition of a channel for
which one gets b 6= 0, supporting our observation that Lorentz normal decom-
position does not take Choi-Jamiołkowski states to something proportional to a
Choi-Jamiołkowski state. Consider the following Kraus rank three channel and
its RE matrix, both written in the Pauli basis:
Ê =




1 0 0 0
0 − 1
3
0 0
0 0 −1
3
0
2
3
0 0 1
3




, (B.1)
and
RE =




1 0 0 0
0 −1
3
0 0
0 0 1
3
0
2
3
0 0 1
3




. (B.2)
Using the algorithm introduced in Ref. [VDD01] to calculate RE ’s Lorentz de-
composition into orthochronous proper Lorentz transformations we obtain
L1 =
1
γ1




4 0 0 1
0 −γ1 0 0
0 0 −γ1 0
1 0 0 4




, (B.3)
97
98 Chapter B. On Lorentz normal forms of Choi-Jamiolkowski state
L2 =
1
γ2




89+9
√
97 0 0 −8
0 −γ2 0 0
0 0 −γ2 0
−8 0 0 89+9
√
97




,
and
ΣE =
1
γ3






√
11+ 109√
97
0 0 −
√
97+1√
89
√
97+873
0 − γ3
3
0 0
0 0
γ3
3
0
√
1+ 49√
97
0 0
√
−1+ 49√
97






with γ1 =
√
15, γ2 = 3
√
178
√
97+1746, and γ3 =
√
30. Although the central ma-
trix ΣE is not exactly of the form eq. (3.17), it is equivalent. To see this notice that
the derivation of the theorem 2 in [VDD01] considers only decompositions into
proper orthochronous Lorentz transformations. But to obtain the desired form, the
authors change signs until they get eq. (3.17); this cannot be done without chang-
ing Lorentz transformations. If we relax the condition over L1,2 of being proper
and orthochronous, we can bring ΣE to the desired form by conjugating ΣE with
G = diag(1,1,1,−1):
G−1ΣE G =
1
γ3






√
11+ 109√
97
0 0
√
97+1√
89
√
97+873
0 − γ3
3
0 0
0 0
γ3
3
0
−
√
1+ 49√
97
0 0
√
−1+ 49√
97






.
In both cases (taking ΣE or G−1ΣE G as the normal form of RE ), the corresponding
channel is not proportional to a trace-preserving one since b 6= 0, see eq. (3.17).
This completes the counterexample.
Appendix C
Articles
C.1 Article: Divisibility of qubit channels and dynamical
maps
Quantum 3, 144 (2019). Click to go to the webpage, Click to go to arXiv.
99
Divisibility of qubit channels and dynamical maps
David Davalos1, Mario Ziman2,3, and Carlos Pineda1,4
1Instituto de Física, Universidad Nacional Autónoma de México, México, D.F., México
2Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, Bratislava 84511, Slovakia
3Faculty of Informatics, Masaryk University, Botanická 68a, 60200 Brno, Czech Republic
4Faculty of Physics, University of Vienna, 1090 Vienna, Austria
May 7, 2019
The concept of divisibility of dynamical
maps is used to introduce an analogous con-
cept for quantum channels by analyzing the
simulability of channels by means of dynami-
cal maps. In particular, this is addressed for
Lindblad divisible, completely positive divisi-
ble and positive divisible dynamical maps. The
corresponding L-divisible, CP-divisible and P-
divisible subsets of channels are characterized
(exploiting the results by Wolf et al. [25]) and
visualized for the case of qubit channels. We
discuss the general inclusions among divisibil-
ity sets and show several equivalences for qubit
channels. To this end we study the conditions
of L-divisibility for finite dimensional chan-
nels, especially the cases with negative eigen-
values, extending and completing the results
of Ref. [26]. Furthermore we show that tran-
sitions between every two of the defined divis-
ibility sets are allowed. We explore particular
examples of dynamical maps to compare these
concepts. Finally, we show that every divisible
but not infinitesimal divisible qubit channel (in
positive maps) is entanglement-breaking, and
open the question if something similar occurs
for higher dimensions.
1 Introduction
The advent of quantum technologies opens questions
aiming for deeper understanding of the fundamental
physics beyond the idealized case of isolated quantum
systems. Also the well established Born-Markov ap-
proximation used to describe open quantum systems
(e.g. relaxation process such as spontaneous decay) is
of limited use and a more general framework of open
system dynamics is demanded. Recent efforts in this
area have given rise to relatively novel research sub-
jects - non-markovianity and divisibility.
Non-markovianity is a characteristic of continuous
time evolutions of quantum systems (quantum dy-
namical maps), whereas divisibility refers to prop-
erties of system’s transformations (discrete quantum
David Davalos: davidphysdavalos@gmail.com
processes) over a fixed time interval (quantum chan-
nels). The non-markovianity aims to capture and de-
scribe the back-action of the system’s environment on
the system’s future time evolution. Such phenomena
is identified as emergence of memory effects [1, 18, 22].
On the other side, the divisibility questions the possi-
bility of splitting a given quantum channel into a con-
catenation of other quantum channels. In this work
we will investigate the relation between these two no-
tions.
Our goal is to understand the possible forms of
the dynamics standing behind the observed quantum
channels, specially in regard to their divisibility prop-
erties which in turn determine their markovian or non-
markovian nature. In particular, we provide charac-
terization of the subsets of qubit channels depending
on their divisibility properties and implementation by
means of dynamical maps. An attempt to charac-
terize the set of channels belonging to one-parameter
semigroups induced by (time-independent) Lindblad
master equations has been already done in Ref. [26].
However, it has drawbacks when dealing with chan-
nels with negative determinants. Using the results of
Ref. [5] and Ref. [3], we will extend the analysis of [26]
also for channels with negative eigenvalues.
The paper is organized as follows: In section 2 we
give the formal definition of quantum channels and of
quantum dynamical maps, and some of their proper-
ties. We discuss the meaning of divisibility for each
object and discuss the known inclusions and equiva-
lences between divisibility types. In section 3 we dis-
cuss properties and representations of qubit channels
and their divisibility. We introduce a useful theorem
to decide L-divisibility, which is in turn valid for any
finite dimension. In section 4 we discuss the possi-
ble transition that can be occur between divisibility
types, and show two examples of dynamical maps and
their transitions. Finally in Section 5 we summarize
our results and discuss open questions.
Accepted in Quantum 2019-04-24, click title to verify 1
2 Basic definitions and divisibility
2.1 Channels and divisibility classes
We shall study transformations of a physical system
associated with a complex Hilbert space Hd of di-
mension d. In particular, we consider linear maps
on bounded operators, B(Hd), that for the finite-
dimensional case coincides with the set of trace-class
operators that accommodate the subset of density op-
erators representing the quantum states of the system.
We say a linear map E : B(Hd) → B(Hd) is positive,
if it maps positive operators into positive operators,
i.e. X ≥ 0 implies E [X] ≥ 0. Quantum channels are
associated with elements of the convex set C of com-
pletely positive trace-preserving linear maps (CPTP)
transforming density matrices into density matrices,
i.e. E : B(H) → B(H) such that tr(E [X]) = tr(X)
for all X ∈ B(H), and all its extensions idn ⊗ E are
positive maps for all n > 1, where idn is the iden-
tity channel on a n-dimensional quantum system. In
general a channel has the form E [X] =
∑
i KiXK†
i .
The minimum number of operators Ki required in the
previous expression is called the Kraus rank of E .
Let us introduce two subsets of channels. First,
we say a channel is unital if it preserves the identity
operator, i.e. E [✶] = ✶. Unital channels have a simple
parametrization which will be useful for our purposes.
Second, if E [X] = UXU† for some unitary operator
U (meaning UU† = U†U = ✶), we say the channel is
unitary.
A quantum channel E is called indivisible if it can-
not be written as a concatenation of two non-unitary
channels, namely, if E = E1E2 implies that either E1,
or E2, exclusively, is a unitary channel. If the channel
is not indivisible, it is said to be divisible. We denote
the set of divisible channels by Cdiv and that of in-
divisible channels by Cdiv. Following this definition,
unitary channels are divisible, because for them both
(decomposing) channels E1,2 must be unitary. The
concept of indivisible channels resembles the concept
of prime numbers: unitary channels play the role of
unity (which are not indivisible/prime), i.e. a compo-
sition of indivisible and a unitary channel results in
an indivisible channel.
We now define the set of infinitely divisible channels
(C∞) and the set of infinitesimal divisible channels
(CInf). Infinitely divisible channels, in some sense op-
posite to indivisible channels, are defined as channels
E for which there exist for all n = 1, 2, 3, . . . a channel
An such that E = (An)n. Now, consider channels E
that may be written as products of channels close to
identity, i.e. such that for all ǫ > 0 there exists a finite
set of channels εj with ||id − εj || ≤ ǫ and E =
∏
j εj .
Its closure determines the set of infinitesimal divisible
channels CInf.
2.2 Quantum dynamical maps and more divis-
ibility classes
The next sets of channels are going to be defined using
three types of dynamical maps. A quantum dynami-
cal map is identified with a continuous parametrized
curve drawn inside the set of channels starting at
the identity channel, i.e. a one-parametric function
t 7→ Et ∈ C for all t belonging to an interval with min-
imum element 0 and satisfying the initial condition
E0 = id. Let Et,s = E−1
t Es be the linear map describ-
ing the state transformations within the time interval
[t, s], whenever E−1
t exists.
• A given quantum dynamical map is called CP-
divisible if for all t < s the map Et,s is a channel.
• A given quantum dynamical map is called P-
divisible [22] if Et,s is a positive trace-preserving
linear map for all t < s.
• A given quantum dynamical map is called L-
divisible if it is induced by a time-independent
Lindblad master equation [7, 14, 16], i.e. Et = etL
with
L(ρ) = i[ρ, H]+
∑
α,β
Gαβ
(
FαρF †
β − 1
2
{F †
βFα, ρ}
)
,
where H = H† ∈ B(Hd) is known as Hamilto-
nian, {Fα} are hermitian and form an orthonor-
mal basis of the operator space B(Hd), and Gαβ
constitutes a hermitian positive semi-definite ma-
trix.
If we allow the Lindblad generator L to depend
on time, we recover the set of CP-divisible quan-
tum dynamical maps as the resulting dynamical maps
Et = T̂e
∫
t
0
L(τ)dτ (T̂ denotes the time-ordering op-
erator) are compositions of infinitesimal completely-
positive maps [1, 7, 14, 16]. Notice that there is a
hierarchy for quantum dynamical maps: L-divisible
quantum dynamical maps are CP-divisible which in
turn are P-divisible.
Using the introduced families of quantum dynam-
ical maps we can now classify quantum channels ac-
cording to whether they can be implemented by the
aforementioned kinds of quantum dynamical maps.
We define subsets CL, CCP, and CP of L-divisible,
CP-divisible and P-divisible channels, respectively. In
particular, we say E ∈ CL if it belongs to the clo-
sure of a L-divisible quantum dynamical map. Let us
stress that the requirement of the existence of Lind-
blad generator L such that E = eL is not sufficient
and closure is necessary. For example, the evolution
governed by L(ρ) = i[ρ, H]+γ[H, [H, ̺]] results [27] in
the diagonalization of states in the energy eigenbasis
of H. Such transformation Ediag is not invertible, thus
(by definition) L = log Ediag does not exist (contains
infinities). Analogously, we say E ∈ CCP (E ∈ CP)
Accepted in Quantum 2019-04-24, click title to verify 2
if there exists a CP-divisible (P-divisible) dynamical
map Et such that E = Et (with arbitrary precision) for
some value of t (including t = ∞).
We now recall how to verify whether a channel is
L-divisible since we will build upon the method for
some of our results. Verifying whether E ∈ CL [26]
requires evaluation of the channel’s logarithms, how-
ever, the matrix logarithm is defined only for invert-
ible matrices and it is not unique. In fact, we need
to check if at least one of its branches has the Lind-
blad form. It was shown in [5] that E is L-divisible
if and only if there exists L such that: exp L = E , is
hermitian preserving, trace-preserving and condition-
ally completely positive (ccp). Thus, we are looking
for logarithm satisfying L(X†) = L(X)† (hermiticity
preserving), L∗(✶) = 0 (trace-preserving), and
(✶ − ω)(id ⊗ L)[ω](✶ − ω) ≥ 0 (1)
(ccp condition), where ω = 1
d
∑d
j,k=1 |j ⊗ j〉〈k ⊗ k| is
the projector onto a maximally entangled state. To
the best of our knowledge, there is no general method
to verify if a channel is P or CP divisible, with some
exceptions [25].
2.3 Relation between channel divisibility
classes
Due to the inclusion set relations between the three
kinds of dynamical maps discussed in the previous
sections, we can see that CL ⊂ CCP ⊂ CP. Simi-
larly, from the earlier definitions, one can easily ar-
gue that C∞ ⊂ CInf ⊂ Cdiv. By the definition of L-
divisibility it is trivial to see that in general CL ⊂ C∞.
Indeed Denisov has shown in [4] that infinitely divis-
ible channels can be written as E = E0eL, with L a
Lindblad generator, and E0 idempotent operator such
that E0LE0 = E0L. Further, it was shown in Ref. [25]
that E ∈ Cinf implies det E ≥ 0 and also that E can
be approximated by
∏
j eLj , i.e. CCP = Cinf . In other
words, the positivity of determinant is necessary for
the channel to be (in the closure of) channels attain-
able by CP-divisible dynamical maps. In summary,
we have the following relations between sets (see also
Fig. 1):
C∞ ⊂ CInf ⊂ Cdiv
⊆ =
CL ⊂ CCP ⊂ CP
. (2)
The relation between CP and Cdiv is unknown, al-
though it is clear that Cdiv ⊂ CP is not possible since
channels in CP are not necessarily divisible in CP
maps. The intersection of CP and Cdiv is not empty
since CCP ⊆ Cdiv and CCP ⊆ CP. Later on we will
investigate if CP ⊆ Cdiv or not.
C
C
L
C
div
C
P
C
CP
C
∞
C
P
∩ Cdiv
Unitary
channels
Figure 1: Scheme illustrating the different sets of quantum
channels for a given dimension, discussed in sec. 2. In partic-
ular, the inclusion relations presented in Eq. (2) are depicted.
3 Qubit channels
3.1 Representations
Using the Pauli basis 1√
2
{1, σx, σy, σz}, and the stan-
dard Hilbert-Schmidt inner product, the real repre-
sentation for qubit channels is given by [10, 20]:
Ê =
(
1 ~0T
~t ∆
)
. (3)
This describes the action of the channel in the Bloch
sphere picture in which the points ~r are identified with
density operators ̺~r = 1
2 (I + ~r · ~σ). We will write
E = (∆,~t) meaning that E(ρ~r) = ρ∆~r+~t.
In order to study qubit channels with simpler ex-
pressions, we will consider a decomposition in uni-
taries such that
E = U1DU2. (4)
This can be performed by decomposing ∆ in ro-
tation matrices, i.e. ∆ = R1DR2, where D =
diag(λ1, λ2, λ3) is diagonal and the rotations R1,2 ∈
SO(3) (of the Bloch sphere) correspond to the uni-
tary channels U1,2. This decomposition should not
be confused with the singular value decomposition.
The latter allows decompositions that include, say,
total reflections. Such operations do not correspond
to unitaries over a qubit, in fact they are not CPTP.
Therefore the channel D, in the Pauli basis, is given
by
D̂ =
(
1 ~0T
~τ D
)
, (5)
where ∆ = R1DR2 and ~τ = RT
1
~t. The latter describes
the shift of the center of the Bloch sphere under the
action of D. The parameters ~λ determine the length
of semi-axes of the Bloch ellipsoid, being the defor-
mation of Bloch sphere under the action of E . From
Accepted in Quantum 2019-04-24, click title to verify 3
now we will call the form D, special orthogonal normal
form.
We shall develop a geometric intuition in the space
determined by the possible values of these three pa-
rameters. For an arbitrary channel, complete positiv-
ity implies that the possible set of lambdas lives in-
side the tetrahedron with corners (1, 1, 1), (1, −1, −1),
(−1, 1, −1) and (−1, −1, 1), see Fig. 2. For unital
channels, all points in the tetrahedron are allowed,
but for non-unital channels more restrictive condi-
tions arise. In Fig. 8 we present a visualization of
the permitted values of ~λ for a particular nontrivial
value of ~τ , and in [2] the steps to study the general
case from an algebraic point of view are presented.
For the unital case, the corner ~λ = (1, 1, 1) corre-
sponds to the identity channel, ~λ = (1, −1, −1) to
σx
~λ = (−1, 1, −1) to σy and ~λ = (−1, −1, 1) to σz
(Kraus rank 1 operations). Points in the edges corre-
spond to Kraus rank 2 operations, points in the faces
to Kraus rank 3 operations and in the interior of the
tetrahedron to Kraus rank 4 operations.
In addition to this decomposition, following the def-
inition of divisibility, concatenation with unitaries of a
given quantum channel do not change the divisibility
character of the latter. Thus orthogonal normal forms
are useful to study divisibility since, following also the
properties of CP and CCP introduced in sec. 3.1, im-
mediately one has:
Theorem 1 (Divisibility of special orthogo-
nal normal forms). Let E a qubit quantum chan-
nel and D its special orthogonal normal form, E be-
longs to CX if and only if D does, where X =
{“Div”, “P”, “CP”}.
There is another another parametrization for qubit
channels called Lorentz normal decomposition [23, 24]
which is specially useful to characterize infinitesimal
divisibility CInf, and geometric aspects of entangle-
ment [15]. This decomposition is derived from the
theorem 3 of Ref. [24], which essentially states that for
a qubit state ρ = 1
4
∑3
i,j=0 Rijσi⊗σj the matrix R can
be decomposed as R = L1ΣLT
2 . Here L1,2 are proper
orthochronous Lorentz transformations and Σ is ei-
ther Σ = diag (s0, s1, s2, s3) with s0 ≥ s1 ≥ s2 ≥ |s3|,
or
Σ =




