Fermat's Library | q-2019-05-20-144 annotated/explained version.

Divisibility of qubit channels and dynamical maps

David Davalos

, Mario Ziman

2,3

, and Carlos Pineda

1,4

Instituto de Física, Universidad Nacional Autónoma de México, México, D.F., México

Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, Bratislava 84511, Slovakia

Faculty of Informatics, Masaryk University, Botanická 68a, 60200 Brno, Czech Republic

Faculty of Physics, University of Vienna, 1090 Vienna, Austria

May 9, 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

arXiv:1812.11437v2 [quant-ph] 7 May 2019

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 H

of di-

mension d. In particular, we consider linear maps

on bounded operators, B(H

), 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(H

) → B(H

) 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 id

⊗ E are

positive maps for all n > 1, where id

is the iden-

tity channel on a n-dimensional quantum system. In

general a channel has the form E[X] =

†

The minimum number of operators K

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] = U XU

†

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 = E

implies that either E

or E

, 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 C

div

and that of in-

divisible channels by C

div

. Following this deﬁnition,

unitary channels are divisible, because for them both

(decomposing) channels E

1,2

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

∞

) and the set of inﬁnitesimal divisible channels

Inf

). 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

such that E = (A

)

. Now, consider channels E

that may be written as products of channels close to

identity, i.e. such that for all  > 0 there exists a ﬁnite

set of channels ε

with ||id − ε

|| ≤  and E =

Its closure determines the set of inﬁnitesimal divisible

channels C

Inf

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→ E

∈ C for all t belonging to an interval with min-

imum element 0 and satisfying the initial condition

= id. Let E

t,s

= E

−1

be the linear map describ-

ing the state transformations within the time interval

[t, s], whenever E

−1

exists.

• A given quantum dynamical map is called CP-

divisible if for all t < s the map E

t,s

is a channel.

• A given quantum dynamical map is called P-

divisible [22] if E

t,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. E

= e

with

L(ρ) = i[ρ, H]+

α,β

αβ



ρF

†

−

†

, ρ}



where H = H

†

∈ B(H

) is known as Hamilto-

nian, {F

} are hermitian and form an orthonor-

mal basis of the operator space B(H

), and G

αβ

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

L(τ)dτ

(

T 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 C

, C

, and C

of L-divisible,

CP-divisible and P-divisible channels, respectively. In

particular, we say E ∈ C

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 = e

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 E

diag

is not invertible, thus

(by deﬁnition) L = log E

diag

does not exist (contains

inﬁnities). Analogously, we say E ∈ C

(E ∈ C

)

Accepted in Quantum 2019-04-24, click title to verify 2

if there exists a CP-divisible (P-divisible) dynamical

map E

such that E = E

(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 ∈ C

[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 =

j,k=1

|j ⊗ jihk ⊗ 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 C

⊂ C

. Simi-

larly, from the earlier deﬁnitions, one can easily ar-

gue that C

∞

⊂ C

Inf

⊂ C

div

. By the deﬁnition of L-

divisibility it is trivial to see that in general C

⊂ C

∞

Indeed Denisov has shown in [4] that inﬁnitely divis-

ible channels can be written as E = E

, with L a

Lindblad generator, and E

idempotent operator such

that E

= E

L. Further, it was shown in Ref. [25]

that E ∈ C

inf

implies det E ≥ 0 and also that E can

be approximated by

, i.e. C

= C

inf

. 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

Inf

⊂ C

div

⊆

⊂ C

. (2)

The relation between C

and C

div

is unknown, al-

though it is clear that C

div

⊂ C

is not possible since

channels in C

are not necessarily divisible in CP

maps. The intersection of C

and C

div

is not empty

since C

⊆ C

div

and C

⊆ C

. Later on we will

investigate if C

⊆ C

div

or not.

div

∞

∩ C

div

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, σ

, σ

}, and the stan-

dard Hilbert-Schmidt inner product, the real repre-

sentation for qubit channels is given by [10, 20]:

E =



t ∆



. (3)

This describes the action of the channel in the Bloch

sphere picture in which the points ~r are identiﬁed with

density operators %

(I + ~r · ~σ). We will write

E = (∆,

t) meaning that E(ρ

) = ρ

∆~r+

In order to study qubit channels with simpler ex-

pressions, we will consider a decomposition in uni-

taries such that

E = U

. (4)