a 0 0 b
0 d 0 0
0 0 −d 0
c 0 0 −b + c + a




. (6)
In theorem 8 of Ref. [23] the authors make a simi-
lar claim, exploiting the Choi-Jamiołkowski isomor-
phism. They forced b = 0 in order to have normal
forms proportional to trace-preserving operations in
the case of Kraus rank deficient ones, see Eq. (6). The
latter is equivalent to saying that the decomposition
of Choi-Jamiołkowski states leads to states that are
proportional to Choi-Jamiołkowski states. We didn’t
find a good argument to justify such an assumption
λ1
λ2
λ3
Figure 2: Tetrahedron of Pauli channels. The corners cor-
respond to unitary Pauli operations (✶, σx,y,z) while the
rest can be written as convex combinations of them. The
bipyramid in blue corresponds to channels with λi > 0∀i, i.e.
channels of the positive octant belonging to CP. The whole
set CP includes three other bipyramids corresponding to the
other vertexes of tetrahedron, i.e. CP enjoys the symmetries
of the tetrahedron, see Eq. (8). The faces of the bipyramids
matching the corners of the tetrahedron are subsets of the
faces of the tetrahedron, i.e. contain Kraus rank three chan-
nels. Such channels are then CPbut also Cdiv, showing that
the intersection shown in Fig. 1 is not empty.
Accepted in Quantum 2019-04-24, click title to verify 4
and found a counterexample (see appendix A). Thus
we propose a restricted version of their theorem:
Theorem 2 (Restricted Lorentz normal form
for qubit quantum channels). For any full Kraus
rank qubit channel E there exists rank-one completely
positive maps T1, T2 such that T = T1ET2 is propor-
tional to
(
1 ~0T
~0 Λ
)
, (7)
where Λ = diag(s1, s2, s3) with 1 ≥ s1 ≥ s2 ≥ |s3|.
The channel T is called the Lorentz normal form of
the channel E . For unital qubit channels D coincides
with Λ, thus in such case the form of (7) holds for any
Kraus rank.
3.2 Divisibility
In this subsection we will recall the criteria to decide
if a qubit channel belongs to Cdiv, CP and CCP follow-
ing [25]. We shall start with some general statements,
and then focus on different types of channels (unital,
diagonal non-unital and general ones). We will also
discuss in detail the characterization of CL, which en-
tails a higher complexity.
It was shown ([25], Theorem 11) that full Kraus
rank channels are divisible (Cdiv). This simply means
that all points in the interior of the set of channels cor-
respond to divisible channels. Moreover, according to
Theorem 23 of the same reference, qubit channels are
indivisible if and only if they have Kraus rank three
and diagonal Lorentz normal form. Notice that since
we dispute the theorem upon which such statement is
based, the classification might be inaccurate, see the
appendix. It follows from the definition that for qubit
channels E is divisible if and only if D is divisible. To
test if D is divisible, we check that all eigenvalues of
its Choi matrix are different from zero.
A non-negative determinant of E is a necessary con-
dition for a general channel to belong to CP ([25],
Proposition 15). For qubits, this is also sufficient
([25], Theorem 25), and given that det D = det E ,
the condition for qubit channels simply reads
det E = λ1λ2λ3 ≥ 0. (8)
However, to our knowledge, a simple condition for
arbitrary dimension is yet unknown.
With respect to testing for CP-divisibility we re-
strict the discussion to qubit channels. To character-
ize CP-divisible channels it is useful to consider the
Lorentz normal form for channels. A full Kraus rank
qubit channel E belongs to CCP if and only if it has
diagonal Lorentz normal form with
s2
min ≥ s1s2s3 > 0 (9)
where smin is the smallest of s1, s2 and s3, see theo-
rem 2 and [26]. For Kraus deficient channels the per-
tinent theorems are based on Kraus deficient Lorentz
normal forms that according to our appendix should
be reviewed.
Deciding L-divisibility, as mentioned above, is
equivalent to proving the existence of a hermiticity
preserving generator which additionally fulfills the ccp
condition.
To prove the former we recall that every hermitic-
ity preserving operator has a real matrix representa-
tion when choosing a hermitian basis. Since quantum
channels preserve hermiticity, the problem is reduced
on finding a real logarithm log Ê given a real matrix
Ê . This problem was already solved by Culver [3] who
characterized completely the existence of real loga-
rithms of real matrices. For diagonalizable matrices
the results can be summarized as follows:
Theorem 3 (Existence of hermiticity preserv-
ing generator). A non-singular matrix with real en-
tries Ê has a real generator (i.e. a log Ê has real en-
tries) if and only if the spectrum fulfills the following
conditions:
(i) negative eigenvalues are even-fold degenerated;
(ii) complex eigenvalues come in complex conjugate
pairs.
Let us examine this theorem for the particular case
of qubits. In this case this theorem means that real
logarithm(s) of Ê exist if and only if E has either only
positive eigenvalues, one positive and two complex, or
one positive and two equal non-positive eigenvalues,
apart from the trivial eigenvalue equal to one. No-
tice that quantum channels with complex eigenvalues
will fulfill the last condition immediately since they
preserve hermiticity.
We now continue discussing the multiplicity of
the solutions of log Ê , as finding an appropriate
parametrization is essential to test for the ccp con-
dition, see Eq. (1). If Ê has positive degenerated,
negative, or complex eigenvalues, its real logarithms
are not unique, and are spanned by real logarithm
branches [3]. In case of having negative eigenval-
ues, it turns out that real logarithms always have a
non-continuous parametrization, in addition to real
branches due to the freedom of the Jordan normal
form transformation matrices. Given a real repre-
sentation of E , i.e. Ê , the Jordan form is given by
Ê = wJw−1 = w̃Jw̃−1, where w = w̃K with K be-
longing to a continuum of matrices that commutes
with J [3]. In the case of diagonalizable matrices, if
there are no degeneracies, K commutes with log(J).
Finally let us note that if a channel belongs to CL,
unitary conjugations can bring it to CInf \CL and vice
versa.
3.3 Unital channels
We shall start our study of unital qubit channels, by
considering Pauli channels, defined as convex combi-
Accepted in Quantum 2019-04-24, click title to verify 5
nations of the unitaries σi:
EPauli[ρ] =
3
∑
i=0
piσiρσi, (10)
where σ0 = ✶ and pi ≥ 0 with
∑
i pi = 1. The spe-
cial orthogonal normal form of a Pauli channel [see
Eqs. (4) and (5)] has U1 = U2 = id and ~τ = ~0. Thus
Pauli channels are fully characterized only by ~λ. No-
tice that every unital qubit channel can be written as
Eunital = U1EPauliU2. (11)
This implies that arbitrary unital qubit channels can
be expressed as convex combinations of (at most) four
unitary channels.
Following theorem 1 it is straightforward to note
that by characterizing Cdiv, CP and CCP of Pauli
channels, the same conclusions hold for general uni-
tal qubit channels having the same ~λ. Additionally
we can have a one-to-one geometrical view of the di-
visibility sets for Pauli channels given they have a
one-to-one correspondence to the tetrahedron shown
in Fig. 2, defined by the inequalities
1 + λi − λj − λk ≥ 0 (12)
1 + λ1 + λ2 + λ3 ≥ 0 (13)
with i, j and k all different [28].
3.3.1 P-divisibility
Let us discuss the divisibility properties of Pauli chan-
nels. Divisibility in CPTP (Cdiv) is guaranteed for full
Kraus rank channels, i.e. for the interior channels of
the tetrahedron. For Pauli channels this is equiva-
lent to taking only the inequality of equations (13).
The characterization of CP can be done directly using
Eq. (8), as it depends only on ~λ. This set is the in-
tersection of the tetrahedron with the octants where
the product of all λs is positive. In fact, it consists of
four triangular bipyramids starting in each vertex of
the tetrahedron and meeting in its center, see Fig. 2.
Let us study the intersection of this set with the set
of unital entanglement-breaking (EB) channels [28],
forming an octahedron (being the intersection of the
tetrahedron with its space inversion, see Fig. 3). It is
defined by the inequalities
λ1 + λ2 + λ3 ≤ 1
λi − λj − λk ≤ 1, (14)
with i, j and k all different [28], together with
Eq. (13). It follows that unital qubit channels that are
not achieved by P-divisible dynamical maps are nec-
essarily entanglement-breaking (see Fig. 3 and Fig. 7).
In fact this holds for general qubit channels, see sec-
tion 3.4.
λ1
λ2
λ3
Figure 3: Tetrahedron of Pauli channels with the octahe-
dron of entanglement breaking channels shown in red, see
Eq. (14). The blue pyramid inside the octahedron is the
intersection of the bipyramid shown in Fig. 2, with the oc-
tahedron. The complement of the intersections of the four
bipyramids forms the set of divisible but not infinitesimal di-
visible channels in PTP. Thus, a central feature of the figure
is that the set Cdiv\CP is always entanglement-breaking, but
the converse is not true.
3.3.2 CP-divisibility
The subset of CP-divisible Pauli channels, following
Eq. (9) and theorem 2, is determined by the inequal-
ities
0 < λ1λ2λ3 ≤ λ2
min . (15)
They determine a body that is symmetric with respect
to permutation of Pauli unitary channels (i.e. in λj),
hence, the set of CCP of Pauli channels possesses the
symmetries of the tetrahedron. The set CCP\CL is
plotted in Fig. 5. Notice that this set coincides with
the set of unistochastic qubit channels, see Ref. [17].
3.3.3 L-divisibility
Let us now derive the conditions for L-divisibility
of Pauli channels with positive eigenvalues λ1, λ2, λ3
(λ0 = 1). The logarithm of D, induced by the princi-
pal logarithm of its eigenvalues, is thus
L = Kdiag(0, log λ1, log λ2, log λ3)K−1 , (16)
which is real (hermiticity preserving). In case of no-
degeneration the dependency on K vanishes and L
is unique. In such case the ccp conditions log λj −
log λk − log λl ≥ 0 imply
λjλk ≤ λl (17)
for all combinations of mutually different j, k, l. This
set (channels belonging to CL with positive eigenval-
ues) forms a three dimensional manifold, see Fig. 6.
Accepted in Quantum 2019-04-24, click title to verify 6
In case of degeneration, let us label the eigenvalues
η, λ and λ. In this case, the real solution for L is not
unique and is parametrized by real branches in the
degenerate subspace and by the continuous parame-
ters of K [3]. Let us study the principal branch with
K = ✶. Eq. (17) is then reduced to
λ2 ≤ η ≤ 1 . (18)
Therefore, if these inequalities are fulfilled, the gen-
erator has Lindblad form. If not, then a priori
other branches can fulfill the ccp condition and con-
sequently have a Lindblad form. Thus, Eq. (18) pro-
vides a sufficient condition for the channel to be in
CL. We will see it is also necessary.
Indeed, the complete positivity condition requires
η, λ ≤ 1, thus, it remains to verify only the condition
λ2 ≤ η. It holds for the case λ ≤ η. If η ≤ λ, then
this condition coincides with the CP-divisibility con-
dition from Eq. (15). Since CL implies CCP the proof
is completed. In conclusion, the condition in Eq. (17)
is a necessary and sufficient condition for a given Pauli
channel with positive eigenvalues to belong to CL.
Let us stress that the obtained subset of L-divisible
channels does not possess the tetrahedron symme-
tries. In fact, composing D with a σz rotation
Uz = diag(1, −1, −1, 1) results in the Pauli channel
D′ = diag(1, −λ1, −λ2, λ3). Clearly, if λj are positive
(D is L-divisible), then D′ has non-positive eigenval-
ues. Moreover, if all λj are different, then D′ does
not have any real logarithm, therefore, it cannot be
L-divisible. In conclusion, the set of L-divisible uni-
tal qubit channels is not symmetric with respect to
tetrahedron symmetries.
In what follows we will investigate the case of
non-positive eigenvalues. Theorem 3 implies that
that eigenvalues have the form (modulo permuta-
tions) η, −λ, −λ, where η, λ ≥ 0. The correspond-
ing Pauli channels are Dx = diag(1, η, −λ, −λ),
Dy = diag(1, −λ, η, −λ), Dz = diag(1, −λ, −λ, η),
thus forming three two-dimensional regions inside the
tetrahedron. Take, for instance, Dz that specifies a
plane (inside the tetrahedron) containing I, σz and
completely depolarizing channel N = diag(1, 0, 0, 0).
The real logarithms for this case are given by
L = K




0 0 0 0
0 log(λ) (2k + 1)π 0
0 −(2k + 1)π log(λ) 0
0 0 0 log(η)




K−1,
(19)
where k ∈ Z and K, as mentioned above, belongs to a
continuum of matrices that commute with Dz. Note
that L is always non-diagonal. For this case (simi-
larly for Dx and Dy) the ccp condition reduces again
to conditions specified in Eq. (18). Using the same
arguments one arrives to a more general conclusion:
Eq. (17) provides necessary and sufficient conditions
for L − divisibility of a given Pauli channel, and if
it is the case, the principal branch with K = ✶ has
Lindblad form. The set of L-divisible Pauli channels
is illustrated in Fig. 6.
In order to decide L-divisibility of general unital
channels it remains to analyze the case of complex
eigenvalues. The logarithms are parametrized as fol-
lows
Lk,K = wK log (J)k K−1w−1, (20)
where J is the real Jordan form of the (qubit) chan-
nel [3]:
J = diag (1, c) ⊕
(
a −b
b a
)
(21)
with a ± ib being the complex eigenvalues and c > 0.
Let us note that K log (J)k K−1 is reduced to equa-
tions (16) and (19) in the case of real eigenvalues. In
general, the generator is (up to diagonalization):
K log (J)k K−1 = Kdiag (0, log(c)) ⊕
(
log(|z|) arg(z) + 2πk
− arg(z) − 2πk log(|z|)
)
K−1.
with z = a + ib. The non-diagonal block of the loga-
rithm has the same structure as the real Jordan form
of the channel, so K also commutes with log(J)k,
leading to a countable parametric space of hermitian
preserving generators. In fact, generators of diagonal-
izable channels have continuous parametrizations if
and only if they have degenerate eigenvalues; the non-
diagonalizable case can be found elsewhere [3]. Since
we are dealing with a diagonalization, the ccp condi-
tion can be very complicated and depends in general
on k, see Eq. (1). But for the complex case we can
simplify the condition for qubit channels which have
exactly the form presented in Eq. (21), say Êcomplex,
i.e. w = ✶. In such case the ccp condition is reduced
to
a2 + b2 ≤ c ≤ 1. (22)
Note that it does not depend on k and the second
inequality is always fulfilled for CPTP channels.
We can present the conditions for L-divisibility for
the case of complex eigenvalues. The orthogonal nor-
mal form of Êcomplex is D̂ = diag (1, η, λ, λ) with
η = c and λ = sign(ab)
√
a2 + b2. The ccp con-
dition for degenerated eigenvalues, see Eq. (18), is
reduced to Eq. (22) for this case. Therefore, the
L-divisible channels with form Êcomplex are also L-
divisible, up to unitaries. This also applies for chan-
nels arising from changing positions of the 1 × 1
block containing c and the 2 × 2 block containing
a and b in Êcomplex, with orthogonal normal forms
diag (1, λ, η, λ) and diag (1, λ, λ, η). The set contain-
ing them is shown in Fig. 4.
3.3.4 Divisibility relations
Consider a Pauli channel with 0 < λmin = λ1 ≤
λ2 ≤ λ3 < 1, thus, the condition λ1λ2 ≤ λ3 triv-
ially holds. Since λ1λ2 ≤ λ1λ3 ≤ λ2λ3 ≤ λ2, it
Accepted in Quantum 2019-04-24, click title to verify 7
λ1
λ2
λ3
Figure 4: Tetrahedron of Pauli channels, with qubit unital
L-divisible channels of the form Êcomplex (see main text). Note
that the set does not have the symmetries of the tetrahedron.
follows that λ1λ3 ≤ λ2, thus, two (out of three) L-
divisibility conditions hold always for Pauli channels
with positive eigenvalues. Moreover, one may ob-
serve that CP-divisibility condition Eq. (15) reduces
to one of L-divisibility conditions λ2λ3 ≤ λ1. In
conclusion, the conditions of CP-divisibility and L-
divisibility for Pauli channels with positive eigenval-
ues coincide, thus, in this case CCP implies CL.
Concatenating (positive-eigenvalues) Pauli chan-
nels with Dx,y,z one can generate the whole set of
CCP Pauli channels. Using the identity CCP = CInf
and considering Eq. (11)we can formulate the follow-
ing theorem:
Theorem 4 (Infinitesimal divisible unital channels).
Let ECP
unital
be an arbitrary infinitesimal divisible unital
qubit channel. There exists at least one L-divisible
Pauli channel Ẽ, and two unitary conjugations U1 and
U2, such that
ECP
unital = U1ẼU2 .
Notice that if ECP
unital
is invertible, Ẽ = eL.
Let us continue with another equivalence relation
holding for Pauli channels. Regarding infinitely divis-
ibility channels, we know that, in general, CL ⊂ C∞,
however, for Pauli channels the corresponding subsets
coincide.
Theorem 5 (Infinitely divisible Pauli channels). The
set of L-divisible Pauli channels is equivalent to the set
of infinitely divisible Pauli channels.
Proof. A channel is infinitely divisible if and only
if it can be written as E0eL, where E0 is an idem-
potent channel satisfying E0LE0 = E0L and L has
λ1
λ2
λ3
Figure 5: Tetrahedron of Pauli channels with part of the set
of CP-divisible, see Eq. (15), but not L-divisible channels
(CCP\CL) shown in purple. The whole set CCP is obtained
applying the symmetry transformations of the tetrahedron to
the purple volume.
Lindblad form [4]. The only idempotent qubit chan-
nels are contractions of the Bloch sphere into sin-
gle points, diagonalization channels Ediag transform-
ing Bloch sphere into a line connecting a pair of
basis states, and the identity channel. Among the
single-point contractions, the only one that is a Pauli
channel is the contraction of the Bloch sphere into
the complete mixture. In particular, E = N eL =
N for all L. The channel N belongs to the clo-
sure of CL, because a sequence of channels eLn with
L̂n = diag (0, −n, −n, −n) converges to N̂ in the limit
n → ∞. For the case of E0 being the identity channel
we have E = eL, thus, trivially such infinitely divisi-
ble channel E is in CL too. It remains to analyze the
case of diagonalization channels. First, let us note
that the matrix of eL̂ is necessarily of full rank, since
detÊ 6= 0. It follows that the matrix Ê = ÊdiageL̂
has rank two as Êdiag is a rank two matrix, thus, it
takes one of the following forms Êλ
x = diag (1, λ, 0, 0),
Êλ
y = diag (1, 0, λ, 0), Êλ
z = diag (1, 0, 0, λ). The in-
finitely divisibility implies λ > 0 in order to keep the
roots of λ real. In what follows we will show that
Êz belongs to (the closure of) CL. Let us define the
channels Êλ,ǫ
z = diag (1, ǫ, ǫ, λ) with ǫ > 0. The com-
plete positivity and ccp conditions translate into the
inequalities ǫ ≤ 1+λ
2 and ǫ2 ≤ λ, respectively; there-
fore one can always find an ǫ > 0 such that Êλ,ǫ
z is
a L-divisible channel. If we choose ǫ =
√
λ/n with
n ∈ Z
+, the channels Êz,n = diag
(
1,
√
λ/n,
√
λ/n, λ
)
form a sequence of L-divisible channels converging to
Êλ
z when n → ∞. The analogous reasoning implies
that Êλ
x , Êλ
y ∈ CL too. Let us note that one parame-
Accepted in Quantum 2019-04-24, click title to verify 8
λ1
λ2
λ3
Figure 6: Tetrahedron of Pauli channels with the set of L-
divisible channels (or equivalently infinitely divisible, see The-
orem 5) shown in green, see equations (17) and (18). The
solid set corresponds to channels with positive eigenvalues,
and the 2D sets correspond to the negative eigenvalue case.
The point where the four sets meet corresponds to the total
depolarizing channel. Notice that this set does not have the
symmetries of the tetrahedron.
ter family Ez are convex combinations of the complete
diagonalization channel Ê1
z = diag (1, 0, 0, 1) and the
complete mixture contraction N̂ . This completes the
proof.
Finally, let us remark that using theorem 23 of
Ref. [25] we conclude that the intersection CP ∩ Cdiv
depicted in Fig. 1 is not empty. To show this, notice
that applying the mentioned theorem to Pauli chan-
nels we get that the faces of the tetrahedron are indi-
visible in CPTP channels. However, there are chan-
nels with positive determinant inside the faces, for
example diag
(
1, 4
5 , 4
5 , 3
5
)
. Therefore we conclude that
up to unitaries, CP ∩ Cdiv correspond to the union of
the four faces faces of the tetrahedron minus the faces
of the octahedron that intersects with the faces of the
tetrahedron, see Fig. 3. We have to remove such inter-
section since it corresponds to channels with negative
determinant, i.e. not in CP.
To get a detailed picture of the position and inclu-
sions of the divisibility sets, we illustrate in Fig. 7 two
slices of the tetrahedron where different types of di-
visibility are visualized. Notice the non-convexity of
the considered divisibility sets.
3.4 Non-unital qubit channels
Similar to unital channels, using theorem 1 we are
able to characterize Cdiv, CP and CCP by studying
special orthogonal normal forms of non-unital chan-
nels. They are characterized by ~λ and ~τ , see Eq. (5).
Thus we can study if a channel is Cdiv by computing
the rank of its Choi matrix. For this case algebraic
equations are in general fourth order polynomials. In
fact, in Ref. [19] a condition in terms of the eigenval-
ues and ~τ is given. For special cases, however, we can
obtain compact expressions, see Fig. 8. The charac-
terization of CP is given by Eq. (8) (note that it only
depends on ~λ), and CCP is tested, for full Kraus rank
non-unital channels, using Eq. (9), see Ref. [24] for
the calculation of the si’s. For the characterization of
CL we use the results developed at the end of the last
section, see Eqs. (18)-(22).
We can plot illustrative pictures even though the
whole space of qubit channels has 12 parameters. This
can be done using orthogonal normal forms and fixing
~τ , exactly in the same way as the unital case. Re-
call that unitaries only modify CL, leaving the shape
of other sets unchanged. CPTP channels are repre-
sented as a volume inside the tetrahedron presented
in Fig. 2, see Fig. 8. In the later figure we show a slice
corresponding to ~τ = (1/2, 0, 0)
T. Indeed, it has the
same structure of the slices for the unital case, but
deformed, see Fig. 7. A difference with respect to the
unital case is that L-divisible channels with negative
eigenvalues (up to unitaries) are not completely inside
CP-divisible channels. A part of them are inside the
CP channels.
A central feature of Figs. 7 and 8 is that the set
Cdiv \ CP is inside the convex slice of the set of en-
tanglement breaking channels (deformed octahedron).
Indeed, we can proof the following theorem.
Theorem 6 (Entanglement-breaking channels and
divisibility). Consider a qubit channel E. If det Ê < 0,
then E is entanglement-breaking, i.e. all qubit chan-
nels outside CP are entanglement breaking.
Proof. Consider the Choi-Jamiołkowski state of a
channel E written in the factorized Pauli operator ba-
sis [24] τE = 1
4
∑3
jk Rjkσj ⊗ σk and let Ê be its repre-
sentation in the Pauli operator basis. Then the matrix
identity R = ÊΦT with ΦT = diag (1, 1, −1, 1) holds.
Since det Ê < 0 it follows that det R = − det Ê > 0.
Using the aforementioned Lorentz normal decompo-
sition R = LT
1 R̃L2 with det L1,2 > 0 and R̃ diagonal,
see Ref. [24]. The transformations L1,2 correspond to
one-way stochastic local operations and classical com-
munications (SLOCC) of τE , thus, R̃ is an unnormal-
ized two-qubit state with det R̃ > 0. The channel cor-
responding to R̃ (in the Pauli basis) is Ĝ = R̃ΦT/R̃00.
Since the latter is diagonal, then G is a Pauli channel
with det Ĝ < 0. A Pauli channel has a negative deter-
minant, if either all λj are negative, or exactly one of
them is negative. In Ref. [28] it has been shown that
the set of channels with λj < 0 ∀j are entanglement
breaking channels. Now, using the symmetries of the
tetrahedron, one can generate all channels with neg-
ative determinant by concatenating this set with the
Pauli rotations. Therefore every Pauli channel with
Accepted in Quantum 2019-04-24, click title to verify 9
λ1
λ2λ3
 C
L
 C
CP\CL
 C
P\CCP
 C
div\CP
 Cdiv
 EB boundary
Figure 7: We show two slices of the unitary tetrahedron (figure in the left) determined by
∑
i
λi = 0.4 (shown in the center)
and
∑
i
λi = −0.4 (shown in the right). The non-convexity of the divisibility sets can be seen, including the set of indivisible
channels. The convexity of sets C and entanglement breaking channels can also be noticed in the slices. A central feature is
that the set Cdiv \ CP is always inside the octahedron of entanglement breaking channels.
negative determinant is entanglement breaking, thus,
τG is separable and given that SLOCC operations can
not create entanglement [11], we have that τE is sep-
arable, too. This implies that if det Ê < 0, then the
qubit channel E is entanglement-breaking.
4 Divisibility transitions and examples
with dynamical process
The aim of this section is to use illustrative examples
of quantum dynamical processes to show transitions
between divisibility types of the instantaneous chan-
nels. From the slices shown above (see figures 7 and
8) it can be noticed that every transition between the
studied divisibility types is permitted. This is due to
the existence of common borders between all combi-
nations of divisibility sets; we can think of any con-
tinuous line inside the tetrahedron [6] as describing
some quantum dynamical map.
We analyze two examples, the first is an implemen-
tation of the approximate NOT gate, ANOT through-
out a specific collision model [21]. The second is the
well known setting of a two-level atom interacting
with a quantized mode of an optical cavity [9]. We de-
fine a simple function that assigns a particular value
to a channel Et according to divisibility hierarchy, i.e.
δ[E ] =







1 if E ∈ CL ,
2/3 if E ∈ CCP \ CL ,
1/3 if E ∈ CP \ CCP ,
0 if E ∈ C \ CP .
(23)
A similar function can be defined to study the transi-
tion to/from the set of entanglement-breaking chan-
λ1
λ2
λ3
Figure 8: (left) Set of non-unital unital channels up to uni-
taries, defined by ~τ = (1/2, 0, 0), see Eq. (5). This set
lies inside the tetrahedron. For this particular case the
CP conditions reduce to the two inequalities 2 ± 2λ1 ≥
√
1 + 4(λ2 ± λ3)2. A cut corresponding to
∑
i
λi = 0.3
is presented inside and in the right, see Fig. 7 for the color
coding. The structure of divisibility sets presented here has
basically the same structure as for the unital case except for
CL. A part of the channels with negative eigenvalues belong-
ing to CL lies outside CCP \ CL, see green lines. As for the
unital case a central feature is that the channels in Cdiv \ CP
are entanglement breaking channels. Channels in the bound-
ary are not characterized due to the restricted character of
Theorem 2.
Accepted in Quantum 2019-04-24, click title to verify 10
nels, i.e.
χ[E ] =
{
1 if E is entanglement breaking ,
0 if E if not. (24)
The quantum NOT gate is defined as NOT : ρ 7→
✶ − ρ, i.e. it maps pure qubit states to its orthogo-
nal state. Although this map transforms the Bloch
sphere into itself it is not a CPTP map, and the clos-
est CPTP map is ANOT : ρ 7→ (✶ − ρ)/3. This is a
rank-three qubit unital channel, thus, it is indivisible
[25]. Moreover, det ANOT = −1/27 implies that this
channel is not achievable by a P-divisible dynamical
map. It is worth noting that ANOT belongs to Cdiv.
A specific collision model was designed in Ref. [21]
simulating stroboscopically a quantum dynamical
map that implements the quantum NOT gate ANOT
in finite time. The model reads
Et(̺) = cos2(t)̺ + sin2(t)ANOT(̺) +
1
2
sin(2t)F(̺) ,
(25)
where F(̺) = i 1
3
∑
j [σj , ̺]. This quantum dynamical
map achieves the desired gate ANOT at t = π/2.
Let us stress that this dynamical map is unital,
i.e. Et(✶) = ✶ for all t, thus, its orthogonal nor-
mal form can be illustrated inside the tetrahedron of
Pauli channels, see Fig. 9. In Fig. 10 we plot δ[Et],
χ[Et] and the value of the det Et. We see the tran-
sitions CL → CP \ CCP → Cdiv \ CP → Cdiv and
back. Notice that in both plots the trajectory never
goes through the CCP \ CL region. This means that
when the parametrized channels up to rotations be-
long to CL, so do the original ones. The transition be-
tween P-divisible and divisible channels, i.e. CP\CCP
and Cdiv\CP, occurs at the discontinuity in the yel-
low curve in Fig. 9. Let us note that this discontinu-
ity only occurs in the space of ~λ; it is a consequence
of the orthogonal normal decomposition, see Eq. (5).
The complete channel is continuous in the full con-
vex space of qubit CPTP maps. The transition from
CP \ Cdiv and back occurs at times π/3 and 2π/3. It
can also be noted that the transition to entanglement
breaking channels occurs shortly before the channel
enters in the Cdiv \ CP region; likewise, the chan-
nel stops being entanglement breaking shortly after
it leaves the Cdiv \ CP region.
Consider now the dynamical map induced by a two-
level atom interacting with a mode of a boson field.
This model serves as a workhorse to explore a great
variety of phenomena in quantum optics [8]. Using the
well known rotating wave approximation one arrives
to the Jaynes-Cummings model [12], whose Hamilto-
nian is
H =
ωa
2
σz + ωf
(
a†a +
1
2
)
+ g
(
σ−a† + σ+a
)
. (26)
By initializing the environment in a coherent state
|α〉, one gets the familiar collapse and revival setting.
Considering a particular set of parameters shown
λ1
λ2 λ3
Figure 9: (top left) Tetrahedron of Pauli channels with
the trajectory, up to rotations, of the quantum dynamical
map Eq. (25) leading to the ANOT gate, as a yellow curve.
(right) Cut along the plane that contains the trajectory; there
one can see the different regions where the channel passes.
For this case, the characterization of the CL of the channels
induced gives the same conclusions as for the corresponding
Pauli channel, see Eq. (5). The discontinuity in the trajec-
tory is due to the reduced representation of the dynamical
map, see Eq. (25); the trajectory is continuous in the space
of channels. See Fig. 7 for the color coding.
.
0
π
3
π
2
2 π
3
π
0
1
3
2
3
1
δ
[E
t
],
d
et
E
t
,
χ
[E
t
]
t
C
P\CCP
C
CP\CL
C
L
C
div\CP
Figure 10: Evolution of divisibility, determinant, and entan-
glement breaking properties of the map induced by Eq. (25),
see Eq. (23) and Eq. (24). Notice that the channel ANOT,
implemented at t = π/2, has minimum determinant. The
horizontal gray dashed lines show the image of the function
δ, with the divisibility types in the right side. It can be
seen that the dynamical map explores the divisibility sets as
CL → CP \CCP → Cdiv \CP → Cdiv and back. The channels
are entanglement breaking in the expected region.
Accepted in Quantum 2019-04-24, click title to verify 11
3.85 3.9 3.95
0 2 4 6 8 10
0
1
3
2
3
1
δ
[E
t
],
p
e
(t
),
χ
[E
t
]
t
C
P\CCP
C
CP\CL
C
L
C
div\CP
Figure 11: Black and red curves show functions δ and χ of the
channels induced by the Jaynes-Cummings model over a two-
level system, see Eq. (26) with the environment initialized in
a coherent state |α〉. The blue curve shows the probability
of finding the two-level atom in its excited state, pe(t). The
figure shows that the fast oscillations in δ occur roughly at
the same frequency as the ones of pe(t), see the inset. Notice
that there are fast transitions between CP \ CCP and CCP \ CL
occurring in the region of revivals, with a few transitions
between CCP \CP and CL in the second revival. The function
χ shows that during revivals channels are not entanglement
breaking, but we find that channels belonging to Cdiv \CP are
always entanglement breaking, in agreement with theorem 6.
The particular chosen set of parameters are α = 6, g = 10,
ωa = 5, and ωf = 20.
in Fig. 11, we constructed the channels parametrized
by time numerically, and studied their divisibility and
entanglement-breaking properties. In the same fig-
ure we plot functions δ[Et] and χ[Et], together with
the probability of finding the atom in its excited state
pe(t), to study and compare the divisibility properties
with the features of the collapses and revivals. The
probability pe(t) is calculated choosing the ground
state of the free Hamiltonian ωa/2σz of the qubit,
and it is given by [13]:
pe(t) =
〈σz(t)〉 + 1
2
, (27)
where
〈σz(t)〉 = −
∞
∑
n=0
Pn
(
∆2
4Ω2
n
+
(
1 − ∆2
4Ω2
n
)
cos (2Ωnt)
)
,
with Pn = e−|α|2 |α|2n/n!, Ωn =
√
∆2/4 + g2n and
∆ = ωf − ωa the detuning.
The divisibility indicator function δ exhibits an os-
cillating behavior, roughly at the same frequency of
pe(t), see inset in Fig. 11. The figure shows fast peri-
odic transitions between CP \ CCP and CCP \ CL oc-
curring in the region of revivals. There are also few
transitions among CCP \ CP and CL in the second re-
vival. Respect to the entanglement breaking and the
function χ, there are no fast transitions in the former,
and during revivals, channels are not entanglement
breaking. We also observe that channels belonging
to Cdiv \ CP are entanglement breaking, supporting
theorem 6 for the non-unital case.
5 Conclusions
We studied the relations between different types of di-
visibility of time-discrete and time-continuous quan-
tum processes, i.e. channels and dynamical maps, re-
spectively. In particular, we investigated classes of
channels by means of their achievability by dynamical
maps of different divisibility types, and also the divis-
ibility of channels occurring during the time evolu-
tions. Apart from investigating the relations between
these concepts in general, we provided a detailed anal-
ysis for the case of qubit channels.
We implemented the known conditions to decide
CL for the general diagonalizable case, and a discus-
sion of the parametric space of Lindblad generators
was given (clarifying one of the results of the paper
[26]). For unital qubit channels it was shown that
every infinitesimal divisible map can be written as a
concatenation of one CL channel and two unitary con-
jugations. For the particular case of Pauli channels
case, we have shown that the sets of infinitely divis-
ible and L-divisible channels coincide. We made an
interesting observation, connecting the concept of di-
visibility with the quantum information paradigm of
entanglement-breaking channels. We found that di-
visible but not infinitesimal divisible qubit channels,
in PTP maps, are necessarily entanglement-breaking.
We also noted that the intersection of indivisible and
P-divisible channels is not empty. This allows us to
implement indivisible channels with infinitesimal PTP
maps. Finally, we questioned the existence of dynam-
ical transitions between different classes of divisibility
channels. We argued that all the transitions are, in
principle, possible, and exploited two simple models
of dynamical maps to demonstrate these transitions.
They clearly illustrate how the channels evolutions
change from being implementable by markovian dy-
namical maps to non-markovian, and vice versa.
There are several directions how to proceed further
in investigation of divisibility of channels and dynam-
ical maps. Apart from extension of this analysis to
larger-dimensional systems, a plethora of interesting
questions are related to design of efficient verification
procedures of the divisibility classes for channels and
dynamical maps. In this paper we question the divisi-
bility features of snapshots of the evolution, however,
it might be of interest to understand when the time
intervals of dynamical maps implemented by non-
markovian evolutions, can be simulated by markovian
dynamical maps. Also the area of channel divisibility
contains several open structural questions, e.g. the
existence of at most n-divisible channels.
Acknowledgements
We acknowledge Thomas Gorin and Tomáš Rybár for
useful discussions, as well PAEP and RedTC for finan-
cial support. Support by projects CONACyT 285754,
Accepted in Quantum 2019-04-24, click title to verify 12
UNAM-PAPIIT IG100518, IN-107414, APVV-14-
0878 (QETWORK) is acknowledged. CP acknowl-
edges support by PASPA program from DGAPA-
UNAM. MZ acknowledges the support of VEGA
2/0173/17 (MAXAP) and GAČR project no. GA16-
22211S.
A On Lorentz normal forms of Choi-
Jamiolkowski state
In this appendix we compute the Lorentz normal de-
composition of a channel for which one gets b 6= 0,
supporting our observation that Lorentz normal de-
composition does not take Choi-Jamiołkowski states
to something proportional to a Choi-Jamiołkowski
state. Consider the following Kraus rank three chan-
nel and its RE matrix, both written in the Pauli basis:
Ê =




1 0 0 0
0 − 1
3 0 0
0 0 − 1
3 0
2
3 0 0 1
3




, (28)
and
RE =




1 0 0 0
0 − 1
3 0 0
0 0 1
3 0
2
3 0 0 1
3




. (29)
Using the algorithm introduced in Ref. [24] to calcu-
late RE ’s Lorentz decomposition into orthochronous
proper Lorentz transformations we obtain
L1 =
1
γ1




4 0 0 1
0 −γ1 0 0
0 0 −γ1 0
1 0 0 4




, (30)
L2 =
1
γ2




89 + 9
√
97 0 0 −8
0 −γ2 0 0
0 0 −γ2 0
−8 0 0 89 + 9
√
97




,
and
ΣE =
1
γ3






√
11 + 109√
97
0 0 −
√
97+1√
89
√
97+873
0 − γ3
3 0 0
0 0 γ3
3 0
√
1 + 49√
97
0 0
√
−1 + 49√
97






with γ1 =
√
15, γ2 = 3
√
178
√
97 + 1746, and γ3 =√
30. Although the central matrix ΣE is not exactly
of the form Eq. (6), it is equivalent. To see this no-
tice that the derivation of the theorem 2 in [24] con-
siders only decompositions into proper orthochronous
Lorentz transformations. But to obtain the desired
form, the authors change signs until they get Eq. (6);
this cannot be done without changing Lorentz trans-
formations. If we relax the condition over L1,2 of
being proper and orthochronous, we can bring ΣE
to the desired form by conjugating ΣE with G =
diag (1, 1, 1, −1):
G−1ΣEG =
1
γ3






√
11 + 109√
97
0 0
√
97+1√
89
√
97+873
0 − γ3
3 0 0
0 0 γ3
3 0
−
√
1 + 49√
97
0 0
√
−1 + 49√
97






.
In both cases (taking ΣE or G−1ΣEG as the normal
form of RE), the corresponding channel is not pro-
portional to a trace-preserving one since b 6= 0, see
Eq. (6). This completes the counterexample.
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Accepted in Quantum 2019-04-24, click title to verify 14
114 Chapter C. Articles
C.2 Article: Gaussian quantum channels beyond the Gaus-
sian functional form: full characterization of the one-
mode case
The following article was sent to Physical Review A. Click to go to arXiv.
Gaussian quantum channels beyond the Gaussian functional form: full
characterization of the one-mode case
David Davalos,1 Camilo Moreno,2 Juan-Diego Urbina,2 and Carlos Pineda1
1Instituto de F́ısica, Universidad Nacional Autónoma de México, México∗
2Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
We study one-mode Gaussian quantum channels in continuous-variable systems by performing
a black-box characterization using complete positivity and trace preserving conditions, and report
the existence of two subsets that do not have a functional Gaussian form. Our study covers as
particular limit the case of singular channels, thus connecting our results with their known classi-
fication scheme based on canonical forms. Our full characterization of Gaussian channels without
Gaussian functional form is completed by showing how Gaussian states are transformed under these
operations, and by deriving the conditions for the existence of master equations for the non-singular
cases.
PACS numbers: 03.65.Yz, 03.65.Ta, 05.45.Mt
I. INTRODUCTION
Within the theory of continuous-variable quantum sys-
tems (a central topic of study given their role in the
description of physical systems like the electromagnetic
field [1], solids and nano-mechanical systems [2] and
atomic ensembles [3]) the simplest states, both from a
theoretical an experimental point of view, are the so-
called Gaussian states. An operation that transforms
such family of states into itself is called a Gaussian quan-
tum channel (GQC). Even though Gaussian states and
channels form small subsets among general states and
channels, they have proven to be useful in a variate of
tasks such as quantum communication [4], quantum com-
putation [5] and the study of quantum entanglement in
simple [6] and complicated scenarios [7].
Writing Gaussian channels in the position state repre-
sentation is often of theoretical convenience, for instance
for the calculation of position correlation functions. Thus
an obvious way to proceed is to characterize the possible
functional forms of GQC in such representation. First at-
tempts in this direction were given in Ref. [8], but their
ansatz is limited to only Gaussian functional forms (de-
noted simply by Gaussian forms or GF). Going beyond
such restrictive assumption, in the present work we char-
acterize another two possible forms that can arise directly
from the definition of Gaussian channel in the one-mode
case. We thus give a complete characterization of GQC in
position state representation, and study the special case
of singular Gaussian quantum channels (SGQC), i.e. the
operations for which the inverse operation doesn’t exist.
The paper is organized as follows. In section II we
discuss the definition of GQC and introduce functional
forms beyond the GF that emerge from singularities in
the coefficients that define a GQC with GF. In section III
we give a black-box characterization of such channels, us-
ing complete positivity and trace preserving conditions.
∗ davidphysdavalos@gmail.com
In section IV we study functional forms that lead to
SGQC and derive their explicit form. Finally in section V
we derive conditions of existence of master equations and
their explicit forms. We conclude in section VI.
II. GAUSSIAN QUANTUM CHANNELS
Gaussian states are characterized completely by first
(mean) and second (correlations) moments encoded in
the mean vector d⃗ and the covariance matrix σ. There-
fore, a Gaussian state S can be denoted as S = S (σ, d⃗),
where for the one-mode case we have
σ = ( ⟨q̂2⟩ − ⟨q̂⟩2 1
2
⟨q̂p̂ + p̂q̂⟩ − ⟨q̂⟩⟨p̂⟩
1
2
⟨q̂p̂ + p̂q̂⟩ − ⟨q̂⟩⟨p̂⟩ ⟨p̂2⟩ − ⟨p̂⟩2 ) ,
and
d⃗ = (⟨q̂⟩, ⟨p̂⟩)T
with q̂ and p̂ denoting the standard position and momen-
tum (quadrature) operators [9].
To start with, we recall the following definition [10]:
Definition 1 (Gaussian quantum channels). A quantum
channel is Gaussian (GQC) if and only if it transforms
Gaussian states into Gaussian states.
This definition is strictly equivalent to the statement
that any GQC, say G, can be written as
G[ρ] = trE [U (ρ⊗ ρE)U †] (1)
where U is a unitary transformation, acting on a com-
bined global state obtained from enlarging the system
with an environment E, that is generated by a quadratic
bosonic Hamiltonian (i.e. U is a Gaussian unitary) [10].
The environmental initial state ρE is a Gaussian state
and the trace is taken over the environmental degrees of
freedom.
Following definition 1, a GQC is fully characterized
by its action over Gaussian states, and this action is in
ar
X
iv
:1
9
0
2
.0
7
1
6
3
v
1
  