This can be performed by decomposing ∆ in ro-

tation matrices, i.e. ∆ = R

, where D =

diag(λ

, λ

) is diagonal and the rotations R

1,2

∈

SO(3) (of the Bloch sphere) correspond to the uni-

tary channels U

1,2

. 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

D =



~τ D



, (5)

where ∆ = R

and ~τ = R

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

λ = (−1, 1, −1) to σ

and

λ = (−1, −1, 1) to σ

(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 C

and C

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 C

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 C

Inf

, 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 =

i,j=0

⊗σ

the matrix R can

be decomposed as R = L

ΣL

. Here L

1,2

are proper

orthochronous Lorentz transformations and Σ is ei-

ther Σ = diag (s

, s

) with s

≥ s

≥ |s

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

Figure 2: Tetrahedron of Pauli channels. The corners cor-

respond to unitary Pauli operations ( , σ

x,y,z

) while the

rest can be written as convex combinations of them. The

bipyramid in blue corresponds to channels with λ

> 0∀i, i.e.

channels of the positive octant belonging to C

. The whole

set C

includes three other bipyramids corresponding to the

other vertexes of tetrahedron, i.e. C

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 C

but also C

div

, 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 T

, T

such that T = T

is propor-

tional to



0 Λ



, (7)

where Λ = diag(s

, s

) with 1 ≥ s

≥ s

≥ |s

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 C

div

, C

and C

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 C

, which en-

tails a higher complexity.

It was shown ([25], Theorem 11) that full Kraus

rank channels are divisible (C

div

). 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 C

([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 = λ

≥ 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 C

if and only if it has

diagonal Lorentz normal form with

min

≥ s

> 0 (9)

where s

min

is the smallest of s

, s

and s

, 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

E given a real matrix

E. 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

E has a real generator (i.e. a log

E 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

E 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

E, as ﬁnding an appropriate

parametrization is essential to test for the ccp con-

dition, see Eq. (1). If

E 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.

E, the Jordan form is given by

E = wJw

−1

= ˜wJ ˜w

−1

, where w = ˜wK 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 C

unitary conjugations can bring it to C

Inf

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 σ

Pauli

[ρ] =

i=0

ρσ

, (10)

where σ

= and p

≥ 0 with

= 1. The spe-

cial orthogonal normal form of a Pauli channel [see

Eqs. (4) and (5)] has U

= U

= id and ~τ =

0. Thus

Pauli channels are fully characterized only by

λ. No-

tice that every unital qubit channel can be written as

unital

= U

Pauli

. (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 C

div

, C

and C

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 + λ

− λ

≥ 0 (12)

1 + λ

+ λ

≥ 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 (C

div

) 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 C

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

+ λ

≤ 1

− λ

≤ 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.

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 C

div

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 < λ

≤ λ

min

. (15)

They determine a body that is symmetric with respect

to permutation of Pauli unitary channels (i.e. in λ

hence, the set of C

of Pauli channels possesses the

symmetries of the tetrahedron. The set C

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 λ

, λ

(λ

= 1). The logarithm of D, induced by the princi-

pal logarithm of its eigenvalues, is thus

L = Kdiag(0, log λ

, log λ

−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 λ

−

log λ

− log λ

≥ 0 imply

≤ λ

(17)

for all combinations of mutually diﬀerent j, k, l. This

set (channels belonging to C

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

≤ 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

. We will see it is also necessary.

Indeed, the complete positivity condition requires

η, λ ≤ 1, thus, it remains to verify only the condition

≤ 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 C

implies C

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 C

Let us stress that the obtained subset of L-divisible

channels does not possess the tetrahedron symme-

tries. In fact, composing D with a σ

rotation

= diag(1, −1, −1, 1) results in the Pauli channel

= diag(1, −λ

, −λ

, λ

). Clearly, if λ

are positive

(D is L-divisible), then D

has non-positive eigenval-

ues. Moreover, if all λ

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 D

= diag(1, η, −λ, −λ),

= diag(1, −λ, η, −λ), D

= diag(1, −λ, −λ, η),

thus forming three two-dimensional regions inside the

tetrahedron. Take, for instance, D

that speciﬁes a

plane (inside the tetrahedron) containing I, σ

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)