[q
u
an
t-
p
h
] 
 1
9
 F
eb
 2
0
1
9
2
turn defined by affine transformations [10]. Specifically,G = G (T,N, τ⃗) is given by a tuple (T,N, τ⃗) where T
and N are 2 × 2 real matrices with N = NT [10] acting
on Gaussian states according to G (T,N, τ⃗) [S (σ, d⃗)] =
S (TσTT +N,Td⃗ + τ⃗). In the particular case of closed
systems we have N = 0 and T is a symplectic matrix.
Let us note that although channels with Gaussian form
trivially transform Gaussian states into Gaussian states,
the definition goes beyond GF. Introducing difference
and sum coordinates, x = q2 − q1 and r = (q1 + q2)/2,
and ρ(x, r) = ⟨r − x
2
∣ ρ̂ ∣r + x
2
⟩, a quantum channel
ρf (xf , rf) = ∫
R2
dxidriJ(xf , xi; rf , ri)ρi (ri, xi) , (2)
maps an initial ρ̂i into a final ρ̂f state linearly through
the kernel J(xf , xi; rf , ri). In order to see how a chan-
nel without GF can be constructed as a limiting case
of a quantum channel with GF, consider the general
parametrization of the later as given in [12]
JG(xf , xi; rf , ri) = b3
2π
exp [ı(b1xfrf + b2xfri + b3xirf
+ b4xiri + c1xf + c2xi) − a1x2
f − a2xfxi − a3x2
i ], (3)
where all coefficients are real and no quadratic terms in
ri,f are allowed. Now it is easy to see that if the co-
efficients of the quadratic form in the exponent of J in
eq. (3) depend on a parameter ǫ such that for ǫ→ 0 they
scale as an ∝ ǫ−1 and bn ∝ ǫ−1/2, then
lim
ǫ→0
JG(xf , xi; rf , ri) =N δ(αxf −βxi)eΣ′(xf ,xi;rf ,ri), (4)
where α, β ∈ R and Σ′(xf , xi; rf , ri) is a quadratic form
that now admits quadratic terms in ri,f . This is the
first example of a δGQC, namely a Gaussian quantum
channel that contains Dirac-delta functions in its coordi-
nate representation. This particular example is not only
of academic interest. Physically, it can be implemented
by means of the ubiquitous quantum Brownian motion
model for harmonic systems (damped harmonic oscilla-
tor) [13]. In such system δGQC occur at isolated points
of time, defined in the limit of the antisymmetric position
autocorrelation function tending to zero.
Since the form of eq. (4) admits quadratic terms in
ri,f , it suggest that a form with two deltas can exist and
can be defined with an appropriate limit. In order to
avoid working with such limits, in this work we provide
a black-box characterization of general GQCs without
Gaussian form. In particular we study channels that can
arise when singularities on the coefficients of Gaussian
forms GF occur, that lead immediately to singular Gaus-
sian operations. We characterize which forms in δGQC
lead to valid quantum channels, and under which condi-
tions singular operations lead to valid singular quantum
channels (SGQC). We will show that only two possible
forms of δGQC hold according to trace preserving (TP)
and hermiticity preserving (HP) conditions. The chan-
nel of eq. (4) is one of these forms, as expected. Later on
we will impose complete positivity in order to have valid
GQC, i.e. complete positive and trace preserving (CPTP)
Gaussian operations, going beyond previous characteri-
zations of GQC [12].
III. COMPLETE POSITIVE AND
TRACE-PRESERVING δ−GAUSSIAN
OPERATIONS
Let us introduce the ansätze for the possible forms
of GQC in the position representation, to perform the
black-box characterization. Following eq. (1) and taking
the continuous variable representation of difference and
sum coordinates, the trace becomes an integral over posi-
tion variables of the environment. Then we end up with
a Fourier transform of a multivariate Gaussian, having
for one mode the following structures: a Gaussian form
eq. (3), a Gaussian form multiplied with one-dimensional
delta or a Gaussian form multiplied by a two-dimensional
delta. Thus, in order to start with the black-box charac-
terization, we shall propose the following general Gaus-
sian operations with one and two deltas, respectively
JI(xf , rf ;xi, ri) =NIδ(α⃗Tv⃗f + β⃗Tv⃗i)eΣ(xf ,xi;rf ,ri), (5)
JII(xf , rf ;xi, ri) =NIIδ(Av⃗f −Bv⃗i)eΣ(xf ,xi;rf ,ri), (6)
where the exponent reads
Σ(xf , xi; rf , ri) = ı(b1xfrf + b2xfri + b3xirf
+ b4xiri + c1xf + c2xi + d1rf + d2ri)
−a1x2
f − a2xfxi − a3x2
i − e1r2f − e2rfri − e3r2i , (7)
v⃗i,j = (ri,j , xi,j) and NI,II are normalization constants.
They provide, together with eq. (3) all possible ansätze
for GQC. Note that the coefficients in the exponential of
every form must be finite, otherwise the functional form
can be modified.
Let us study now CPTP conditions, since complete pos-
itivity implies positivity and in turn it implies hermiticity
preserving (HP). For sum and difference coordinates HP
reads
J(−xf , rf ;−xi, ri) = J(xf , rf ;xi, ri)∗. (8)
Following this equation, it is easy to note that the coef-
ficients an, bn, cn, en must be real with dn = 0, as well
the entries of matrices (and vectors) A, B, α⃗, β⃗. The
factor concerning the delta function of eq. (5), is reduced
into two possible combinations variables: i) δ(αxf −βxi)
and ii) δ(αrf −βri). For the case of eq. (6), the prefactor
concerning the two-dimensional delta is reduced to iii)
δ(γrf − ηri)δ(αxf − βxi). Let us now analyze the trace
preserving condition (TP), which for continuous variable
systems reads
∫
R
drfJ(xf = 0, rf ;xi, ri) = δ(xi). (9)
3
This condition immediately discards ii) from the above
combinations of deltas, thus we end up with cases i) and
iii). For case i) TP reads
NI ∫ drfδ(−βxi)eΣ = NI∣β∣
√
π
e1
δ(xi)e e2
2
r2
i
4e1 e−e3r
2
i , (10)
thus the relation between the coefficients assumes the
form
e22
4e1
− e3 = 0, (11)
and the normalization constant NI = ∣β∣√ e1
π
with β ≠ 0
and e1 > 0. For case iii) the trace-preserving condition
reads
NII ∫ drfδ(γrf − ηri)δ(−βxi)eΣ
= NII∣βγ∣δ(xi)e−e1( ηγ )2r2i −e2 η
γ
r2i −e3r2i .
Thus the following relation between en coefficients must
be fulfilled
e1
⎛⎝ηγ⎞⎠
2 + e2 η
γ
+ e3 = 0, (12)
with γ, β ≠ 0 and NII = ∣βγ∣. In the particular case of
η = 0, eq. (12) is reduced to e3 = 0. As expected from
the analysis of limits above, we showed that δGQC admit
quadratic terms in ri,j .
Up to this point we have hermitian and trace preserv-
ing Gaussian operations; to derive the remaining CPTP
conditions, it is useful to write Wigner’s function and
Wigner’s characteristic function, which we now derive.
The representation of the Wigner’s characteristic func-
tion reads
χ(k⃗) = tr [ρD(k⃗)]
= exp [−1
2
k⃗T (ΩσΩT) k⃗ − ı (Ω⟨x̂⟩)T k⃗] (13)
and its relation with Wigner’s function
W (x) = ∫
R2
dx⃗e−ıx⃗
TΩk⃗χ (k⃗) (14)
= ∫
R
eıpxdx ⟨r − x
2
∣ ρ̂ ∣r + x
2
⟩ . (15)
where k⃗ = (k1, k2)T, x⃗ = (r, p)T and h̵ = 1 (we are using
natural units). Using the previous equations to construct
Wigner and Wigner’s characteristic functions of the ini-
tial and final states, and substituting them in eq. (2), it
is straightforward to get the propagator in the Wigner’s
characteristic function representation
J̃ (k⃗f , k⃗i) = ∫
R6
dΓK(l⃗)J(v⃗f , v⃗i), (16)
where the transformation kernel reads
K(l⃗) = 1(2π)3 e[ı(k
f
2
rf−kf
1
pf−ki
2ri+ki
1pi−pixi+pfxf)],
with
dΓ = dpfdpidxfdxidrfdri and
l⃗ = (pf , pi, xf , xi, rf , ri)T .
By elementary integration of eq. (16) one can show that
for both cases
J̃I,III (k⃗f , k⃗i) = δ (ki1 − α
β
kf1) δ (ki2 − φ⃗T
I,IIIk⃗f) ePI,III(k⃗f ),
(17)
where PI,III(k⃗f) = ∑2
i,j=1 P
(I,III)
ij kfi k
f
j +∑2
i=1 P
(I,III)
0i kfi with
P
(I,III)
ij = P (I,III)ji . For case i) we obtain
P
(I)
11 = −((αβ )
2 (a3 + b23
4e1
) + α
β
(a2 + 1
2
b1b3
e1
) + a1 + b21
4e1
) ,
P
(I)
12 = −(αβ b3
2e1
+ b1
2e1
) ,
P
(I)
22 = − 1
4e1
. (18)
For case iii) we have
P
(III)
11 = −((α
β
)2 a3 + α
β
a2 + a1) ,
P
(III)
12 = P (III)
22 = 0, (19)
And for both cases we have P
(I,III)
01 = ı (α
β
c2 + c1) and
P
(I,III)
02 = 0. Vectors φ⃗ are given by
φ⃗I = (α
β
(b4 − b3e2
2e1
) − b1e2
2e1
+ b2,− e2
2e1
)T ,
φ⃗III = (α
β
η
γ
b3 + α
β
b4 + η
γ
b1 + b2, η
γ
)T . (20)
We are now in position to write explicitly the condi-
tions for complete positivity. Having a Gaussian opera-
tion characterized by (T,N, τ⃗), the CP condition can be
expressed in terms of the matrix
C =N + ıΩ − ıTΩTT, (21)
where Ω = ( 0 1−1 0
) is the symplectic matrix. An op-
eration G (T,N, τ⃗) is CP if and only if C ≥ 0 [10, 14].
Applying the propagator on a test characteristic func-
tion, eq. (13), it is easy to compute the corresponding
tuples. For both cases we get (TI,III,NI,III, τ⃗I,III):
NI,III = 2( −P22 P12
P12 −P11
) ,
τ⃗I,III = (0, ıP (I,III)01 )T , (22)
4
while for case i) matrix T is given by
TI = ( e2
2e1
0
φ⃗I,1 −α
β
) , (23)
where φ⃗I,1 denotes the first component of vector φ⃗I, see
eq. (20). The complete positive condition is given by the
inequalities raised from the eigenvalues of matrix eq. (21)
±
√
α2e22 + 4αβe2e1 + 4β2e21 (4P (I)12
2 + (P (I)11 − P (I)22 )2 + 1)
2βe1− (P (I)11 + P (I)22 ) ≥ 0. (24)
For case iii) matrix T is
TIII = ( − η
γ
0
φ⃗III,1 −α
β
) , (25)
and complete positivity conditions read
±
√(βγ − αη)2 + β2γ2P
(III)
11
2
βγ
− P (III)11 ≥ 0. (26)
Note that in both cases complete positivity conditions do
not depend on φ⃗.
IV. ALLOWED SINGULAR FORMS
There are two classes of Gaussian singular channels.
Since the inverse of a Gaussian channel G (T,N, τ⃗) isG (T−1,−T−1NT−T ,−T−1τ⃗), its existence rests on the
invertibility of T. Thus studying the rank of the lat-
ter we are able to explore singular forms. We are going
to use the classification of one-mode channels developed
by Holevo [15].
For singular channels there are two classes character-
ized by its canonical form [16], i.e. any channel can be
obtained by applying Gaussian unitaries before and after
the canonical form. The class called “A1” correspond to
singular channels with Rank (T) = 0 and coincide with
the family of total depolarizing channels. The class “A2”
is characterized by Rank (T) = 1. Both channels are
entanglement-breaking [16].
Before analyzing the functional forms constructed in
this work, let us study channels with GF. The tuple of the
affine transformation, corresponding to the propagator
JG, eq. (3), were introduced in Ref. [12] up to some typos.
Our calculation for this tuple is
TG = ( − b4
b3
1
b3
b1b4
b3
− b2 − b1
b3
) ,
NG = ⎛⎝
2a3
b2
3
a2
b3
− 2a3b1
b2
3
a2
b3
− 2a3b1
b2
3
−2(−a3b
2
1
b2
3
+ a2b1
b3
− a1)
⎞⎠ ,
τ⃗G = (−c2
b3
,
b1c2
b3
− c1)T . (27)
It is straightforward to check that for b2 = 0, TG is singu-
lar with Rank (TG) = 1, i.e. it belongs to class A2. Due
to the full support of Gaussian functions, it was surpris-
ing that Gaussian channels with GF have singular limit.
In this case the singular behavior arises from the lack of
a Fourier factor for xfri. This is the only singular case
for GF.
Now we analyze functional forms derived in sec. III.
The complete positivity conditions of the form J̃III, pre-
sented in eq. (26), have no solution for α → 0 and/or
γ → 0, thus this form cannot lead to singular channels.
This is not the case for J̃I, eq. (17), which leads to sin-
gular operations belonging to class A2 for
αe2 = 0, (28)
and to class A1 for
e2 = α = b2 = 0. (29)
For the latter, the complete positivity conditions read
e1 ≤ a1. (30)
By using an initial state characterized by σi and d⃗i we
can compute the explicit dependence of the final states
on the initial parameters. For channels belonging to class
A2 [see eq. (27) with b2 = 0 and eq. (23) with e2α = 0] the
final state only depends one combination of the compo-
nents of σi, and in one combination of the components of
d⃗i, i.e. ∑mn lmn (σi)mn and ∑m nm (d⃗i)m, respectively,
where lmn and nm depend on the channel parameters.
See the appendix for the explicit formulas and fig. 1 for an
schematic description of the final states. From such com-
binations it is obvious that we cannot solve for the initial
state parameters given a final state as expected; this is
because the parametric space dimension is reduced from
5 to 2. The channel belonging to A1 [see eq. (23) with
e2 = α = b2 = 0 and eq. (30)] maps every initial state to
a single one characterized by σf =N and d⃗f = (0,−c1)T,
see fig. 2 for a schematic description.
According to our ansätze [see equations (5) and 6)], we
conclude that one-mode SGQC can only have the func-
tional forms given in eq. (3) and eq. (5). This is the
central result our work and can be stated as:
Theorem 1 (One-mode singular Gaussian channels). A
one-mode Gaussian quantum channel is singular if and
only if it has one of the following functional forms in the
position space representation
1. b3
2π
exp [ı(b1xfrf + b3xirf
+ b4xiri + c1xf + c2xi) − a1x2
f − a2xfxi − a3x2
i ],
2. ∣β∣√e1/πδ(αxf − βxi)× exp [ − a2xfxi − a1x2
f − a3x2
i
+ı(b2xfri + b3rfxi + b1rfxf + b4rixi + c1xf + c2xi)
−e1r2f − e2rfri − e22r
2
i
4e1
], with e2α = 0.
5
Corollary 1 (Singular classes). A one-mode singular
Gaussian channel belongs to class A1 if and only if its
position representation has the following form:
√
e1/πδ(xi) exp [−a1x2
f +ı(b2xfri+b1rfxf +c1xf)−e1r2f ].
Otherwise the channel belongs to class A2.
Since channels on each class are connected each other
by unitary conjugations [15], a consequence of the the-
orem and the subsequent corollary is that the set of al-
lowed forms must remain invariant under unitary conju-
gations. To show this we must know the possible func-
tional forms of Gaussian unitaries. They are given by
following lemma for one mode
Lemma 1 (One-mode Gaussian unitaries). Gaussian
unitaries can have only GF or the one given by eq. (6).
Proof. Recalling that for a unitary GQC, Tmust be sym-
plectic (TΩTT = Ω) and N = 0. However, an inspec-
tion to eq. (18) lead us to note that N ≠ 0 unless e1 di-
verges. Thus Gaussian unitaries cannot have the form JI
[see eq. (5)]. An inspection of matrices T and N of GQC
with GF [see eq. (27)] and the ones for JII [see equations
(19) and (25)] lead us to note the following two observa-
tions: (i) in both cases we have N = 0 for an = 0 ∀n;
(ii) the matrix T is symplectic for GF when b2 = b3, and
when αη = βγ for JII. In particular the identity map has
the last form. This completes the proof.
One can now compute the concatenations of the
SGQCs with Gaussian unitaries. This can be done
straightforward using the well known formulas for Gaus-
sian integrals and the Fourier transform of the Dirac
delta. Given that the calculation is elementary, and for
sake of brevity, we present only the resulting forms of
each concatenation. To show this compactly we intro-
duce the following abbreviations: Singular channels be-
longing to class A2 with form JI and with α = 0, e2 = 0
and α = e2 = 0, will be denoted as δαA2
, δe2A2
and δα,e2A2
, re-
spectively; singular channels belonging to the same class
but with GF will be denoted as GA2
; channels belong-
ing to class A1 will be denoted as δA1
; finally Gaussian
unitaries with GF will be denoted as GU and the ones
with form JII as δU . Writing the concatenation of two
channels in the position representation as
J(f)(xf , rf ;xi, ri) =
∫
R2
dx′dr′J (1) (xf , rf ;x
′, r′)J(2) (x′, r′;xi, ri) , (31)
the resulting functional forms for J(f) are given in table I.
As expected, the table shows that the integral has only
the forms stated by our theorem. Additionally it shows
the cases when unitaries change the functional form of
class A2, while for class A1 J(f) has always the unique
form enunciated by the corollary.
J(1) J(2) J(f)
δαA2
GU GA2
GU δαA2
δαA2
δαA2
δU δαA2
δU δαA2
δαA2
δe2A2
GU δe2A2
GU δe2A2
GA2
δe2A2
δU δe2A2
δU δe2A2
δe2A2
GU , δ
α,e2
A2
δ
α,e2
A2
δ
α,e2
A2
GU GA2
δU , δ
α,e2
A2
δ
α,e2
A2
δ
α,e2
A2
δU δ
α,e2
A2
δU ,GU δA1
δA1
δA1
δU ,GU δA1
TABLE I. The first and second columns show the functional
forms of J(1) and J(2), respectively. The last column shows
the resulting form of the concatenation of them [see eq. (31)].
See main text for symbol coding.
V. EXISTENCE OF MASTER EQUATIONS
In this section we show the conditions under which
master equations, associated with the channels derived
in sec. III, exist. To be more precise, we study if the
functional forms derived above parametrize channels be-
longing to one-parameter differentiable families of GQCs.
As a first step, we let the coefficients of forms presented
in equations (5) and (6) to depend on time. Later we de-
rive the conditions under which they bring any quantum
state ρ(x, r; t) to ρ(x, r; t + ǫ) (with ǫ > 0 and t ∈ [0,∞))
smoothly, while holding the specific functional form of
the channel, i.e.
ρ(x, r; t + ǫ) = ρ(x, r; t) + ǫLt [ρ(x, r; t)] +O(ǫ2), (32)
where both ρ(x, r; t) and ρ(x, r; t+ǫ) are propagated from
t = 0 with channels either with the form JI or JII, and Lt
is a bounded superoperator in the state subspace. This is
basically the problem of the existence of a master equa-
tion
∂tρ(x, r; t) = Lt [ρ(x, r; t)] , (33)
for such functional forms. Thus the problem is reduced
to prove the existence of the linear generator Lt, also
known as Liouvillian.
To do this we use an ansatz proposed in Ref. [17] to
investigate the existence and derive the master equation
6
Class A2
r
p
(σf )11 (s1)
(σf )22 (s1)
(σf )12 (s1)
~di
7→
~df
(s2
)
(
σi,
~di
)
FIG. 1. Schematic picture of the channels belonging to class
A2, acting on Wigner functions of Gaussian states. The ex-
plicit dependence of the final state in terms of the combina-
tions s1 and s2 are presented in the appendix. As well the
formulas for si depending on the form of the channel. The pic-
tured coordinate system corresponds to the position variable
r and its conjugate momentum.
for GFs,
L = Lc(t) + (∂x, ∂r)X(t)⎛⎝∂x∂r
⎞⎠
+ (x, r)Y(t)⎛⎝∂x∂r
⎞⎠ + (x, r)Z(t)⎛⎝xr⎞⎠ (34)
where Lc(t) is a complex function and
X(t) = ⎛⎝Xxx(t) Xxr(t)
Xrx(t) Xrr(t)
⎞⎠ (35)
is a complex matrix as well as Y(t) and Z(t), whose
entries are defined in a similar way as in eq. (35). Note
that X(t) and Z(t) can always be chosen symmetric, i.e.
Xxr = Xrs and Zxr = Zrx. Thus we must determine 11
time-dependent functions from eq. (34). This ansatz is
also appropriate to study the functional forms introduced
in this work, given that the left hand side of eq. (33) only
involves quadratic polynomials in x, r, ∂/∂x and ∂/∂r,
as in the GF case.
Notice that singular channels do not admit a master
equation since its existence implies that channels with the
functional form involved can be found arbitrarily close
from the identity channel. This is not possible for sin-
gular channels due to the continuity of the determinant
of the matrix T. This fact can be also shown using the
ansatz of eq. (34), one finds infinitely Liuville operators,
thus the master equation is not well defined.
For the non-singular cases presented in equations (5)
and (6), the condition for the existence of a master equa-
Class A1
r
p
1
2e1
2a1 +
b
2
1
2e1
−
b1
2e1
(0,−c1)
T
(
σi,
~di
)
FIG. 2. Schematic picture of the channels belonging to class
A1, acting on Wigner functions of Gaussian states. Every
channel of this class maps every initial quantum state, in par-
ticular GSs characterized by (σi, d⃗i), to a Gaussian state that
depends only on the channel parameters. We indicate in the
figure the values of the corresponding components of the first
and second moments of the final Gaussian state. The pictured
coordinate system corresponds to the position variable r and
its conjugate momentum.
tion is obtained as follows. (i) Substitute the ansatz
of eq. (34) in the right hand side of the eq. (33). (ii)
Define ρ(x, r; t) using eq. (2), given an initial condi-
tion ρ(x, r; 0), for each functional form JI,II. (iii) Take
ρf(xf , rf) → ρ(x, r; t) and ρi(xi, ri) → ρ(x, r; 0). Fi-
nally, (iv) compare both sides of eq. (33). Defining
A(t) = α(t)/β(t) and B(t) = γ(t)/η(t), the conclusion
is that for both JI and JII, a master equations exist if
c(t)∝ A(t) (36)
holds, where c(t) = c1(t) + A(t)c2(t). Additionally, for
the form JI the solutions for the matrices X(t), Y(t)
7
and Z(t) are given by
Xxx =Xxr = Yrx = Zrr = 0,
Yxx = Ȧ
A
,
Lc = Yrr = ė1
e1
− ė2
e2
,
Xrr = ė1
4e21
− ė2
2e1e2
,
Yxr = ı(λ1ė2
e1e2
+ λ2Ȧ
e2A
− λ1ė1
2e21
− λ̇2
e2
) ,
Zxx = λ2
1
2
⎛⎝ ė2
e1e2
− ė1
2e21
⎞⎠ + λ1
e2
⎛⎝λ2Ȧ
A
− λ̇2
⎞⎠ + 2λ3
Ȧ
A
− λ̇3,
Zxr = ı⎛⎝ ȦA⎛⎝e1λ2
e2
− λ1
2
⎞⎠ + λ̇1
2
− λ̇2e1
e2
+ λ2
2
⎛⎝ ė2e2 −
ė1
e1
⎞⎠⎞⎠ ,
(37)
where we have defined the following coefficients: λ1 =
b1 +Ab3, λ2 = b2 +Ab4 and λ3 = a1 +Aa2 +A2a3.
For the form JII the solutions are the following
Lc =Xxx =Xxr =Xrr = Zrr = 0,
Yrx = Yxr = 0,
Yxx = Ȧ
A
, Yrr = Ḃ
B
.
Zxx = a2(t)Ȧ(t) + 2a1(t)Ȧ(t)
A(t) −A(t)2
−ȧ3(t) −A(t)ȧ2(t) − ȧ1(t),
Zxr = ı⎛⎝12 λ̇ − λ
2
⎛⎝ ȦA + Ḃ
B
⎞⎠⎞⎠ ,
(38)
where λ = b1 +Ab3 +B(b2 +Ab4).
VI. CONCLUSIONS
In this work we have critically reviewed the deceptively
natural idea that Gaussian quantum channels always ad-
mit a Gaussian functional form. To this end, we went
beyond the pioneering characterization of Gaussian chan-
nels with Gaussian form presented in Ref. [12] in two new
directions. First we have shown that, starting from their
most general definition as mapping Gaussian states into
Gaussian states, a more general parametrization of the
coordinate representation of the one-mode case exists,
that admits non-Gaussian functional forms. Second, we
were able to provide a black-box characterization of such
new forms by imposing complete positivity (not consid-
ered in Ref. [12]) and trace preserving conditions. While
our parametrization connects with the analysis done by
Holevo [16] in the particular cases where besides hav-
ing a non-Gaussian form the channel is also singular, it
also allows the study of Gaussian unitaries, thus pro-
viding similar classification schemes. We completed the
classification of the new types of channels by deriving
the form of the Liouvillian super operator that gener-
ates their time evolution in the form of a master equa-
tion. Surprisingly, Gaussian quantum channels without
Gaussian form can be experimentally addressed by means
of the celebrated Caldeira-Legget model for the quan-
tum damped harmonic oscillator, where the new types
of channels described here naturally appear in the sub-
ohmic regime.
ACKNOWLEDGEMENTS
We acknowledge PAEP and RedTC for financial sup-
port. Support by projects CONACyT 285754, UNAM-
PAPIIT IG100518 is acknowledged. CP acknowledges
support by PASPA program from DGAPA-UNAM. CAM
and JDU acknowledge financial support from the German
Academic Exchange Service (DAAD). We are thankful to
the University of Vienna where part of this project was
done.
Appendix A: Explicit formulas for class A2
The explicit formulas of the final states for channels of
class A2 with the form presented in eq. (6) with e2 = 0
are
(σf)11 = 1
2e1
,
(σf)22 = (αβ )
2 ( b23
2e1
+ 2a3) + α
β
(2a2 + b1b3
e1
)+
2a1 + b21
2e1
+ s1,
(σf)12 = −αβ b3
2e1
− b1
2e1
,
d⃗f (s3) = (0,−α
β
c2 − c1 + s2)T , (A1)
where
s1 = (b22 + 2αβ b2b4 + (α
β
)2 b24)(σi)11
− 2(α
β
b2 + (α
β
)2 b4)(σi)12 + (αβ )
2 (σi)22 ,
s2 = (α
β
b4 + b2)(di)1 − α
β
(di)2. (A2)
8
The explicit formulas of the final states for channels of
class A2 with the form presented in eq. (6) with α = 0 are
(σf)11 = e22
4e21
(σi)11 + 1
2e1
,
(σf)12 = (b2e22e1
− b1e
2
2
4e21
)(σi)11 − b1
2e1
,
(σf)22 = 2a1 + (b2 − b1e2
2e1
)2 (σi)11 + b21
2e1
, (A3)
and
d⃗f = ( e2
2e1
(d⃗i)1 ,(b2 − b1e2
2e1
)(d⃗i)1 − c1)
T
. (A4)
The explicit formulas of the final states for channels of
class A2 with Gaussian form are
(σf)11 (s1) = 2a3
b23
+ s1,
(σf)12 (s1) = a2
b3
− 2a3b1
b23
− b1s1,
(σf)22 (s1) = b1 (b3 (b1b3s1 − 2a2) + 2a3b1)
b23
+ 2a1,
d⃗f (s2) = (s2 − c2
b3
, b1 (c2
b3
− s2) − c1)T , (A5)
where
s1 = b24
b23
(σi)11 − 2b4
b23
(σi)12 + 1
b23
(σi)22 ,
s2 = 1
b3
(di)2 − b4
b3
(di)1 . (A6)
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C.3. Article: Positivity and Complete positivity of differentiable quantum
processes 123
C.3 Article: Positivity and Complete positivity of differ-
entiable quantum processes
Physics Letters A 383, 23 (2019). Click to go to the webpage, Click to go to
arXiv.
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Physics Letters A ••• (••••) •••–•••
Contents lists available at ScienceDirect
Physics Letters A
www.elsevier.com/locate/pla
Positivity and complete positivity of differentiable quantum processes
Gustavo Montes Cabrera a, David Davalos b, Thomas Gorin a,∗
a Departamento de Física, Universidad de Guadalajara, Guadalajara, Jalisco, Mexico
b Instituto de Física, Universidad Nacional Autónoma de México, Ciudad de México, Mexico
a r t i c l e i n f o a b s t r a c t
Article history:
Received 7 February 2019
Received in revised form 27 May 2019
Accepted 28 May 2019
Available online xxxx
Communicated by M.G.A. Paris
Keywords:
Quantum process
Divisibility
Quantum non-Markovianity
We study quantum processes, as one parameter families of differentiable completely positive and trace 
preserving (CPTP) maps. Using different representations of the generator, and the Sylvester criterion for 
positive semi-definite matrices, we obtain conditions for the divisibility of the process into completely 
positive (CP-divisibility) and positive (P-divisibility) infinitesimal maps. Both concepts are directly related 
to the definition of quantum non-Markovianity. For the single qubit case we show that CP- and P-
divisibility only depend on the dissipation matrix in the master equation form of the generator. We 
then discuss three classes of processes where the criteria for the different types of divisibility result in 
simple geometric inequalities, among these the class of non-unital anisotropic Pauli channels.
 2019 Elsevier B.V. All rights reserved.
1. Introduction
Non-Markovianity of quantum processes has been a topic of 
increasing interest during approximately the last ten years [1–3]. 
Starting with the papers by Breuer et al. [4] and Rivas et al. [5], a 
definition of quantum Markovianity has been reduced to the ques-
tion whether all intermediate quantum maps are physically real-
izable; this induces a characterization that is more closely related 
to the Chapman-Kolmogorov condition than to the full definition 
of classical Markovianity [6]. For differentiable quantum processes, 
the question of divisibility into physically realizable quantum maps 
can be further reduced to the analysis of the time dependent gen-
erator of the process. This is the approach taken for this work.
The concept of divisibility has been introduced in Refs. [7,8]. 
In its original form, it refers to the condition that all intermediate 
maps are completely positive (CP-divisibility). However, one may 
as well consider P-divisibility, where it is sufficient that the inter-
mediate maps are positive [9,10]. Note that it has been shown in 
Ref. [11] that P-divisibility provides a direct connection to classi-
cal Markovianity. If an intermediate map is positive but not com-
pletely positive, one may observe information backflow for entan-
gled states between system and some ancillary system, but not in 
the system alone [5,12].
In this work, we derive general criteria for the positivity and 
the complete positivity (of the local infinitessimal intermediate 
map). In particular, for single qubit processes we show that both, 
CP-divisibility and P-divisibility conditions, only depend on the dis-
* Corresponding author.
E-mail address: thomas.gorin@cucei.udg.mx (T. Gorin).
sipation matrix of the master equation. We identify three different 
classes of single qubit processes, where the criteria for CP- and 
P-divisibility are reduced to simple explicit geometric inequalities. 
One of these classes consists of processes where the Choi-matrix 
has the shape of an X (i.e. all non-zero elements are located on 
the diagonal or the anti-diagonal). Many examples considered in 
the literature of quantum non-Markovianity are of this type. A 
second class consists of those processes, where the Choi-matrix 
has the shape of an O . The third class is that of the non-unital 
anisotropic Pauli channels. While criteria applicable to the gener-
ators have been studied in the context of CP-divisibility, see for 
instance Ref. [13,29], this has rarely been done for P-divisibility. 
The complete positivity of non-unital anisotropic Pauli channels, as 
such, has been considered in Refs. [30] and [31].
Explicit analytical criteria are valuable for the construction of 
Markovian approximations to a non-Markovian process as pro-
posed in Ref. [8] and more specifically in Ref. [5]. Another area 
of applications is that of quantum process tomography [14–16], 
where it is important to identify the independent parameters 
which are to be determined. Finally, it may be of interest to iden-
tify quantum channels, which are P-divisible but not CP-divisible 
as processes where non-Markovianity may be identified as a gen-
uine quantum effect [17].
Our work relies on a few general results which have been de-
rived previously: (i) The Kossakowski theorem, which establishes 
the equivalence between positivity and contractivity (for the do-
main of Helstrom matrices), see Ref. [12] (and references therein), 
(ii) necessary and sufficient criteria which can be applied di-
rectly to the time dependent generator of the quantum process, 
see Ref. [9] for positivity and Ref. [8] for complete positivity, (iii) 
https://doi.org/10.1016/j.physleta.2019.05.049
0375-9601/ 2019 Elsevier B.V. All rights reserved.
JID:PLA AID:25754 /SCO Doctopic: Quantum physics [m5G; v1.260; Prn:5/06/2019; 16:24] P.2 (1-10)
2 G. Montes Cabrera et al. / Physics Letters A ••• (••••) •••–•••
Sylvester’s criterion for definite and semi-definite positivity [18,
19].
The paper is organized as follows: In Sec. 2 we discuss the de-
scription of quantum processes in terms of their generators and 
the general conditions for CP- and P-divisibility in terms of the 
generator. In Sec. 3 we analyze these conditions for general sin-
gle qubit processes. In Sec. 4 we present classes of single qubit 
processes, where the conditions for P- and CP-divisibility can be 
solved analytically. In Sec. 5 we present our conclusions.
2. Differentiable quantum processes
In this section we introduce differentiable quantum processes 
and the definitions of P-divisibility and CP-divisibility. For both 
properties, we present criteria which can be applied directly to the 
generator of the quantum process in question.
2.1. Processes and generators
Let us denote a quantum process t , (∀t ∈ R+
0 ), as a one-
parameter family of differentiable (with respect to t) completely 
positive and trace preserving linear maps (CPTP-maps), with 0 =
1, the identity. For simplicity, we assume that the corresponding 
Hilbert space is of finite dimension, dim(H) = d < ∞. The quan-
tum process t can be defined equivalently by the generator Lt , 
such that
d
dt
t = Lt t , 0 = 1 . (1)
One natural question to ask would be the following: What prop-
erties must be fulfilled by Lt in order to produce a valid quan-
tum process of CPTP maps? Very recently this question has been 
addressed in Ref. [20]. In the present work, our objective is differ-
ent. Assuming that Lt generates a valid quantum process, we ask 
whether that process is CP-divisible and/or P-divisible.
Note that for a given quantum process t , we can compute its 
generator as
Lt = dt
dt
−1
t . (2)
In what follows we will assume that t is invertible. Where this 
is not the case (if at all, this typically happens at isolated points 
in time), one has to proceed with care [21]. In order to derive 
P-divisibility and CP-divisibility criteria in terms of the generator, 
we need to relate Lt to the intermediate quantum map,
t+δ,t = t+δ −1
t , δ > 0 . (3)
This can be achieved by considering an infinitesimal intermediate 
quantum map, such that
Lt = lim
δ→0
δ−1 (
t+δ,t − 1
)
. (4)
Choi-matrix representation. A direct method to represent linear 
quantum maps (this includes generators such as Lt ) consists in 
embedding the state space into the vector space Md×d of com-
plex quadratic matrices of dimension d. In that case, the ele-
ments { |i〉〈 j| }1≤i, j≤d form an orthonormal basis with respect to 
the Hilbert-Schmidt scalar product 〈A, B〉 = tr(A† B), and their im-
ages under the quantum map uniquely define that map. Arranging 
these images in a d × d block-matrix results in the Choi matrix 
representation. Formally, the Choi-matrix representation [22] of a 
linear map  in Md×d can be defined as
C =
∑
i, j
|i〉〈 j| ⊗ [ |i〉〈 j| ] . (5)
It has the following remarkable properties: (i) C = C
†
 iff [†] =
[] for every bounded operator , (ii) C ≥ 0 iff  is complete 
positive, (iii) tr (C) = d if  preserves the trace [22,23].
Master equation. The generator obtained in Eq. (4) preserves Her-
miticity by construction, thus we can bring it to the following 
standard form [8,24] (see Appendix E for a detailed derivation):
d
dt
̺ = Lt[̺] , (6)
Lt[̺] = −i [H,̺] +
d2−1
∑
i, j=1
D i j
(
F i ̺ F
†
j −
1
2
{
F
†
j F i , ̺
}
)
.
In this expression, Planck’s constant h̄ has been absorbed into the 
Hamiltonian H . The matrix D is Hermitian, and the set {F i}1≤i≤d2
forms an orthonormal basis in the space of operators, such that 
tr(F †
i F j) = δi j . In addition, the operators are chosen such that 
tr(F i) = 0, except for the last element, which is given by Fd2 =
1/
√
d.
In this work, we will use Eq. (6) as one possible representation 
of the generator Lt , at some arbitrary, fixed time t . We call this 
representation the “master equation representation” of Lt , and D
the “dissipation matrix”. Note that an intermediate quantum pro-
cess t+δ,t ≈ 1 + δLt , as defined in Eq. (3), is CPTP if and only 
if D is positive semidefinite [2,13]. In this case, t becomes a 
valid CPTP map [11]. Finally, if the generator is time-independent, 
t becomes a one-parameter semigroup in the space of CPTP 
maps [25–27].
2.2. Markovianity: P-divisibility vs. CP-divisibility
In this subsection we present the definitions for the P-divis-
ibility and the CP-divisibility of quantum processes. We use the 
term “Markovianity” in cases, where we want to refer to both 
properties, indistinctively.
CP-divisibility. A process t is called CP-divisible if and only if 
the intermediate map t+δ,t as defined in Eq. (3) is CPTP for 
all t, δ ∈ R
+
0 . Complete positivity of a quantum map  is conve-
niently verified using the Choi matrix representation, introduced 
in Eq. (5).
P-divisibility. A process t is called P-divisible if and only if the 
intermediate map t+δ,t as defined in Eq. (3) is PTP (positivity 
and trace preserving) for all t, δ ∈ R
+
0 .
Positivity of a Hermiticity and trace preserving quantum map 
 is more complicated to verify. In that case, one has to show 
that [̺] ≥ 0 for all density matrices ̺. In practice, it is sufficient 
to check the condition for all density matrices representing pure 
states.
Local complete positivity. Following Refs. [8], and [28], let CL be the 
Choi-matrix representation of the generator Lt , and let C⊥ be a 
matrix representation of CL in the subspace orthogonal to the Bell 
state
|B〉 = 1√
d
∑
i
|ii〉 , (7)
where d is the dimension of the Hilbert space. Then, the quantum 
process t is locally CP at time t , if and only if
C⊥ ≥ 0 . (8)
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Therefore the process t is CP-divisible, if and only if it is locally 
CP for all t ∈ R
+
0 . Note that in Ref. [28], it has been shown that C⊥
is unitarily equivalent to the dissipation matrix D (see Appendix E
for a detailed derivation).
Local positivity. A quantum process is locally positive at time t , if 
and only if for all orthogonal states |ψ〉, |φ〉 ∈ H it holds that
〈ψ |Lt[ |φ〉〈φ| ]ψ〉 ≥ 0 . (9)
Similar to the CP case, it holds that a quantum process t is P-
divisible if and only if it is locally positive for all t ∈ R
+
0 [9]. The 
equivalence between local positivity and P-divisibility follows from 
Eq. (4):
〈ψ |t+δ,t[ |φ〉〈φ| ] |ψ〉 ≥ 0
⇔ δ 〈ψ |Lt[ |φ〉〈φ| ] |ψ〉 + 〈ψ |φ〉 〈φ|ψ〉 ≥ 0 . (10)
In the limit δ → 0, this can only happen if ψ and φ are orthog-
onal, 〈ψ |φ〉 = 0. In fact, if |〈ψ |φ〉|2 > 0, it might very well be 
that 〈ψ | Lt [ |φ〉〈φ| ] |ψ〉 < 0 even if the process is P-divisible in the 
neighborhood of that point.
To summarize, we may express both properties CP-divisibility 
and P-divisibility in terms of local conditions which have to be 
fulfilled by the generator Lt . In what follows, we analyze these in 
more detail. To avoid overly cumbersome terminology, we denote 
generators which fulfill Eq. (9) and/or Eq. (8) simply as “positive” 
and/or “completely positive generators”.
3. Single qubit processes
In the case of single qubit processes, the Bloch vector represen-
tation is yet another method to represent quantum channels and 
their generators. In Sec. 3.1 we discuss the following three repre-
sentations: (i) the master equation, (ii) the Choi-matrix, and (iii) 
the Bloch vector representation and how they are related one-to-
another. In Sec. 3.2, we derive explicit criteria for local positivity 
and local complete positivity in terms of the dissipation matrix D .
3.1. Equivalent representations
Choi matrix representation. For our purposes, the Choi matrix repre-
sentation will be the most useful. A CPTP-map , which belongs 
to a quantum process, may be parametrized as
C =
(
[ |0〉〈0| ] [ |0〉〈1| ]
[ |1〉〈0| ] [ |1〉〈1| ]
)
=
⎛
⎜
⎜
⎝
1− r1 y∗
1 x∗ 1− z∗1
y1 r1 z2 −x∗
x z∗2 r2 y∗
2
1− z1 −x y2 1− r2
⎞
⎟
⎟
⎠
. (11)
The structure of C is due to the fact that  must preserve Her-
miticity and the trace. We have chosen the parametrization in such 
a way that the parameters r1, r2 , y1, y2, x, z1, z2 as functions of 
time are all zero at t = 0.
Note that any intermediate map t+δ,t is at least Hermitic-
ity and trace preserving. Therefore, Eq. (4) implies that the Choi-
matrix representation of the generator Lt must be Hermitian, and 
in all blocks, the partial trace must be equal to zero. That leaves 
us with the following parametrization:
CL =
⎛
⎜
⎜
⎝
−q1 Y ∗
1 X∗ −Z∗
1
Y1 q1 Z2 −X∗
X Z∗
2 q2 Y ∗
2
−Z1 −X Y2 −q2
⎞
⎟
⎟
⎠
. (12)
In general, there is no simple relation between the parametrization 
used here, and that of Eq. (11). This is because the expression for 
the generator Lt includes the inverse of t .
Master equation representation. Note that every generator Lt of a 
Hermiticity and trace preserving quantum process, can be written 
in the form of Eq. (6), with Hermitian matrices H and D . There-
fore, we may calculate the Choi-representation of the generator, by 
inserting ̺ = |i〉〈 j| into the RHS of Eq. (6), and compare the result 
to the general form in Eq. (12). For the calculation, we choose the 
following orthonormal operator basis {F i}1≤i≤d2 :
F1 = 1√
2
(
|0〉〈0| − |1〉〈1|
)
, F2 = |0〉〈1| ,
F3 = |1〉〈0| , and F4 = 1/
√
2 . (13)
As a result, we obtain a linear one-to-one correspondence between 
the parameters used in the master equation representation and 
those, used in the Choi representation:
⎛
⎝
q1
q2
ReZ1
⎞
⎠ =
⎛
⎝
0 0 1
0 1 0
1 1/2 1/2
⎞
⎠
⎛
⎝
D11
D22
D33
⎞
⎠ ,
Im Z1 = H22 − H11 ,
Z2 = D32 ,
⎛
⎝
Y1
Y2
X∗
⎞
⎠ =
⎛
⎝
−
√
2/4
√
18/4 −i
−
√
18/4
√
2/4 i√
2/4
√
2/4 i
⎞
⎠
⎛
⎝
D12
D31
H21
⎞
⎠ . (14)
As one might have expected, the quantity H11 + H22 is irrelevant 
for the representation of the generator, and may be set equal to 