−1

(19)

where k ∈ Z and K, as mentioned above, belongs to a

continuum of matrices that commute with D

. Note

that L is always non-diagonal. For this case (simi-

larly for D

and D

) 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

k,K

= wK log (J)

−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)

−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)

−1

= Kdiag (0, log(c)) ⊕



log(|z|) arg(z) + 2πk

−arg(z) − 2πk log(|z|)



−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)

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

complex

i.e. w = . In such case the ccp condition is reduced

+ b

≤ 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

complex

D = diag (1, η, λ, λ) with

η = c and λ = sign(ab)

√

+ b

. 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

complex

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

complex

, 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 < λ

min

= λ

≤

≤ λ

< 1, thus, the condition λ

≤ λ

triv-

ially holds. Since λ

≤ λ

, it

Accepted in Quantum 2019-04-24, click title to verify 7

Figure 4: Tetrahedron of Pauli channels, with qubit unital

L-divisible channels of the form

complex

(see main text). Note

that the set does not have the symmetries of the tetrahedron.

follows that λ

≤ λ

, 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 λ

≤ λ

. In

conclusion, the conditions of CP-divisibility and L-

divisibility for Pauli channels with positive eigenval-

ues coincide, thus, in this case C

implies C

Concatenating (positive-eigenvalues) Pauli chan-

nels with D

x,y,z

one can generate the whole set of

Pauli channels. Using the identity C

= C

Inf

and considering Eq. (11)we can formulate the follow-

ing theorem:

Theorem 4 (Inﬁnitesimal divisible unital channels).

Let E

unital

be an arbitrary inﬁnitesimal divisible unital

qubit channel. There exists at least one L-divisible

Pauli channel

E, and two unitary conjugations U

and

, such that

unital

= U

Notice that if E

unital

is invertible,

E = e

Let us continue with another equivalence relation

holding for Pauli channels. Regarding inﬁnitely divis-

ibility channels, we know that, in general, C

⊂ 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 E

, where E

is an idem-

potent channel satisfying E

= E

L and L has

Figure 5: Tetrahedron of Pauli channels with part of the set

of CP-divisible, see Eq. (15), but not L-divisible channels

) shown in purple. The whole set C

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 E

diag

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 = Ne

N for all L. The channel N belongs to the clo-

sure of C

, because a sequence of channels e

with

= diag (0, −n, −n, −n) converges to

N in the limit

n → ∞. For the case of E

being the identity channel

we have E = e

, thus, trivially such inﬁnitely divisi-

ble channel E is in C

too. It remains to analyze the

case of diagonalization channels. First, let us note

that the matrix of e

is necessarily of full rank, since

det

E 6= 0. It follows that the matrix

E =

diag

has rank two as

diag

is a rank two matrix, thus, it

takes one of the following forms

= diag (1, λ, 0, 0),

= diag (1, 0, λ, 0),

= 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

belongs to (the closure of) C

. Let us deﬁne the

channels

λ,

= diag (1, , , λ) with  > 0. The com-

plete positivity and ccp conditions translate into the

inequalities  ≤

1+λ

and 

≤ λ, respectively; there-

fore one can always ﬁnd an  > 0 such that

λ,

a L-divisible channel. If we choose  =

√

λ/n with

n ∈ Z

, the channels

z,n

= diag



√

λ/n,

√

λ/n, λ



form a sequence of L-divisible channels converging to

when n → ∞. The analogous reasoning implies

that

∈ C

too. Let us note that one parame-

Accepted in Quantum 2019-04-24, click title to verify 8

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 E

are convex combinations of the complete

diagonalization channel

= diag (1, 0, 0, 1) and the

complete mixture contraction

N. This completes the

proof.

Finally, let us remark that using theorem 23 of

Ref. [25] we conclude that the intersection C

∩ C

div

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





. Therefore we conclude that

up to unitaries, C

∩ C

div

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 C

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 C

div

, C

and C

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 C

div

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 C

is given by Eq. (8) (note that it only

depends on

λ), and C

is tested, for full Kraus rank

non-unital channels, using Eq. (9), see Ref. [24] for

the calculation of the s

’s. For the characterization of

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 C

, 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)

. 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

channels.