zero without loss of generality. Then, Eq. (14) is clearly an invert-
ible linear system of equations.
Bloch vector representation. Any qubit density matrix can be written 
in terms of the Pauli matrices and the identity matrix 1 as follows:
̺ = 1
2
(
v0 1+
3
∑
j=1
v j σ j
)
, (15)
where v0 = 1 and v = (v1, v2, v3) is a vector in R3 of norm ‖v‖ ≤
1. Any Hermiticity and trace preserving quantum map  can then 
be written as an affine transformation [32]
 : v → v ′ = R v +t , (16)
where R is a real not necessarily symmetric square matrix and t
is a real three-dimensional vector. The coefficients of R and t are 
given by
t j =
1
2
tr
(
σ j Lt[1 ]
)
, R jk = 1
2
tr
(
σ j Lt[σk ]
)
. (17)
For the generator with the Choi-matrix representation given in 
Eq. (12), we find
R =
⎛
⎝
Re(Z2 − Z1) Im(Z1 + Z2) Re(Y1 − Y2)
Im(Z2 − Z1) −Re(Z1 + Z2) Im(Y1 − Y2)
2Re(X) −2 Im(X) −q1 − q2
⎞
⎠ ,
t =
⎛
⎝
Re(Y1 + Y2)
Im(Y1 + Y2)
q2 − q1
⎞
⎠ . (18)
Again, it is easy to verify that the relation between this Bloch vec-
tor representation and the Choi representation is invertible.
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3.2. Criteria for positivity and complete positivity
Local complete positivity. In order to verify if the Choi-matrix (as a 
linear transformation) projected onto the orthogonal subspace of 
|φB〉〈φB|, is positive, we choose the orthonormal states
|ψ1〉 = 1√
2
⎡
⎢
⎢
⎣
1
0
0
−1
⎤
⎥
⎥
⎦
, |ψ2〉 =
⎡
⎢
⎢
⎣
0
0
1
0
⎤
⎥
⎥
⎦
, |ψ3〉 =
⎡
⎢
⎢
⎣
0
1
0
0
⎤
⎥
⎥
⎦
, (19)
to span that subspace. Then we obtain for the matrix representa-
tion of the Choi matrix of Lt , projected on that subspace:
C⊥ =
⎛
⎜
⎜
⎝
Re(Z1) − q1+q2
2
X∗−Y2√
2
X+Y ∗
1√
2
X−Y ∗
2√
2
q2 Z∗
2
X∗+Y1√
2
Z2 q1
⎞
⎟
⎟
⎠
=
⎛
⎝
D11 D12 D13
D21 D22 D23
D31 D32 D33
⎞
⎠ . (20)
The second equality is obtained by solving Eq. (14) for the matrix 
elements D i j . It simply means that C⊥ = D .
We may now use the Sylvester criterion to check whether D ≥
0 or not. A general discussion of that criterion can be found in 
the text book [18]; the present positive semidefinite case has been 
treated in Ref. [19]. In that case, the statement is the following: A 
Hermitian matrix is positive semidefinite if and only if all principal 
minors are larger or equal to zero. Hence, for D ≥ 0, it must hold:
D11, D22, D33 ≥ 0 , D11D33 − |D31|2 ≥ 0 ,
D11D22 − |D21|2 ≥ 0 , D33D22 − |D32|2 ≥ 0 ,
D11D22D33 + 2Re(D12D23D31) ≥
D11|D32|2 + D22|D31|2 + D33|D21|2 . (21)
Local positivity. According to the criterion in Eq. (9), we need to 
verify that 〈ψ | L[ |φ〉〈φ| ] |ψ〉 ≥ 0 for all |ψ〉 ⊥ |φ〉. Such general 
orthonormal states may be written as the column vectors of a uni-
tary matrix, taken from the group SU (2). Removing an ineffective 
global phase we find:
|ψ〉 =
(
cos(θ/2)
eiβ sin(θ/2)
)
, |φ〉 =
( − sin(θ/2)
eiβ cos(θ/2)
)
.
Hence, we consider p(θ, β) = 〈ψ | L[ |φ〉〈φ| ] |ψ〉 as a function of θ
and β . Therefore, we may say that the Lt is positive at time t , if 
and only if p(θ, β) ≥ 0 for all θ and β . Using the parametrization 
of Eq. (12), p(θ, β) may be written as
p(θ,β) = q1 + q2
2
cos2 θ + q2 − q1
2
cos θ + A
2
sin2 θ
+ Re
[
(Y1 + Y2)e−iβ
]
2
sin θ (22)
+ Re
[
(Y2 − Y1)e−iβ − 2X eiβ
]
2
sin θ cos θ ,
where A = Re[Z1 − Z2 e−2iβ ]. In terms of the master equation pa-
rameters, we find
R = D22 + D33 , Y1 + Y2 =
√
2 (D21 − D13) ,
S = D33 − D22 , A1 = D11 − D33 + D22
2
− Re D23 ,
Y2 − Y1 − 2 X∗ = −
√
2 (D21 + D13) , (23)
such that
2 p(θ,β) = R + S cos θ +
(
D11 − R
2
)
sin2 θ
+ Re
[
− D23 e
−2iβ sin θ +
√
2 (D21 − D13)e
−iβ
−
√
2 (D21 + D13)e
−iβ cos θ
]
sin θ . (24)
This shows that positivity, just as complete positivity, only depends 
on the dissipation matrix D .
In general, one should try to find all minima of this function 
and verify that those are non-negative. Since the domain of p(θ, β)
is a torus without boundaries, it is sufficient to find the critical 
points where the partial derivatives ∂p/∂θ and ∂p/∂β are both 
equal to zero. The corresponding equations may be reduced to a 
root-finding problem for 4’th order polynomials. Thus analytical 
expressions may be obtained in principle, even so they are proba-
bly not very useful. Still, numerical evaluations are pretty straight 
forward to implement. In Sec. 4 we will discuss different classes 
of generators, where particularly simple analytical solutions can be 
found.
4. Examples
In this section, we consider three different classes of generators. 
For each class, the set of positive (completely positive) generators 
is interpreted as a region in a certain parameter space (a subspace 
of the 9-dimensional vector space of dissipation matrices). In gen-
eral, these regions must be convex, since the respective criteria 
involve expectation values of some linear matrix which represents 
the generator. Hence, if we consider the expectation value of any 
convex combination of two generators, it immediately decomposes 
into the corresponding convex combination of expectation values. 
Unless stated otherwise, we analyze the criteria for positivity and 
complete positivity in terms of the dissipation matrix D .
4.1. X-shaped quantum channels and generators
The term “X-shape” refers to the case, where the non-zero ele-
ments in the Choi matrix appear to form the letter “X”, that means 
that Y1 = Y2 = X = 0 in Eq. (12). Hence,
CL =
⎛
⎜
⎜
⎝
−q1 0 0 −Z∗
1
0 q1 Z2 0
0 Z∗
2 q2 0
−Z1 0 0 −q2
⎞
⎟
⎟
⎠
. (25)
In this case, the X-shape of the generator implies the X-shape of 
the quantum channel, and vice versa. Many important models lead 
to quantum channels of that type [2,4]. In terms of the Bloch vec-
tor representation, the X-shape implies that the dynamics along 
the z-axis is independent from that in the (x, y)-plane [13].
According to Eq. (14) the X-shape of the Choi matrix CL im-
plies for H and D from the master equation representation in 
Eq. (6): H12 = 0, D13 = D12 = 0 as well as
q1 = D22 , q2 = D33 , Z2 = D23
and Z1 = i (H22 − H11) + D11 + D33 + D22
2
. (26)
For the matrix C⊥ we thus obtain:
C⊥ =
⎛
⎝
D11 0 0
0 D22 D23
0 D32 D33
⎞
⎠ . (27)
Complete positivity. Considering all principal minors of the dissipa-
tion matrix in Eq. (27), we find
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Fig. 1. The parameter space D22, D33 ≥ 0 for visualizing the regions of positivity 
and complete positivity for the X-shaped generator, for |D32| = 1. For complete 
positivity, the point (D22, D33) must lie above the black dashed line, while D11 ≥ 0
is required. For positivity, the allowed region for (D22, D33) depends on D11: For 
D11 ≥ 1, it is the whole quadrant; for D11 = 1/2 the allowed region consists of 
the dark green and blue areas; for D11 = 0 it consists of the blue areas above the 
black dashed line; and for D11 = −1/2 it consists of the dark blue area alone. (For 
interpretation of the colors in the figure(s), the reader is referred to the web version 
of this article.)
D11, D22, D33 ≥ 0 , D22 D33 − |D23|2 ≥ 0 ,
D11 D22 ≥ 0 , D11
[
D22 D33 − |D23|2
]
≥ 0 . (28)
Removing redundant inequalities, we are left with
D11, D22, D33 ≥ 0 , D22 D33 ≥ |D23|2 . (29)
Positivity. From Eq. (24) we find:
2p(θ,β) = R + S cos θ + A sin2 θ ≥ 0 , (30)
where A = D11 − R
2
− Re
[
D23 e
−2iβ ]
,
R = D22+D33 , and S = D33−D22 . This inequality must hold for all 
values of θ and β , parametrizing the quantum state to which the 
generator is applied. Thus, we only need to verify if the minimum 
of this expression is larger than zero. As far as β is concerned, this 
means that we may replace A by its minimum (as a function of 
β), which is given by Amin = D11 − R/2 − |D23|. We are then left 
with the condition
∀ θ : R + S cos θ + Amin sin2 θ ≥ 0 . (31)
This condition is further evaluated in Appendix A. As a result, we 
find that the conditions for positivity become
D22, D33 ≥ 0 , (32)
and if D11 < |D23|, in addition
∣
∣ |D23| − D11
∣
∣ ≤
√
D22 D33 . (33)
In Fig. 1, we show the parameter space D22, D33 ≥ 0 for vi-
sualizing the regions of positivity and complete positivity for the 
X-shaped generator. For complete positivity, the inequalities to ful-
fill are given in Eq. (29), which states independent conditions on 
D11 on the one hand and D22, D33, |D23|2 on the other. For posi-
tivity, by contrast, the conditions on D22 and D33 depend on D11 . 
Here, we observe an interesting behavior: As D11 approaches zero 
from above, the region of positivity becomes more and more sim-
ilar to the region of complete positivity, until they coincide for 
D11 = 0. When D11 becomes negative, complete positivity is vio-
lated while positivity is still maintained sufficiently far away from 
the black dashed line.
4.2. O-shaped quantum channels
Here, we consider another subset of single qubit generators, 
which also allow for an analytic solution. These are in some sense 
complementary to the X-shaped channels, These are obtained from 
the general case by setting q1 = q2 = q, Y1 = −Y2 = Y , and Z2 = 0. 
The Choi matrix, representing the generator resembles an O , in-
stead of an X , that is why we call them O -shaped channels.
CL =
⎡
⎢
⎢
⎣
−q Y ∗ X∗ −Z∗
1
Y q 0 −X∗
X 0 q −Y ∗
−Z1 −X −Y −q
⎤
⎥
⎥
⎦
(34)
According to Eq. (14), this implies for the matrix elements of H
and D from the master equation (6):
q = D33 , Z1 = i (H22 − H11) + D11 + D33 ,
Y = −i H21 + D13√
2
, X∗ = i H21 + D13√
2
, (35)
with D22 = D33 , D21 = D13 , and D23 = 0. The matrix for verifying 
complete positivity reads
C⊥ =
⎛
⎝
D11 D31 D13
D13 D22 0
D31 0 D22
⎞
⎠ , (36)
Complete positivity. Expressed in terms of the dissipation matrix, 
considering all principal minors.
D11, D22 ≥ 0 , D11D22 − |D13|2 ≥ 0 ,
D11 D
2
22 − D13D31D22 − D31D22D13 ≥ 0 (37)
This can be reduced to
D11, D22 ≥ 0 , D11D22 ≥ 2 |D13|2 . (38)
Positivity. Under the conditions mentioned above, the function 
2 p(θ, β) from Eq. (24) becomes (in terms of the master equation 
parameters)
2p(θ,β) = 2 D22 + (D11 − D22) sin2 θ
− 2
√
2Re
[
D13 e
−iβ]
sin θ cos θ ≥ 0 . (39)
Again, it is possible to derive the conditions for positivity, which do 
no longer involve the angles θ and β . The respective calculation is 
outlined in Appendix B, with the result [see Eq. (B.4)]
3 D22 + D11 ≥ 0 , D22 (D11 + D22) ≥ |D13|2 . (40)
In Fig. 2, we show the different regions of positivity and com-
plete positivity in the parameter space of D22, D11 . We distinguish 
two qualitatively different cases, |D13| = 0 and |D13| = 1. In both 
cases, the region of positivity is considerably larger than the region 
for complete positivity.
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Fig. 2. The parameter space D22, D11 for visualizing the regions of positivity 
[Eq. (40)] and complete positivity [Eq. (38)] for the O -shaped generator. For |D13| =
0, the region of complete positivity is simply the positive quadrant D22, D11 ≥ 0, 
while the region for positivity is the whole colored region. For |D13| = 1, the region 
of complete positivity is colored in orange, the region of positivity is dark green and 
orange.
4.3. Non-unital anisotropic Pauli channels
Here, t is given as an affine transformation of state vec-
tors in the Bloch sphere [32] (see the corresponding paragraph in 
Sec. 3.1):
t : v → v ′ = R v + s , (41)
where R is a real diagonal matrix and s a real vector.
R =
⎛
⎝
R11 0 0
0 R22 0
0 0 R33
⎞
⎠ , s =
⎛
⎝
s1
s2
s3
⎞
⎠ . (42)
Using the general formula, Eq. (2), for constructing the generator, 
we find
LP : v → v ′ = dR
dt
[
R−1 (v − s)
]
+ ds
dt
. (43)
Hence, the generator for this Pauli channel is given by the affine 
transformation v → v ′ = RLP v +tLP , with
RLP =
⎛
⎝
−γ1 0 0
0 −γ2 0
0 0 −γ3
⎞
⎠ , tLP =
⎛
⎝
τ1
τ2
τ3
⎞
⎠ ,
where γ j =
−1
R j j
dR j j
dt
, τ j =
d s j
dt
+ γ j s j . (44)
The Choi matrix representation of LP is obtained by inverting 
Eq. (18), with the result
CLP = 1
2
⎛
⎜
⎜
⎝
−γ3 + τ3 τ1 − iτ2 0 −γ1 − γ2
τ1 + iτ2 γ3 − τ3 γ2 − γ1 0
0 γ2 − γ1 γ3 + τ3 τ1 − iτ2
−γ1 − γ2 0 τ1 + iτ2 −γ3 − τ3
⎞
⎟
⎟
⎠
. (45)
In what follows, we compute the positivity and the complete pos-
itivity condition in terms of the parameters γ j and τ j , since this 
allows for relatively simple geometric interpretations. In the case 
of complete positivity, this has been worked out previously, in 
Ref. [29]. For the parametrization in terms of the master equa-
tion (6), we obtain from Eq. (12) and (14):
Fig. 3. Parameter space of γ1 , γ2 and γ3 . For the generator LP in Eq. (43) to be 
positive, all elements γ j must be positive (blue transparent color). For LP to be 
completely positive, the elements γ j must fulfill the conditions in Eq. (48). The 
corresponding region is colored in orange. Note that we show a cut through the 
regions of positivity and complete positivity which really extend towards arbitrary 
large positive values.
D22 = γ3 − τ3
2
, D33 = γ3 + τ3
2
, H22 = H11 ,
D11 = γ1 + γ2 − γ3
2
, D23 = γ2 − γ1
2
, H12 = 0 ,
D21 = τ1 + iτ2
2
√
2
= −D13 . (46)
This yields
C⊥ = 1
2
⎛
⎝
γ1 + γ2 − γ3 w∗ −w
w γ3 − τ3 γ2 − γ1
−w∗ γ2 − γ1 γ3 + τ3
⎞
⎠ , (47)
with w = (τ1 + iτ2)/
√
2.
Complete positivity. The complete derivation can be found in Ap-
pendix C. It yields separate conditions for the diagonal elements 
γ j and the vector τ . For the diagonal elements γ j we find:
∀ i = j = k = i : |γi − γ j| ≤ γk ≤ γi + γ j . (48)
The corresponding region in the parameter space of the elements 
γ j is depicted as a orange region in Fig. 3. Assuming these con-
ditions are fulfilled, the vector τ must lie inside the following 
ellipsoid:
τ 2
1
a21
+ τ 2
2
a22
+ τ 2
3
a23
≤ 1 , a1 = γ 2
1 − (γ2 − γ3)
2 ,
a2 = γ 2
2 − (γ1 − γ3)
2 , a3 = γ 2
3 − (γ1 − γ2)
2 . (49)
The regions of τ where the generator LP fulfills the conditions of 
complete positivity are shown in Fig. 4 in orange. Note that in this 
figure, we consider two particular cases, where γ1 = γ2 such that 
the resulting ellipsoid as defined above is symmetric with respect 
to the τ3 axis.
Positivity. In the general expression for p(θ, β) in Eq. (24), we re-
place the parameters with those from the Pauli channel, given in 
Eq. (46). This yields
2 p(θ,β) = γ3 cos2 θ +
[
γ1 cos2 β + γ2 sin2 β
]
sin2 θ
+ τ3 cos θ +
(
τ1 cosβ + τ2 sinβ
)
sin θ . (50)
We can express the general inequality 2 p(θ, β) ≥ 0 in a geometric 
form:
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Fig. 4. Comparison of the region of positivity and complete positivity in the param-
eter space of τ for γ1 = γ2 . The black solid line shows the ellipsoid τ = γ er , the 
orange region shows the region of complete positivity, the green region (including 
orange) the region of positivity. In panel (a), γ1 = 0.255, γ3 = 0.49 which amounts 
to an ellipsoid of the shape of a rugby ball, in panel (b) γ1 = 0.495, γ3 = 0.01 where 
the ellipsoid looks more like a pancake.
er =
⎛
⎝
sin θ cosβ
sin θ sinβ
cos θ
⎞
⎠ : er ·
(
γ er + τ
)
≥ 0 , (51)
where γ is the diagonal matrix with elements γ j .
The interpretation of this result is easy: −γ er + τ is the image 
of er under the generator LP . Thus an infinitesimal intermediate 
map would yield
t,t+δ : er → er ′ = er + δ LP[er] .
In order to have ‖er ′‖ ≤ 1, the image under the generator must 
be pointing towards the center of the Bloch sphere, i.e. the scalar 
product between −γ er + τ and er must be negative. Multiplying 
the resulting inequality by minus one, we find
∀er : er ·
(
γ er − τ
)
≥ 0 .
This relation is equivalent to the inequality in Eq. (51), as can be 
seen by replacing er by −er .
As shown in Appendix D, the set of τ for which the Pauli gen-
erator LP is positive, i.e. the inequality in Eq. (51) holds, is the 
convex region, which contains the origin and is limited by the sur-
face [see Eq. (D.2)]
T = {τ (θ,β) = (er · γ er) er − 2γ er } . (52)
In Fig. 4, we show the region in τ -space which corresponds to 
positivity and complete positivity of the Pauli channel generator 
LP . We consider two cases: γ1 = γ2 = 0.255, γ3 = 0.49 in panel 
(a), and γ1 = γ2 = 0.495, γ3 = 0.01 in panel (b). In the yellow 
triangle shown in Fig. 3, these points are located near the up-
per horizontal line (a) and near the lower corner (b), respectively. 
Choosing γ1 = γ2 leads to regions of (complete) positivity, which 
are symmetric with respect to the τ3-axis, which allows us to show 
two-dimensional projections. We find that the regions of positivity 
and complete positivity are always contained in ellipsoid with the 
parametrization τ (θ, β) = γ er . As required, the region of complete 
positivity (orange) is fully contained in the region of positivity 
(olive green). In panel (a), we show a case where the ellipsoid γ er
resemble roughly a rugby ball. In that case, the are only rather thin 
stripes near the border of the ellipsoid, where the generator is not 
positive any more. In panel (b), the ellipsoid has the shape of a flat 
pancake, and the region of positivity in the center is much smaller.
5. Conclusions
In order to determine whether a given differentiable quantum 
process is CP-divisible and/or P-divisible, we derive criteria to be 
applied to the generator of the process. For the single qubit case, 
we discuss three common representations of the generator and 
work out the one-to-one mappings between them. We find criteria 
for CP- and P-divisibility, which can be expressed as inequalities 
in terms of the elements of the dissipation matrix. In the CP case, 
we avoid solving an eigenvalue problem by using the principal mi-
nor test for semidefinite matrices. In the P case, the corresponding 
inequality must be fulfilled for a whole two-parameter family of 
functions, which leads to an optimization problem without explicit 
general solution.
We then discuss three different classes of generators, where our 
criteria do yield explicit results: the familiar X-shaped channels 
where the elements of the Choi matrix are non-zero in the diago-
nal and the anti-diagonal, only; the so called O -shaped channels, 
where C23 = 0, C11 = C44 and C12 = −C34; and most importantly 
the non-unital Pauli channels.
Besides its general value, as for instance the positivity criteria 
for the Pauli channel, we expect our results to prove useful in the 
area of quantum process tomography and the construction of op-
timal P-divisible or CP-divisible approximations to non-Markovian 
quantum processes. In particular there, the renouncement on the 
calculation of higher order roots may help to find analytical or 
semi-analytical solutions.
Acknowledgements
We gratefully acknowledge Sergey Filippov for useful discus-
sions, as well as Carlos Pineda for valuable comments on the 
manuscript.
Appendix A. Positivity of X-shaped generators
The condition in Eq. (31) can be expressed equivalently in terms 
of the variable x = cos θ as follows:
∀ x ∈ [−1,1] :
f (x) = (R − Amin) x
2 + S x+ Amin ≥ 0 . (A.1)
First note the following obviously necessary conditions
f (0) : Amin ≥ 0 and
f (±1) : R ± S ≥ 0 ⇔ 0 ≤ |S| ≤ R . (A.2)
To find the necessary and sufficient conditions, we will divide the 
problem in two cases: (i)  = R − Amin ≤ 0 and (ii)  > 0.
In case (i) the conditions in Eq. (A.2) are also sufficient as can 
be seen as follows: f (x) is convex, such that for any x1, x2 and 
0 < λ < 1:
f (λ x1 + (1 − λ) x2) ≥ λ f (x1) + (1− λ) f (x2) .
Choosing x1 = −1 and x2 = 1, we find
f (1 − 2λ) ≥ λ f (−1) + (1− λ) f (1) ,
which implies that f (x) ≥ 0 in the interval (−1, 1).
In case (ii)  > 0, the conditions in Eq. (A.2) are not sufficient. 
In this case, positivity requires that either f (x) has no zeros, or its 
zeros
x1,2 = − S
2
±
√
S2
2
− 4 Amin

,
are lying both to the left or both to the right of the interval (−1.1). 
This can be expressed as
|S| ≤ 2
√
Amin  or |S| ≥ 2 +
√
S2 − 4Amin  .
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The inequality to the right is equivalent to
|S| ≥ 2 and ( |S| − 2 )2 ≥ S2 − 4Amin  ,
which is equivalent to
|S| ≥ 2 and |S| ≤ R ,
where |S| ≤ R had already been identified as a necessary condi-
tion, previously. Therefore, in case (ii) the necessary and sufficient 
conditions for positivity are Amin ≥ 0 , 0 ≤ |S| ≤ R and
|S| ≤ 2
√
Amin  or |S| ≥ 2 .
It turns out that for  ≤ Amin it holds that 2 
√
Amin  < 2 such 
that the two conditions cancel each other, i.e. one of the two 
conditions is always fulfilled. For  > Amin which is equivalent 
to 2 > R , by contrast, implies that |S| ≥ 2 cannot hold, such 
that |S| ≤ 2 
√
Amin  must be fulfilled. To summarize, the neces-
sary and sufficient conditions for positivity are as follows:
• Amin = Re Z1 − |Z2| ≥ 0 , 0 ≤ |S| ≤ R .
• In addition, if R > 2Amin:
|S| ≤ 2
√
Amin (R − Amin) .
For the parametrization in terms of the master equation, we find 
that positivity only depends on the dissipation matrix D . Since
R = D33 + D22 , S = D22 − D33 , (A.3)
the condition 0 ≤ |S| ≤ R implies that both D22 and D33 must be 
larger than or equal to zero. Furthermore, with
Amin = D11 + D33 + D22
2
− |D32| , (A.4)
the condition R > 2 Amin implies that |D32| > D11 . Finally,
Amin(R − Amin) =
[D33 + D22
2
−
(
|D32| − D11
)
]
×
[ D33 + D22
2
+
(
|D32| − D11
)
]
= (D33 + D22)
2
4
−
(
|D32| − D11
)2
, (A.5)
such that |S| ≤ 2 
√
Amin (R − Amin) is equivalent to
|D22 − D33|2 ≤ (D33 + D22)
2 − 4
(
|D32| − D11
)2
⇔
(
|D32| − D11
)2 ≤ D33 D22 . (A.6)
To summarize, in this parametrization, the conditions for positivity 
read
D22, D33 ≥ 0 , D11 − |D32| +
D33 + D22
2
≥ 0 (A.7)
and if D11 < |D32|, in addition
∣
∣ |D32| − D11
∣
∣ ≤
√
D33 D22 . (A.8)
Note that this last inequality implies the second inequality of 
Eq. (A.7), which can therefore be ignored.
Appendix B. Positivity of O -shaped generators
We start from the condition for positivity in Eq. (39). Us-
ing the trigonometric identities 2 sin2 θ = 1 − cos2θ and sin2θ =
2 sin θ cos θ , Eq. (39) becomes
2p(θ,β) = 3D22 + D11
2
+ D22 − D11
2
cos 2θ
−
√
2Re
(
D12 e
−iβ )
sin 2θ ≥ 0 . (B.1)
This expression is minimized with respect to β , simply by making 
sure that Re( D12 e−iβ ) = ± |D12|. In other words: 2p(θ, β) ≥ 0 for 
all β and θ is equivalent to
3D22 + D11
2
+ D22 − D11
2
cos 2θ ±
√
2 |D12| sin 2θ ≥ 0 . (B.2)
This condition is equivalent to
3D22 + D11
2
≥ 0 and
(3D22 + D11)
2
4
≥ (D22 − D11)
2
4
+ 2|D12|2 (B.3)
These two inequalities are equivalent to
3D22 + D11 ≥ 0 and D22 (D22 + D11) ≥ |D12|2 (B.4)
Appendix C. Complete positivity of the non-unital anisotropic 
Pauli channel
For the generator LP to be completely positive, the matrix C⊥
given in Eq. (47) must fulfill the inequalities in Eq. (21). In the 
present case, this yields three sets of inequalities:
γ1 + γ2 − γ3 ≥ 0 , γ3 − τ3 ≥ 0 , γ3 + τ3 ≥ 0 ,
(γ3 − τ3)(γ3 + τ3) − (γ2 − γ1)
2 ≥ 0 , (C.1)
(γ1 + γ2 − γ3)(γ3 − τ3) − |w|2 ≥ 0 ,
(γ1 + γ2 − γ3)(γ3 + τ3) − |w|2 ≥ 0 , (C.2)
and
(γ1 + γ2 − γ3)
[
(γ3 − τ3)(γ3 + τ3) − (γ2 − γ1)
2]
− w
[
w∗ (γ3 + τ3) + w (γ2 − γ1)
]
− w∗ [
w∗ (γ2 − γ1) + w (γ3 − τ3)
]
≥ 0 , (C.3)
where w = (τ1 + i τ2)/
√
2. From Eq. (C.1), we find
γ3 ≥ |τ3| ≥ 0 , γ1 + γ2 ≥ γ3 , (γ2 − γ1)
2 ≤ γ 2
3 − τ 2
3 ,
which yields the following conditions as necessary conditions 
(since we set τ3 = 0 to arrive there):
γ1,γ2,γ3 ≥ 0 , |γ2 − γ1| ≤ γ3 ≤ γ1 + γ2 . (C.4)
It is easy to verify that these inequalities are invariant under any 
permutation of indices; see Fig. 3. The remaining conditions, may 
be interpreted as conditions for the vector τ . These consist of the 
inequalities in Eq. (C.2) together with
|τ3| ≤
√
γ 2
3 − (γ2 − γ1)2 , and (C.5)
(γ1 + γ2 − γ3) (γ2 + γ3 − γ1) (γ3 + γ1 − γ2) ≥
(γ1 + γ2 − γ3) τ 2
3 + (γ2 + γ3 − γ1) τ 2
1 + (γ3 + γ1 − γ2) τ 2
2 .
(C.6)
In Appendix C.1 we demonstrate that condition (C.6) implies all 
other conditions for the vector τ , which can therefore be omitted. 
Reorganizing the terms in Eq. (C.6), we arrive at
τ 2
1
a21
+ τ 2
2
a22
+ τ 2
3
a23
≤ 1 , a1 = γ 2
1 − (γ2 − γ3)
2 ,
a2 = γ 2
2 − (γ1 − γ3)
2 , a3 = γ 2
3 − (γ1 − γ2)
2 . (C.7)
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Appendix C.1. Omissible inequalities for τ
In what follows, we demonstrate that Eq. (C.5) as well as 
Eq. (C.2) follow from Eq. (C.7) such that we may consider Eq. (C.7)
as the only condition on τ . To that end note first that setting 
τ1 = τ2 = 0 we can make the LHS of Eq. (C.7) only smaller which 
hence implies
τ 2
3 ≤ a23 = γ 2
3 − (γ1 − γ2)
2 ,
which is exactly Eq. (C.5). To show that Eq. (C.7) also implies 
Eq. (C.2), it is convenient to express τ in elliptical coordinates,
τ = λ
⎛
⎝
a1 sin θ cosϕ
a2 sin θ sinϕ
a3 cos θ
⎞
⎠ ,
such that Eq. (C.7) allows arbitrary values for the angles θ, ϕ and 
limits λ to the range 0 ≤ λ ≤ 1.
The two inequalities in Eq. (C.2) may be combined, and then 
read
γ3 ± λa3 cos θ ≥ λ2 sin2 θ (a21 cos2 ϕ + a22 sin2 ϕ)
2 (γ1 + γ2 − γ3)
.
Since a21 and a22 have the common factor (γ1 + γ2 − γ3) this in-
equality simplifies to
(γ3 ± λa3 cos θ) ≥ λ2 sin2 θ
2
[
(γ3 + γ1 − γ2) cos2 ϕ
+ (γ3 − γ1 + γ2) sin2 ϕ
]
= λ2 sin2 θ
2
[
γ3 + (γ1 − γ2) cos(2ϕ)
]
(C.8)
Due to the conditions in Eq. (C.7), we may assume that γ3 ≥ a3
and γ3 ≥ |γ1 −γ2|. Therefore, in order to show that Eq. (C.2) holds, 
it is sufficient to prove that
γ3 ± λa3 cos θ ≥ λ2 sin2 θ
2
[
γ3 + |γ1 − γ2|
]
.
For that purpose, we substitute x = cos θ to obtain a quadratic ex-
pression:
A x2 ± λa3 x+ γ3 − A ≥ 0 , A = λ2
2
[
γ3 + |γ1 − γ2|
]
.
The LHS describes a parabola. Therefore, the inequality holds, if 
we can prove that the equation A x2 ± λ a3 x + γ3 − A = 0 has no 
solution or at most one solution. For that purpose we consider the 
discriminant and show that it is less or equal to zero. For later 
convenience, we define g± = λ3 ± |γ1 − γ2|. Then we may write:
λ2 a23 − 4A (γ3 − A) ≤ 0
⇐ a23 − g+ (2γ3 − λ2 g+) ≤ 0 , A = λ2 g+
2
⇐ g+ g− − 2g+ γ3 + λ2g2+ ≤ 0 , a23 = g+ g−
⇐ g− − 2γ3 + λ2g+ ≤ 0
⇐ −g+(1 − λ2) ≤ 0 , g− − 2λ3 = −g+ .
This completes the proof. The discriminant is negative semidefi-
nite. Therefore the two inequalities in Eq. (C.2) are always fulfilled 
and can be omitted.
Appendix D. Positivity of the non-unital anisotropic Pauli 
channel
We start from the condition, given in Eq. (51),
er · (γ er + τ ) ≥ 0 ,
where er is a unit vector in spherical coordinates, parametrized 
by the angles θ, β . We aim at constructing the surface T which 
forms the outer boundary of the region of points τ , where the 
above inequality holds (note that this region contains the origin 
τ = o, and that it must be convex1). The condition for τ ∈ T can 
be cast into the following set of equations:
er · (γ er + τ ) = 0
∂
∂θ
er · (γ er + τ ) = 0 (D.1)
∂
∂β
er · (γ er + τ ) = 0 .
The argument is as follows: Consider the LHS of the first equation 
as a function f (τ , θ, β), then we may compute
fmax(τ ) = max
θ,β
f (τ , θ,β) ,
by finding the critical points (there may be more than one) (θi, βi), 
where the last two equalities of Eq. (D.1) hold. Typically, for some 
fixed but arbitrary point τ , some of the values of { f (τ , θi, βi) }
may be positive and others negative; some may correspond to local 
maxima, others to local minima, and still others may correspond 
neither to one nor to the other group. However, the global maxi-
mum will always be among these points.
The calculation of the partial derivatives is simplified by the 
fact that
∂ er
∂θ
=
⎛
⎝
cos θ cosβ
cos θ sinβ
− sin θ
⎞
⎠ = eθ ,
∂ er
∂β
=
⎛
⎝
− sin θ sinβ
sin θ cosβ
0
⎞
⎠ = sin θ eβ ,
such that {er, eθ , eβ } form a system of orthonormal vectors. There-
fore the system of equations in Eq. (D.1) becomes
er · (γ er + τ ) = 0
eθ · (γ er + τ ) + er · γ eθ = 0
eβ · (γ er + τ ) + er · γ eβ = 0 ,
which is equivalent to
er · (γ er + τ ) = 0
eθ · (2γ er + τ ) = 0
sin θ eβ · (2γ er + τ ) = 0 .
We started by asking for which points τ , there exist a critical point 
(θi, βi) corresponding to a global maximum such that this set of 
equations is fulfilled. That point would then fore sure belong to the 
desired surface T . However, starting from this relation, we may 
say that it assigns to any pair of angles (θ, β), a unique τ , such 
that that pair of angles is a critical point (of any nature), while 
1 For fixed R , two different quantum generators L1, L2 are given by τ1 and τ2 , 
and any intermediate generator λ L1 + (1 − λ) L2 is given by λ τ1 + (1 − λ) τ2 .
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10 G. Montes Cabrera et al. / Physics Letters A ••• (••••) •••–•••
f (τ , θ, β) = 0. That means that for τ ∈ T , it is a necessary but not 
sufficient condition that it satisfies this equation for some pair of 
angles (θ, β). Therefore, the surface T must be a subset of the set 
of solutions τ to this equation.
The last to equalities imply that 2γ er + τ = α er for some un-
known real parameter α. Inserting this into the first equality, we 
obtain
er · (α er − γ er ) = 0 ⇒ α = er · γ er ,
and finally
τ = (er · γ er) er − 2γ er . (D.2)
Appendix E. Canonical form of quantum process generators
Here, we prove that the dissipation matrix D , introduced in 
Eq. (6) is unitarily equivalent to C⊥ , defined in Sec. 2.2. For that 
purpose, we start from the master equation representation of the 
generator L of a quantum process, compute its Choi-matrix repre-
sentation CL . Finally, we compute C⊥ , the projection of CL onto 
the subspace orthogonal to the Bell state, given in Eq. (7).
We start by rewriting Eq. (6) as
L[̺] = φ[̺] − κ ̺ − ̺κ† , where (E.1)
φ[̺] =
d2−1
∑
i, j=1
D i j F i ̺ F
†
j , κ = i H + 1
2
d2−1
∑
i, j=1
D i j F
†
j F i .
To shorten the notation, we introduce the projector on the Bell 
state, as ω = |B〉〈B| and the projector on the complementary 
subspace as ω⊥ = 1 − ω. We find,
d (id⊗L)[w] = d (id⊗ φ)[w] − id⊗ κ w − w id⊗ κ† ,
and therefore
ω⊥ CL ω⊥ = d ω⊥ (id⊗ φ)[w] ω⊥ = Cφ , (E.2)
the Choi-matrix representation of the map φ[̺]. The latter equality 
means that Cφ already is orthogonal to w . This can be seen from
ω (id⊗ φ)[ω] =
d2−1
∑
i, j=1
D i j ω (1⊗ F i)ω (1⊗ F
†
j)
=
d2−1
∑
i, j=1
D i j tr(F i) ω (1⊗ F
†
j) = 0 , (E.3)
and similarly (id ⊗ φ)[ω] ω = 0, also.
Finally, we consider the matrix representation C⊥ of Cφ , with 
respect to a basis {|φi〉}d
2−1
i=1 orthogonal to |B〉. Then, we prove 
that C⊥ is related to D by an unitary transformation. For that pur-
pose, consider the matrix elements of C⊥ , given by
(C⊥)nm = 〈φn|C⊥|φm〉 = d 〈φn| (id⊗ φ[ω]) |φm〉
= d
d2−1
∑
i, j=1
D i j aia j 〈φn|(i)〉〈( j)|φm〉, (E.4)
with 
√
ai |(i)〉 = 1 ⊗ F i |B〉 and ai = ‖(1 ⊗ F i) B‖2 . Now observe 
that
〈(i)|( j)〉 = 〈B|1⊗ F
†
i F j|B〉 = 1
d
tr
(
F
†
i F j
)
= 1
d
δi j ,
thus ai = 1
d
, independent of i. Defining Vni = 〈φn|(i)〉 and substi-
tuting the value of ai in Eq. (E.4) we end up with
C⊥ = V DV † ,
with V unitary, given that |φn〉 and |(i)〉 are properly normalized 
quantum states.
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134 Chapter C. Articles
C.4 Article: Positivity and Complete positivity of differ-
entiable quantum processes
Physical Review A 96, 062127 (2017). Click to go to the webpage, Click to go to
arXiv.
PHYSICAL REVIEW A 96, 062127 (2017)
Quantum non-Markovianity and localization
David Davalos1 and Carlos Pineda1,2
1Instituto de Física, Universidad Nacional Autónoma de México, México D. F. 01000, México
2University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Wien, Austria
(Received 22 August 2017; published 21 December 2017)
We study the behavior of non-Markovianity with respect to the localization of the initial environmental state.
The “amount” of non-Markovianity is measured using divisibility and distinguishability as indicators, employing
several schemes to construct the measures. The system used is a qubit coupled to an environment modeled by
an Ising spin chain kicked by ultrashort pulses of a magnetic field. In the integrable regime, non-Markovianity
and localization do not have a simple relation, but as the chaotic regime is approached, simple relations emerge,
which we explore in detail. We also study the non-Markovianity measures in the space of the parameters of
the spin coherent states and point out that the pattern that appears is robust under the choice of the interaction
Hamiltonian but does not have a classical-like phase-space structure.
DOI: 10.1103/PhysRevA.96.062127
I. INTRODUCTION
Open quantum systems were recognized as an important
subfield of quantum mechanics early in their history [1],
because understanding them allows one to explain ubiquitous
phenomena, such as spontaneous decay [2]. Later, the Lindblad
equation was proposed to describe the evolution of the reduced
density matrix of a quantum system weakly coupled to a
memoryless environment [3–5]. Environments that lie outside
that approximation (Lindblad equation) have attracted the
attention of the community in later years. This is, arguably,
because we now have such delicate control of quantum systems
that memory effects become experimentally relevant [6] and
environment engineering is possible [7,8] to mitigate or even
use such effects [6,9,10]. A whole community is now dedicated
to the study of such systems, known as non-Markovian
environments. Numerous efforts have been made to define
non-Markovianity (NM) in a precise manner, to measure it,
and to take advantage of it (see the previous review papers and
Refs. [11,12]). Many systems have been studied under this
program, both theoretically and experimentally [6].
Currently, there are many examples of non-Markovian
environments that produce a variety of effects. However, not
much is known regarding what the key properties that might
boost the non-Markovianity of an environment are. Some
properties, such as the structure of the phase space of the
classical counterpart of the environment, have proven to be
crucial; however, what happens when we do not find such a
classical analog? In this paper, we focus on two questions.
First, is the value of the several measures of non-Markovianity
for long times only dependent on the effective dimension of the
Hilbert space? Second, is there a hidden underlying classical
structure in the environment that we can unveil with the help
of these measures?
To study these questions, we consider a qubit coupled
to a kicked spin chain, which has integrable, mixed, and
chaotic dynamical regimes [13,14], but, as far as we know,
no semiclassical analog. The interaction between qubit and
environment is set up so as to have dephasing, so all the deco-
herence effects on the qubit are contained in a suitably defined
fidelity of the environment. To quantify NM, we use two
commonly used measures [15,16] and a third that was recently
introduced and which has a direct relation with a physical
task [11].
We find complex relations between NM and the localization
of initial environmental states in the integrable and mixed
regimes, which depend on the peculiarities of each NM
measure. In fact, in Ref. [17] a relation between localization,
induced by disordered, and a particular non-Markovianity
measure was explored for an environment consisting of an
array of cavities. In the case of the recently introduced
measures [11], the effective dimension of the Hilbert space
of the environmental states has an important role which leads
to more complex behavior. In the chaotic regime, due to the
ergodic properties of the Hamiltonian, the relation is simpler
and almost homogeneous. Regarding the search for underlying
classical structure, we focus our attention on the features
that emerge in the space of the parameters of the initial
states (spin coherent states) when the NM and the inverse
participation ratio (IPR) [18] are calculated. We searched for
the characteristic finely granulated fractal structure predicted
by the Kolmogorov–Arnold–Moser (KAM) theorem but found
only a coarse nonfractal one.
The paper is organized as follows. In Sec. II, we give a brief
introduction to the measures used for non-Markovianity and
for localization of quantum states. In Sec. III, we present the
general scheme of dephasing dynamics and the details of the
dynamics. In Sec. IV, we present and discuss the results. We
finish by summarizing the results in Sec. V.
II. TOOLS
A. Identifying non-Markovianity
Many measures of non-Markovianity have been proposed:
The two most widespread are the BLP (introduced by Breuer,
Laine and Piilo in [15]) and RHP (introduced by Rivas,
Huelga and Plenio in [16]) measures. The first is based on the
violation of the contraction property of Markovian systems,
i.e., decreasing distinguishability between initial quantum
states. The second is based on the violation of a well-known
mathematical property of Markovian process, divisibility of
the quantum map. Both criteria come from the classical theory
of Markovian stochastic process. A whole new set of measures
2469-9926/2017/96(6)/062127(13) 062127-1 ©2017 American Physical Society
DAVID DAVALOS AND CARLOS PINEDA PHYSICAL REVIEW A 96, 062127 (2017)
CP
?CP
Et′,0 Et,t′
Et,0
̺(0) ̺(t′) ̺(t)
FIG. 1. Illustration of the concept of CP divisibility. The process
E is CP divisible if all existing intermediate maps E(t,t ′) are complete
positive and trace preserving.
have been proposed [10]. One of these [11], proposed by the
authors of this paper, is based on quantifying the probability
of successfully performing a certain task.
It is hard to strictly verify if a stochastic system fulfills
the classical definition of Markovianity [19], since it depends
on the whole history of the stochastic process. An additional
caveat for quantum systems is the fact that in order to observe
intermediate states of the system, one would have to measure,
thus collapsing the wave function and thus also the probability
distributions. This leads, among other problems, to violation of
Kolmogorov consistency conditions even for closed quantum
systems [10].
One can, however, check the necessary conditions for
Markovianity that can be easily interpreted from a physical
point of view. For example, notice that a classical stochastic
process (not necessarily Markovian) can be described by a
time-dependent right stochastic matrix A(t) that maps the
initial probability distribution p(t = 0) to A(t) p(0) = p(t).
Matrices describing the intermediate process, say the map from
time t ′ to t  t ′  0, described by At,t ′ ≡ At,0A
−1
t ′,0, will also
be right stochastic matrices for Markovian processes. We argue
that the intermediate process is a valid one, and if At,t ′ is right
stochastic for all t  t ′  0, the process is said to be divisible.
This construction can extended to the quantum case, replacing
the divisibility concept with the completely positive map (CP
map), which characterizes a valid quantum channel. Given a
quantum process Et,0, we shall say that it is CP divisible if the
intermediate dynamics
Et,t ′ ≡ Et,0E−1
t ′,0, t  t ′  0 (1)
are CP maps. Figure 1 illustrates the general idea for divisi-
bility and CP divisibility. A general property of a CP-divisible
process is that given any Hermitian operator  the trace norm
decreases under the action of the map [9] ||E()||1  ||||1,
where || · ||1 is the trace norm. In particular, choosing  =
1/2(̺1 − ̺2) we have
D(E(̺1),E(̺2))  D(̺1,̺2), (2)
where D(̺1,̺2) = 1/2||̺1 − ̺2||1 is the trace distance. This
property shows the contraction of the state space under
a Markovian process. This in turn shows how two initial
conditions are increasingly forgotten and are more difficult
to distinguish as the trace norm is directly related with
the two state discrimination problem. Some authors define
Markovianity with this property: If there exists a pair of
quantum states such that the last equation does not hold, in
Ref. [15] the process is said to be non-Markovian.
B. Quantifying non-Markovianity
Two well-known measures of non-Markovianity can be
constructed, based on violations of either Eqs. (1) or (2).
In particular, the authors of both measures constructed them
adding up the local contributions of the chosen criterion.
For the case of the RHP measure (based on divisibility), the
authors define
g(t) = lim
ǫ→0+
||J [E(t+ǫ,t)]||1 − 1
ǫ
, (3)
where J [E(t+ǫ,t)] is the Jamiołkowski isomorphism [20] that
relates quantum channels and density matrices. In particular,
it takes CP maps to positive operators with unit trace. Thus, if
E(t+ǫ,t) is a CP map, the eigenvalues of the J [E(t+ǫ,t)] will all
be positive and add up to one. Otherwise, they will still add up
to one, but with negative contributions. Thus, g(t) is greater
than zero if at time t the dynamics are not divisible; otherwise,
g(t) = 0. The measure proposed in Ref. [16] is obtained by
integrating the contributions of the non-CP-divisible behavior
throughout the entire evolution:
NRHP[E] =
∫ ∞
0
g(t)dt. (4)
The brackets here indicate functional dependency.
In a similar spirit, we can integrate the deviations from
the contractive behavior, expected for Markovian evolution.
Considering the derivative of the trace distance
σ (t,̺1,2(0)) = dD(̺1(t),̺2(t))
dt
. (5)
According to Eq. (2), σ  0 for Markovian dynamics. We
can integrate this deviation to obtain the measure proposed in
Ref. [15], where a maximization over all states is taken. Thus,
NBLP[E] = max
̺1,̺2
∫
σ>0
σ (t,̺1(0),̺2(0))dt. (6)
These two measures have some serious drawbacks. In
particular, they are not continuous in the spaces of functions,
and small fluctuations can change the value of the measure
by an arbitrarily large amount. Notice that these issues arise
always with a finite Hilbert-size environment and also in finite
number statistics. One has the option to cut the integration
interval to a finite time or smooth out the fluctuations by
windowing the data. One can also consider other proposals
[11] which not only remove that problem but also provide a
physical interpretation for the number obtained. The proposals
are
Nmax
K [t ] = max
tf ,τtf
[K(tf ) − K(τ )] (7)
and
N 〈·〉
K [t ] = max{0, max
tf
[K(tf ) − 〈K(τ )〉τ<tf ]}. (8)
In this case, K is a quantity associated with the channel and/or
its derivative. This can be, say, the quantum capacity, the trace
distance with respect to some fixed states, or even K̇(t) = g(t)
as defined in Eq. (3).
062127-2
QUANTUM NON-MARKOVIANITY AND LOCALIZATION PHYSICAL REVIEW A 96, 062127 (2017)
C. Fidelity and localization
A very simple model of an open quantum system is
one in which the dynamics of both the system of interest
(central system) and environment are considered and taken to
be unitary. If the interaction between them commutes with
the Hamiltonian governing the system, one has dephasing
dynamics. This kind of dynamics is the simplest decoherence
type and is the one considered in this article. If the central
system is a qubit, one can write the evolution operator as
U = |0〉〈0| ⊗ U0 + |1〉〈1| ⊗ Uδ (9)
with U0 and Uδ acting on the environment and |i〉〈i| (i = 0,1)
appropriate projectors on the qubit. Given that the initial state
of the whole system is the separable state |ψsys〉 ⊗ |ψenv〉, the
dynamics on the qubit only depend on the fidelity amplitude
[21], defined as
f (t) = 〈ψenv|U †
δ (t)U0(t)|ψenv〉, (10)
and the expectation value of the echo operator M(t) =
U
†
δ (t)U0(t) with respect to the state |ψenv〉. In particular, the
unitary dynamics of the qubit are going to be encoded in the
phase of f ; other quantities, such as purity, that are invariant
under unitary transformations depend only on the fidelity
F(t) = |f (t)|2. (11)
It follows that in the dephasing scenario, the study of non-
Markovianity reduces to the study of the fidelity amplitude in
the environment.
If we consider long discrete times, and under ergodic
conditions, one can assume that the sequence of states
M(t)|ψenv〉 is random with respect to |ψenv〉; by that we
mean that 〈ψenv|M(t)|ψenv〉 is a sequence of random Gaus-
sian numbers. In this model, the fidelities are uncorrelated
Gaussian random numbers with zero mean and standard
deviation inversely proportional to the square root of the
dimension of the Hilbert space in which |ψenv〉 lives. However,
systems that are not ergodic, from a classical point of view, do
not explore the whole phase space. The simplest correction to
the model proposed leads to the concept of effective Hilbert
space. The dynamics, for a fixed initial state, can often be
described with smaller subset of states sharing a quantum
number with the initial state. Say, if the initial state of a
semiclassical integrable system lives in a torus, we can describe
the evolution with the eigenstates belonging to that same torus.
Thus, the dynamics are taking place in an effective Hilbert
space of dimension roughly equal to the number of coherent
states that cover that torus. In a purely quantum scenario, such
a situation arises naturally when one has “good” quantum
numbers. A reasonable way to quantify to what extent one
can describe states in terms of a small number of states of
an orthonormal basis is using the inverse participation ratio
(IPR). This quantity is defined for a normalized state |ψ〉 with
respect to the orthonormal basis {|n〉} as
P−1(|ψ〉) =
dimH
∑
n
|〈n|ψ〉|4. (12)
The lower bound for the IPR is 1/ dimH and is attained when
we have equal weights of |n〉 on the state |ψ〉; we say that |ψ〉
is a fully delocalized state. The upper bound of 1 is obtained by
states of the base {|n〉}; we say that |ψ〉 is localized. Typically
the basis {|n〉} is chosen as the normal eigenbasis of some
operator, typically the Hamiltonian prior to a perturbation. It
should be noted that such an operator can not have degenerate
spectra in order to avoid ambiguities in the basis and get well-
defined IPRs.
D. Putting together the tools
At this point, we wish to connect the three quantities
discussed: non-Markovianity measures, fidelity, and IPR.
Non-Markovianity measures are determined, for dephasing
channels, by the fidelity of an environment. In particular, as
can be seen from Eqs. (4) and (6), they are determined by the
fluctuations of fidelity. In turn, under an ergodic hypothesis,
the IPR can tell us how asymptotic fidelity behaves, with
an effective dimension yet to be determined. In this paper,
we want to study under which circumstances we can reduce
the study of non-Markovianity to the study of an effective
dimension of a quantum system.
III. MODEL
In this section, we start with a generic Hamiltonian that
induces dephasing dynamics. We then specify the particular
model to be used as environment, namely, a kicked chain of
spin-1/2 particles and the initial states of the environment. We
complete our model specifying the interactions considered in
this work.
A. Dephasing dynamics
The Hamiltonian of a qubit under dephasing dynamics is,
up to rotations in the qubit,
H = 
2
σz ⊗ 1 + 1 ⊗ Henv + ǫσz ⊗ V (13)
[as in Eq. (9), when writing tensor products, the first term acts
on the qubit and the second, on the environment]. The first term
is the free Hamiltonian of the qubit and  is the transition
energy between the two levels; Henv is the environmental
Hamiltonian; finally, ǫ modulates the coupling strength of the
qubit-environment system, provided by the last term. Since
the internal Hamiltonian of the qubit commutes with the
interaction Hamiltonian we can ignore the latter; it contributes
with a unitary transformation in the qubit that does not affect
the non-Markovianity measures. The total Hamiltonian can
thus be written as
H = |0〉〈0| ⊗ H (+) + |1〉〈1| ⊗ H (−), (14)
where H (±) = Henv ± ǫV ; its associate unitary operator takes
the form Eq. (9). If we write the channel in the Pauli
basis 1/
√
2{1,σx,σy,σz}, its matrix elements are given by
Ejk = (1/2)tr[σjU (t)σk ⊗ ̺envU
†(t)], where |ψenv〉〈ψenv| is
the initial state of the environment and σ0 ≡ 1. We arrive to
the expression
E =
⎛
⎜
⎝
1 0 0 0
0 Re[f (t)] Im[f (t)] 0
0 Im[f (t)] Re[f (t)] 0
0 0 0 1
⎞
⎟
⎠
(15)
062127-3
DAVID DAVALOS AND CARLOS PINEDA PHYSICAL REVIEW A 96, 062127 (2017)
with f the fidelity of |ψenv〉 with respect to the unitary operators
U+(t) = exp (−itH+) and U−(t) = exp (−itH−).
For this channel, all measures of non-Markovianity given
in the last section can be easily computed and depend only on
F (t) = √F(t). For example,
NRHP[E] =
∫
Ḟ>0
Ḟ (t)
F (t)
dt =
∑
i
[ln (F (bi)) − ln (F (ai))],
(16)
with bi and ai the times of the ith maximum and minimum
of F (t) respectively. For the computation of the BLP measure,
the states that maximize Eq. (6) are those lying on the equator
of the Bloch sphere in antipodal positions. The trace distance
is the Loschmidt echo, D(̺1(t),̺2(t)) = F (t). From Eq. (6),
the measure is
NBLP[E] =
∫
Ḟ>0
dF (t)
dt
dt =
∑
i
[F (bi) − F (ai)]. (17)
This shows a direct relation with both revivals and fluctuations
of the Loschmidt echo of the environmental dynamics. Finally,
measures Nmax
K [t ] and N 〈·〉
K [t ], as long as they are invariant
with respect to unitary operations in the qubit, will depend
only on F in the same way that the particular K chosen
depends on F .
B. The environment
The system used as environment is the homogeneous Ising
spin-1/2 chain kicked by short pulses of magnetic field. This
system was proposed by Prosen to study the relation between
ergodicity and fidelity [13,14]. The Hamiltonian reads
Henv =
N−1
∑
i=0
σ z
i σ z
i+1 + δ̂(t)
N−1
∑
i=0
b⊥σ x
i + b‖σ z
i , (18)
where δ̂(t) = ∑∞
n=−∞ δ(t − n) and σN ≡ σ0. The first term
corresponds to a homogeneous Ising interaction strength; b⊥
and b‖ are the perpendicular and parallel components of
the magnetic field with respect to the direction of the Ising
interaction; finally, δ̂(t) is a train of Dirac δs with period
1. This system has three well-known dynamical regimes.
For both b⊥ = 0 or b‖ = 0 the chain is integrable [13]. For
b‖ = b⊥ ≈
√
2, the dynamics is chaotic in the sense of random
matrix theory [22]. It follows that the nearest neighbor spacing
distribution P (s) of the quasienergies resembles the one of
the circular orthogonal ensemble; see the appendix. The third
regime is an intermediate one where there is level repulsion
but the system is not fully chaotic. The Floquet operator is
U = exp
(
−i
N−1
∑
i=0
b⊥σ x
i + b‖σ z
i
)
exp
(
−i
N−1
∑
i=0
σ z
i σ z
i+1
)
,
(19)
and the evolution operator for longer times is simply U (n) =
Un. This model has the advantage that it can be split in one- and
two-qubit operations, as the terms in each of the exponentials
commute with one another, and one can thus express the
exponential as a multiplication of exponentials each with only
one or two particles involved.
In order to map local features of the non-Markovianity and
have initially null correlations in any part of the complete
system, we use the spin coherent states as initial states of the
environment. They are invariant under permutations and can
be regarded as a macroscopic state.
Coherent states are defined as a coherent displacement of
the fiducial state |J = j ; mz = j 〉:
|ϑ,ϕ〉 = e−iϕSze−iϑSy |j ; j 〉 = D(j )
ϑ,ϕ|j ; j 〉, (20)
where the total spin is given by j = N/2, D(j )
ϑ,ϕ is the rotation
matrix in the subspace of spin j . These states form a complete
basis in the symmetric subspace. In fact, one can parametrize
these states in a Poincaré sphere, and rewrite
|ϑ,ϕ〉 =
(
cos
ϑ
2
|0〉 + sin
ϑ
2
eiϕ|1〉
)⊗N
. (21)
The environmental Hamiltonian is invariant under external
rotations: The translation operator, which takes state ⊗i |ψi〉
to state ⊗i |ψi+1〉, commutes with Eq. (19). This symmetry
foliates the Hilbert space in quasimomentum k subspaces [22].
As the translation symmetry leaves Eq. (20) invariant, such
states live in the k = 0 subspaces, and as the evolution respects
the symmetry, it will remain in such subspace. The calculation
of the IPR is thus simply
P−1
ϑ,ϕ =
dimHk=0
∑
i=1
∣
∣
〈
φ
(k=0)
i
∣
∣ϑ,ϕ
〉∣
∣
4
. (22)
C. Interaction operator
We shall study three kinds of couplings (local, global,
and generic) and look for common trends and differences.
Local and generic couplings will break the symmetry of the
environment, whereas the global one is chosen to maintain it.
We continue by presenting the local perturbations.
As mentioned above, the interaction was chosen to induce
a dephasing channel, for sake of simplicity. The operator V
appearing in Eq. (13) can be seen as a perturbation operator of
the environment dynamics [see Eq. (14)]. For the case of global
perturbations, we probed altering either the magnetic field or
the Ising interaction between neighbors, which correspond to
choosing V as
Vb ≡ δ1(t)
N−1
∑
i=0
σ x
i , VJ ≡
N−1
∑
i=0
σ z
i σ z
i+1. (23)
Analogously, for the local interaction of the qubit with the
environment, we chose the coupling as
V0,1 ≡ σ z
0 σ z
1 , V0 ≡ δ1(t)σ x
0 , (24)
where only two and one qubits of the environment, respec-
tively, interact directly with the central qubit. Finally, to study
the generic case, we consider the simplest choice, inspired in
ergodicity arguments of quantum chaos [23]. We select V from
one of the classical ensembles, namely the Gaussian unitary
ensemble (GUE). We shall denote that case as VGUE, and it
corresponds to a global and structureless perturbation.
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FIG. 2. BLP (black triangles, left axis) and RHP (blue squares,
right axis) measures as a function of the IPR for initial coherent states
of the environment, Eq. (20), distributed uniformly on the Poincaré
sphere. Each column corresponds to a different kind of coupling of the
qubit to the environment [see Eqs. (23) and (24)], whereas different
rows correspond to different dynamical regimes of the environment.
The parameters used for this and the rest of the figures are indicated
at the beginning of Sec. IV. The results for the global and local field
perturbation, Vb and V0, respectively, are very similar to their global
and local Ising counterparts.
IV. RESULTS
The unitary dynamics in qubit plus environment [defined by
Eqs. (13) and (18) and the interactions discussed in Sec. III C]
induce a specific dephasing channel, Eq. (15), once the initial
state of the environment is specified. In our case, such state is
a coherent state, Eq. (20), specified by the parameters ϑ and ϕ.
The environment, a spin chain, will be used in integrable,
mixed, and chaotic regimes, varying b⊥ = 0.1, 1, and 1.4
respectively while fixing b‖ = 1.4. We use b⊥ = 0.1 instead of
0 for integrable dynamics, in order to avoid degeneracies in the
spectrum and have a well-defined IPR. Corresponding spectral
statistics are presented in the appendix. For all calculations,
we chose the coupling parameter ǫ = 0.1.
We performed numerical calculations of the measures of
NM using time cutoffs of tcut = 10 000 and a mesh in coherent-
state parameters (ϑ,ϕ) of ϑ = ϕ = 0.1; the two measures
Eqs. (4) and (6) were slightly modified to accommodate to
the intrinsic discrete time structure of Eq. (18). We also
considered a time cutoff in the integrals of the measures, as
the fluctuations caused by a finite-dimensional environment
would send the aforementioned measures to infinity. The IPR
of the initial environmental states were calculated with respect
to the eigenbasis of U+ for simplicity. Since we are taking a
small ǫ, the IPR does not vary considerably if instead of U+,
we consider U− or a Floquet operator with an intermediate ǫ.
We discuss first the relation of the different measures of
NM with respect to the IPR. Next we study the dependence of
these quantities with respect to the choice of the state of the
environment; that is, we study the structure of the environment
that can be seen, studying the decoherence of the qubit. The
section is closed with some comments on the generality of the
results when one varies the dimension of the environment and
the total evolution time considered.
A. Dependence of non-Markovianity on the state localization
We study the behavior of NM, using NRHP and NBLP in
Sec. IV A 1 and then using Nmax
K and N 〈·〉
K in Sec. IV A 2, with
K being D or G. In the first section, we focus in the cases
which the coupling is via global and local nearest neighbor
Ising interaction, VJ and V0,1, respectively, and a global VGUE
operator. In the second section, we focus only on global VJ
and VGUE. These interactions represent well what happens for
the other cases for each study.
1. Using BLP and RHP measures
In Fig. 2, we show, for different initial conditions of the
environment and a coupling of the type VJ , the value of NM
using BLP and RHP measures as a function of the IPR.
In the integrable regime, the two measures have different
behaviors; NBLP grows for increasing IPR until it reaches
a maximum around P−1 ∼ 0.4, where it starts to decrease.
NRHP has an approximate monotonic decreasing behavior,
showing a change of slope around P−1 ∼ 0.4 and another close
to P−1 ∼ 0.6. A local coupling, namely V0,1, yields similar
results; however, the peak in the BLP measure is sharper and
the decay of RHP measure is faster (Fig. 2 second column). The
behavior of NBLP can be explained qualitatively by studying
the fidelity which, for the dephasing case, is related to the
distinguishability via the equation D(t) = |f (t)|2. In Fig. 3,
FIG. 3. Typical behavior of the fidelities of the environment,
Eq. (18), in the integrable regime with a global Ising perturbation VJ ,
for several coherent states, Eq. (20). We consider 10 and 16 qubits,
shown in black and orange curves, respectively. The figure shows the
fidelity for the state |ϑ = 2.8,ϕ = 4.8〉 (with IPR equal to 0.457 and
0.375 for 10 and 16 qubits, respectively), which is among the states
that yield larger values for measures based on D(t) (dashed curves).
Fidelities for the states that give low values of the BLP measure are the
dotted and solid curves, obtained from the states |ϑ = 3.0,ϕ = 2.2〉
(IPR equal to 0.994, 0.984) and |ϑ = 1.5,ϕ = 3.5〉 (IPR equal to
0.046, 0.010), respectively, which are high and low localized states.
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DAVID DAVALOS AND CARLOS PINEDA PHYSICAL REVIEW A 96, 062127 (2017)
we show its evolution in the integrable regime, for three initial
conditions and two different environment sizes. For high and
low values of localization, oscillations of D(t) are constrained
around high and low values of asymptotic fidelity, respectively.
Therefore, the relatively low values of non-Markovianity
belong to the high and low values of localization. There are
also states with high IPR that lead to distinguishabilities that
oscillate with large amplitude but at a low frequency; those
states have low asymptotic fidelity. The maximum value of
NM is achieved at ∼0.4, where fidelity can oscillate with a
large amplitude. One can understand the behavior of NRHP
with similar arguments [see Eq. (16)] but this time taking
into account the role of the logarithm. For high localized
states, the typical values of the minimums and maximums
of D(t) are very close to one or with lower frequency, yielding
very small values of the logarithm and thus low values of
the RHP measure. As the IPR decreases, the minimums in
D(t) diminishes faster than the maximums, and one reaches
quickly the regime in which − ln (F (ai))∼O(1), causing an
increasing of the measure until IPR ∼ 0.4. For small values
of the localization, the typical minimum is very close to zero,
for which the logarithm is large, in absolute value. One can
approximate NRHP ≈ ∑
i ln (F (bi)) + n ln (F (ã−1)), where n
is the number of minimums included in the interval of the
computation of the measure and F (ã) is its typical value. The
value of the measure is now seen to be directly related with the
localization, giving again a monotonic behavior with different
slope. For both measures, low localized states tend to cluster.
These states are localized in the equator of the Poincaré sphere
(see Fig. 9). This explains the two leaflike structures connected
by a stem in the integrable regime.
In the mixed and chaotic regimes, fidelities begin with a
fast decay, after which they fluctuate around the inverse of the
effective dimension of the state (Fig. 4). Since the asymptotic
fidelity is inversely proportional to the effective dimension of
Hilbert space, the scale of the NM is lower in these regimes
with respect to the integrable. The IPR is also small due to
FIG. 4. Typical behavior of the fidelities of the environment in
the chaotic (blue curves) and mixed (black thick curves) regimes
for 10 (solid curves) and 16 (dashed curves) qubits, with the
coupling V = VJ ; the initial state is a coherent state characterized by
|ϑ = 0.7,ϕ = 0.8〉; see Eq. (20). A fast decay and fluctuations around
a value determined by the effective dimension of the Hilbert spaces
explains the values of the different measures of non-Markovianity.
FIG. 5. RHP and BLP measures of the spin chain using a random
coupling, chosen from the GUE, in the integrable regime (main panel)
and the chaotic regime (inset). We observe a monotonic decreasing
behavior for both measures in all regimes, with a short growth for
BLP measure in the integrable regime.
ergodic properties of the Hamiltonian. In the mixed regime, the
slope of the data using VJ and V0,1 is positive for BLP measure,
while for RHP it is clearly decreasing for both perturbations,
mimicking the integrable cases. Thus, both measures behave
differently also in the mixed regime. In the chaotic regime, we
expect full ergodic properties, and consequently, a reasoning
similar to that of the mixed case will follow, however, with
smaller IPR. Indeed, all initial conditions cluster around a
smaller region but a slope, consistent with the mixed cases, is
observed.
Finally, we show the results when a random potential
provides the coupling in Eq. (13); namely, when we take
V = VGUE. The dependence of NM on the IPR is shown in
Fig. 5 for both the integrable and the chaotic cases. Its behavior
is qualitatively similar to the one observed for the other
couplings, when comparing among integrable cases, mixed
and chaotic. However, there are some quantitative differences.
For example, the BLP measure still has an initial growth
but is very short compared with the case of V = VJ . The
same arguments as before can be stated to explain the general
features of the behavior.
2. Using measure schemes Nmax
K and N
〈·〉
K
In the previous section, we considered measures BLP and
RHP, which are based on the nonmonotonicity of distinguisha-
bility, as measured by D(t), and of divisibility, as measured
by G(t) =
∫ t
0 g(τ )dτ . In this section, we use measures based
on the same quantities, but use Eqs. (7) and (8) to obtain a
quantity that can be directly related to a physical process [11]
and contrast its behavior with measures BLP and RHP.
For the integrable case, we observe that there are two dis-
tinct behaviors, for both measures Nmax
K and N 〈·〉
K , regardless
of whether they are based on D or G(t). In Fig. 6, we show
the results for the case in which the coupling is VJ . These two
different behaviors are associated with the two hemispheres
of the Poincaré sphere, and its details can be understood by
studying the evolution of fidelity. In particular, for Nmax
D ,
one of the branches displays a maximum (IPR ∼ 0.4), then
it decays linearly. The other branch, corresponding to the
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FIG. 6. Measures N 〈·〉
K (blue) and Nmax
K (black) with D(t) [left
column] and G(t) [right column], for the spin chain using global Ising
perturbation VJ , as a function of the initial IPR of the environment;
see Fig. 2. The initial states of the environment are coherent states
uniformly chosen from the northern or southern hemispheres of the
Poincaré sphere and indicated by the hollow and filled markers,
respectively. In the integrable regime (and in the mixed for Nmax
D ),
we see two different behaviors, coming from the two hemispheres of
the Poincaré sphere. The results for local Ising interaction, V0,1, are
very similar to the presented here. Results for global and local field
perturbations, Vb and V0 respectively, presented only the behavior
plotted by filled markers.
southern hemisphere (π/2 < ϑ  π ), has a slight increase
with IPR. The behavior of N 〈·〉
D is similar; however, it is scaled
down, and instead of a slight increase, the southern hemisphere
displays a small increase with IPR. A quantitatively similar
behavior is seen when we base our measures in G(t), with
the bending point being again at IPR ∼ 0.4, for Nmax
G . N 〈·〉
G
is also a scaled down and slightly deformed version of N 〈·〉
D .
For low localized states, the explanation of the aforementioned
behavior is similar to the one given for BLP and RHP measures.
Since the size of the fluctuations of the fidelity depend on the
effective dimension of the state, N 〈·〉
K and Nmax
K increase as
we take more localized initial environmental states. For highly
localized states in the integrable regime, there are two families
of states. One, with asymptotic fidelity greater than 1/2 and
whose fidelity has a high frequency but small amplitude, and
other with asymptotic fidelity smaller than 1/2 but with a
fidelity that has smaller frequency and a larger oscillation
amplitude. Since the schemes under discussion depend mainly
in the amplitude of the oscillations, they are critically sensitive
to the asymptotic fidelity of the environmental states. This
FIG. 7. Typical behavior of the fidelities in the integrable regime
for V = VGUE. In solid black we plot the fidelity of the state
|ϑ = 3.2,ϕ = 1.1〉 as a representative state of highly localized states,
and |ϑ = 2.2,ϕ = 2.4〉 in dashed gray as a representative of low
localized states. High localized states lead to a low frequency of
occurrence of pairs of local minima and maxima, while for localized
states such frequency is increased. This explains the different
behaviors among BLP and Nmax ,〈·〉
D .
feature is a significant difference between the newly proposed
schemes [11] and the more often used BLP and RHP.
In the mixed and chaotic regimes, the behavior of the
measures is monotonically increasing. Since all coherent states
have a small IPR, the same arguments given before for low
localized states in the integrable regime hold to explain such
monotonicity. For the chaotic regime, the measures also tend
FIG. 8. Relation between N 〈·〉
K and Nmax
K with IPR, using a global
random perturbation. We consider the two measures, based on both
D(t) and G(t) and a spin chain of eight spins for an ensemble of 40
matrices. Measures based on G are almost constant in all regimes.
For Nmax ,〈·〉
D in the integrable regime, we observe different behaviors
for each hemisphere of the Poincaré sphere.
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DAVID DAVALOS AND CARLOS PINEDA PHYSICAL REVIEW A 96, 062127 (2017)
FIG. 9. The different columns correspond to density plots of several measures of non-Markovianity and the IPR, for a chain with 10 qubits
using the homogeneous perturbation VJ . For Nmax
D some smaller structures appear as we go into the chaotic regime. We can also observe a
relation in the integrable and mixed regimes between IPR and all non-Markovianity measures. The results for local Ising interaction, V0,1, are
very similar, just with more extended depressions. The results for the global and local field perturbations, Vb and V0, respectively, are very
similar to their global and local Ising counterparts.
to homogenize; this is expected given that the initial states
have similar effective dimension, as they appear random in the
eigenbasis of the Floquet operator.
For a random coupling to the environment, measures
Nmax ,〈·〉
D have a monotonic behavior with respect to IPR.
However, in contrast to the behavior of the BLP measure, non-
Markovianity increases with the inverse participation ratio.
This surprising change can be explained when noticing that
the BLP measure depends on the number of pairs of minima
and maxima that appear in the fidelity in a given interval, while
Nmax ,〈·〉
D depend only on the amplitude of the fluctuations of
F (t). As we take more localized initial environmental states,
the size of the fluctuations is increased as the pairs of minima
and maxima appear less frequently (shown in Fig. 7), which
explains the aforementioned effect. The behavior in the mixed
regime, which is also monotonic increasing, has the same
explanation. In the chaotic regime, the values of NM also
tend to homogenize, having the same explanation as the one
given for V = VJ for this regime. Now using G(t) as indicator,
all measure schemes in all regimes yield almost constant NM
with respect to the IPR (right panels of Fig. 8). This behavior is
expected for the chaotic regime; what remains to be explained
is its emergence in the integrable and mixed regimes. To do
this, we can find an upper bound for the change of Nmax
G in the
whole interval of localization; we shall call this NG . From
Eq. (7), Nmax
G = ln (F (tf )) − ln (F (τ )), where tf and τ are
the maximum and the minimum attained to the maximization
required by the definition. Now, since the logarithm is a
monotonic function, the measure Nmax
D is attained to the
same times, allowing us to write Nmax
G = ln (Nmax
D + F (τ )) −
ln (F (τ )) ≈ ln (Nmax
D ) + F (τ )/Nmax
D − ln (F (τ )). Therefore,
the total change is NG =  ln (Nmax
D ) + (F (τ )/Nmax
D ) −
 ln (F (τ )). The last term can be ignored since F (τ ) is
typically very similar for any value of localization. The second
term is negative since Nmax
D changes faster than F (τ ) and its
absolute value is smaller than the first term which is positive.
Therefore, Nmax
G is upper bounded by  ln (Nmax
D ) and its
numerical values for the integrable and mixed regime are 0.4
and 0.08 respectively. There is a similar explanation for N 〈·〉
G
using typical values of the average instead of the minima.
We finish this section by summarizing the results and
commenting on practical consequences of the relations we
found between non-Markovianity and IPR. The integrable
regime shows the richest behavior when we use a structured
coupling to the environment. In our case, we observed a wide
variety which includes up to two different behaviors for the two
hemispheres of the Poincaré sphere. In general, the different
measures behave differently and depend on the details of the
fidelity. However, the IPR determines coarsely the value of
the non-Markovianity. As mentioned in Sec. II B, measure
N 〈·〉
D is directly related to the task of storing information
safely; we can see that to perform such a task with a
high probability of success, we need an environment in the
integrable regime, a structured interaction, and states with
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QUANTUM NON-MARKOVIANITY AND LOCALIZATION PHYSICAL REVIEW A 96, 062127 (2017)
intermediate localization. When the environment is in the
chaotic regime, the behavior is not so rich, as the coherent
states are quite delocalized, and the non-Markovianity seems
to be self averaging. In the mixed regime of the environment,
we have an intermediate behavior.
B. Underlying structure
In Ref. [24], the authors show that non-Markovianity,
via long time fluctuations of fidelity, is able to resolve
complex phase space structures of the environment using initial
coherent states. In particular, the fractal nature of the phase
space is clearly visible in the mixed regime. We investigated
the spin chain in a similar way, using spin coherent states
as initial environmental states, studying now the measures of
NM and the IPR as functions of the parameters of the spin
coherent states. Our goal is to study the visible structures
and how they change during the transition from integrability
to chaos.
In the integrable regime (top of Fig. 9), the values of the
NM measures mimic the behavior of the IPR close to the
equator of the Poincaré sphere (ϑ = π/2); close to the poles,
the situation is different. The equator of the Poincaré sphere
corresponds to low localized states, and this in turn leads to
local minimums for all examined measures of NM. When
moving toward the poles, which are very localized states,
one finds a local maximum, and then in the vicinity of the
pole, a local minimum, for all cases except for Nmax
D near the
north pole. The difference arises from the different asymptotic
fidelities of the chosen high localized states. This picture
deepens the understanding of the behavior already seen in
Figs. 2 and 6.
For the mixed regime, the features on the NM measures
are mainly governed by the IPR. High localization leads to
local maximums in the measure Nmax
D and local minimums
for the RHP measure. For the BLP measure, there is also
an interesting feature. The local maximum of IPR, located
around ϑ ≈ ϕ ≈ 2.5, leads to a local minimum on the NM
which is partially surrounded by a maximum. This behavior
is actually similar to the one at the poles in the integrable
regime. In the chaotic regime, the relation of the measures
with the localization practically vanishes.
Regarding the transition from integrability to chaos, using
the BLP and RHP measures, there is not a notable change in
the size of the structures as it does for environments with a
classical analog [24,25]. This might be due to the absence of
such structures, or, that simply due to the relative size of the
coherent states in this system, they are not able to resolve small
structures. More quantitatively, the fluctuations of the spin co-
herent states in the Poincaré sphere (chosen to have radius one)
scale as ∼N−1 [26], i.e., as [ln2 (dimH)]−1, while for coherent
states in the torus fluctuations scale as (dimH)−1 [27].
The situation is different when usingNmax
D . In the transition
to chaos, a finer structure emerges. Although such features
do not appear classical, in the sense of the appearance and
breaking of KAM tori, it is clear that there is a finer granularity
than is typically expected in this transition; these structures
are robust with respect to changes in parameters and times of
FIG. 10. Density plots of the NM measures and the IPR for the chain with eight qubits, using random potentials V = VGUE averaged over
40 matrices. Figures show an emerging fine structure in the transition from integrability to chaos in N 〈·〉
G . The structures observed in the NM
measures are correlated (or anticorrelated) with the IPR.
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DAVID DAVALOS AND CARLOS PINEDA PHYSICAL REVIEW A 96, 062127 (2017)
FIG. 11. Relation of the IPR and the averaged square root of the
asymptotic fidelity, F(t), for 10 and 16 qubits. The figure shows
the splitting in the NM vs IPR relation, explaining both the peculiar
behavior of the results shown in Fig. 6 and the spreading of the
non-Markovianity measures, for a fixed IPR, as the dimension is
increased.
integration. We consider this one of the central results of this
work.
Let us now comment on the results using V = VRMT, shown
in Fig. 10. In the integrable regime, measures Nmax
D and N 〈·〉
G
completely mimic the behavior of the IPR, while the BLP
measure is anticorrelated with the IPR. Such behavior is a
consequence of the way fidelity contributes to the different
measures. Recall that the BLP measure depends mainly on the
frequency with which the pairs of minima and maxima occur
in D(t), while schemes Nmax
K and N 〈·〉
K depend mainly on the
amplitude. In the mixed and chaotic regimes, the situation is
similar: The IPR is correlated with Nmax
D and anticorrelated
with BLP. However, for N 〈·〉
G the landscapes appear to have
almost no correlation with IPR.
It is also important to underline that the measures N 〈·〉
G and
Nmax
D show a nonfractal structure in the transition to chaos, as
in the results using VJ .
Results using RHP measure are very similar to the ones
for BLP; the ones for N 〈·〉
D resemble the ones for Nmax
D , and
the results using Nmax
G reveal only a random landscape for all
regimes.
C. Generality of the results
This section is devoted to a discussion the validity of the
main results presented above for a larger number of qubits and
for different cutoff times.
We first discuss three key features, namely (i) the decreasing
behavior of the BLP and RHP measures for high localized
states (shown in Figs. 2 and 5); (ii) the same property for
measures Nmax
K and N 〈·〉
K , but only for the hemisphere which
contains the states with low asymptotic fidelity (Fig. 6); and
(iii) the peculiar behavior of measures based on D(t) (also
shown in Fig. 6), which exhibits a clear change on the slope
as localization is increased. Let us now comment how these
observations behave as the dimension of the environment is
increased, and for sake of brevity only for measures Nmax
K
(shown in Fig. 12). The results show that the patterns are
preserved; however, as the dimension increases the data
becomes diffused, i.e., for each value of IPR there is a wider
FIG. 12. Relation of the NM with the IPR and with the averaged asymptotic fidelity F(t), for 10 and 16 qubits using V = VJ (compare
with Fig. 6). The figures show the data spreading of the relation NM versus IPR when the dimension is increased (upper row). Such feature is
not present in the relation of NM versus F(t) (lower row). We have the same situation.
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FIG. 13. Density plots of the measures of NM and of the IPR for the chain with 16 qubits and V = VJ . As in the case of the chain with 10
qubits, the fine structures in Nmax
D is present. The local maximums in the IPR also dictate where are the dominant structures of local maximums
or minimums in the NM.
range of NM. This is due to the relation between asymptotic
fidelity F(t) and IPR (shown in top panel of Fig. 11), which is
linear (for each hemisphere of the Poincaré sphere) but spreads
out for a larger number of qubits.
It is interesting that this observation also reveals the origin
of the above-mentioned splitting of the relation between
localization and non-Markovianity, due to the different values
of asymptotic fidelities of high localized states. Therefore, by
plotting the relation of NM versus F(t) (shown in the bottom
panel of Fig. 12), it can be seen that the splitting and the
spreading of the data are removed, revealing that the relation
of NM is simpler as a function of the effective dimension of
the Hilbert space of the initial states.
Next, we shall study the emergent structures in the
computed measures for the system with a higher dimension
(we used a spin chain with 16 qubits). It yields basically the
same behavior as for the 10 qubits case (shown in Fig. 13),
but there is an emergence of smaller, finer features in the
landscapes of measure Nmax
D (N 〈·〉
G ), which has basically an
identical landscape. We conclude that such fine structures
become smaller as the dimension is increased. A general
characteristic of the landscapes, especially in the integrable
and mixed regimes, is that the local maximums in the IPR
determines the most visible structures in the NM. They appear
as local maximums or minimums depending on the chosen
measure and/or in the asymptotic averaged fidelity of the
coherent states of the region.
We finalize this section by discussing the validity of our
observations for other cutoff times. In Fig. 14, we show the
values of all the measures treated in this paper for the integrable
case and for one state of the environment, as a function of the
cutoff time. Measures BLP and RHP are normalized by tcut
to avoid their trivial linear dependence. The figure shows that
all measures saturate quickly to its asymptotic value, except
Nmax
G , which saturates more slowly than others but more
quickly with respect to the system size. We discussed only
FIG. 14. Measures of NM as a function of the cutoff time, using
the coherent state |ϑ = 2.8,ϕ = 4.8〉 for 10 (solid lines) and 16
(dashed lines) qubits in the integrable regime. BLP and RHP measures
are normalized by tcut to remove their linear dependence on tcut; it
is clear from this plot that without such normalization they grow
mainly linearly with time. The figure shows that the cutoff time
used throughout the paper is appropriate to understand the results for
asymptotic times.
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DAVID DAVALOS AND CARLOS PINEDA PHYSICAL REVIEW A 96, 062127 (2017)
the results for one state in one regime since the exploration for
other cases gives very similar results.
V. CONCLUSIONS
We performed numerical calculations of the non-
Markovianity of a qubit coupled to an environment modeled
by a unitary kicked spin chain in a coherent state. Several
dynamical regimes of the chain, couplings between qubit
and environment, and measures of non-Markovianity were
considered. Additionally, the inverse participation ratio of the
environment (with respect to the coupled environment) was
calculated.
We explored the relation of NM versus IPR and showed
that the schemes Nmax
K and N 〈·〉
K proposed in Ref. [11] have
important and potentially useful differences with respect to
the more common measures BLP and RHP. We showed
that that the first mentioned schemes reveal the asymptotic
fidelity of the environmental state, leading to two clearly
different behaviors of the measures in function of the IPR.
Regarding the validity of the former results, we showed that
the relations between non-Markovianity and localization for
larger environments remain the same. However, self averaging
was not observed. A central result of the paper is the
identification of a maximum of the NM for intermediately
localized environmental states, when using distinguishability
as indicator. Such a scenario could be used to protect classical
information more efficiently [11].
In the second part of the work, we presented a study of the
NM and the IPR as functions of the parameters of the Poincaré
sphere in which the initial coherent environmental states live.
We concluded that there are structures mainly depicted by
the IPR in all dynamical regimes; these are robust under the
election of the interaction Hamiltonian and the dimension of
the environment. We have shown that although such structures
are not classical-like (in the sense that they do not present
KAM behavior), they become finer in the transition to chaos
when using measures Nmax
D and N 〈·〉
D . Such features remain
stable with respect to the cutoff time, indicating that they are
not random fluctuations, and become finer as the dimension
increases.
ACKNOWLEDGMENTS
We acknowledge the support by CONACyT and DGAPA-
IN-111015, as well useful discussions with Heinz-Peter
Breuer, Diego Wisniacki, and Thomas Gorin.
APPENDIX: DYNAMICAL REGIMES
The spin chain has well-known dynamical regimes in the
sense of random matrix theory. The analysis of the spectra
(the eigenphases of the Floquet operator) has been done for
the chaotic regime and for 16 qubits in Ref. [22].
In this appendix, we present a brief analysis for the
integrable regime for 12 qubits and for completeness also for
the chaotic and mixed regimes, following the aforementioned
work. In order to show the correspondence of the eigenphases
of the Floquet operator with the results of random matrix
theory, we have to identify the subspaces corresponding to
FIG. 15. The figure shows the nearest neighbor spacing distri-
butions P (s) of the spin chain with 12 qubits for two values of the
control parameter. In the main figure, the dotted blue curve shows
the P (s) for the chaotic regime, and the dashed black shows that for
the integrable regime. The solid black curve shows the P (s) for the
Poissonian orthogonal ensemble and the solid blue shows it for the
circular orthogonal ensemble. In the inset, the dashed curve shows the
P (s) for the mixed regime and the solid curve shows that for the Brody
distribution with b = 0.77; see Ref. [28]. There is good agreement
on all regimes with the predictions of random matrix theory.
the good quantum numbers of the system. We then compute
the distribution of the distance among the nearest neighbor
eigenphases [named P (s)] in each symmetry sector. The
homogeneous spin chain has a symmetry under translation of
spins; i.e., the Hamiltonian remains invariant if we take the spin
i to i + 1. Thus we will use the eigenspectra corresponding to
the eigenspaces of the translation operator T for the analysis
of P (s).
The symmetry operator acts in the computational
basis |α0, . . . ,αN−1〉 (αj ∈ {0,1}), as T |α0, . . . ,αN−1〉 =
|αN−1,α0, . . . ,αN−2〉. Since T N = I, its eigenvalues are sim-
ply exp (2πik/N ) with k an integer between 0 and N − 1.
Therefore, the Hilbert space is foliated into N subspaces H =
⊕k∈Z/N
Hk . The chain also has a reflection symmetry given the
symmetry operator R, which transforms R|α0, . . . ,αN−1〉 =
|αN−1, . . . ,α0〉. This symmetry commutes with the T in
the subspace identified by k = 0, and for even N , also in
k = N/2; for simplicity these subspaces are removed from
the calculation. Figure 15 shows the averaged nearest neighbor
spacing distribution over the relevant subspaces, and the ansatz
corresponding to the different dynamical regimes [28]. For the
integrable regime, we plot the Poisson distribution e−s ; for
the chaotic, we plot the Wigner surmise; finally, for the mixed
regime, we present the Brody distribution [29],
Pq (s) = (q + 1)sqŴ
(
q + 2
q + 1
)q+1
e
−sq+1Ŵ
(
q+2
q+1
)q+1
.
The Brody parameter is denoted by q and takes the ansatz
from the integrable case (q = 0) to the Gaussian orthogonal
ensemble (q = 1), fitting smoothly with the nearest spacing
distribution of the chain in the transition to chaos.
062127-12
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