A central feature of Figs. 7 and 8 is that the set

div

\ C

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

E < 0,

then E is entanglement-breaking, i.e. all qubit chan-

nels outside C

are entanglement breaking.

Proof. Consider the Choi-Jamiołkowski state of a

channel E written in the factorized Pauli operator ba-

sis [24] τ

⊗σ

and let

E be its repre-

sentation in the Pauli operator basis. Then the matrix

identity R =

EΦ

with Φ

= diag (1, 1, −1, 1) holds.

Since det

E < 0 it follows that det R = −det

E > 0.

Using the aforementioned Lorentz normal decompo-

sition R = L

with det L

1,2

> 0 and

R diagonal,

see Ref. [24]. The transformations L

1,2

correspond to

one-way stochastic local operations and classical com-

munications (SLOCC) of τ

, thus,

R is an unnormal-

ized two-qubit state with det

R > 0. The channel cor-

responding to

R (in the Pauli basis) is

G =

RΦ

Since the latter is diagonal, then G is a Pauli channel

with det

G < 0. A Pauli channel has a negative deter-

minant, if either all λ

are negative, or exactly one of

them is negative. In Ref. [28] it has been shown that

the set of channels with λ

< 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

 C

div

 C

div

 EB boundary

Figure 7: We show two slices of the unitary tetrahedron (ﬁgure in the left) determined by

= 0.4 (shown in the center)

and

= −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 C

div

\ C

is always inside the octahedron of entanglement breaking channels.

negative determinant is entanglement breaking, thus,

is separable and given that SLOCC operations can

not create entanglement [11], we have that τ

is sep-

arable, too. This implies that if det

E < 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, A

NOT

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 E

according to divisibility hierarchy, i.e.

δ[E] =











1 if E ∈ C

2/3 if E ∈ C

\ C

1/3 if E ∈ C

\ C

0 if E ∈ C \ C

(23)

A similar function can be deﬁned to study the transi-

tion to/from the set of entanglement-breaking chan-

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 ± 2λ

≥

1 + 4(λ

± λ

)

. A cut corresponding to

= 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

. A part of the channels with negative eigenvalues belong-

ing to C

lies outside C

\ C

, see green lines. As for the

unital case a central feature is that the channels in C

div

\ C

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 A

NOT

: ρ 7→ ( − ρ)/3. This is a

rank-three qubit unital channel, thus, it is indivisible

[25]. Moreover, det A

NOT

= −1/27 implies that this

channel is not achievable by a P-divisible dynamical

map. It is worth noting that A

NOT

belongs to C

div

A speciﬁc collision model was designed in Ref. [21]

simulating stroboscopically a quantum dynamical

map that implements the quantum NOT gate A

NOT

in ﬁnite time. The model reads

(%) = cos

(t)% + sin

(t)A

NOT

(%) +

sin(2t)F(%) ,

(25)

where F(%) = i

[σ

, %]. This quantum dynamical

map achieves the desired gate A

NOT

at t = π/2.

Let us stress that this dynamical map is unital,

i.e. E

( ) = 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[E

χ[E

] and the value of the det E

. We see the tran-

sitions C

→ C

\ C

→ C

div

\ C

→ C

div

and

back. Notice that in both plots the trajectory never

goes through the C

\ C

region. This means that

when the parametrized channels up to rotations be-

long to C

, so do the original ones. The transition be-

tween P-divisible and divisible channels, i.e. C

and C

div

, 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

\ C

div

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 C

div

\ C

region; likewise, the chan-

nel stops being entanglement breaking shortly after

it leaves the C

div

\ C

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 =

+ ω



†

a +



+ g



−

†

+ σ



. (26)

By initializing the environment in a coherent state

|αi, one gets the familiar collapse and revival setting.

Considering a particular set of parameters shown

Figure 9: (top left) Tetrahedron of Pauli channels with

the trajectory, up to rotations, of the quantum dynamical

map Eq. (25) leading to the A

NOT

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 C

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.

2 π

δ[E

], det E

, χ[E

]

div

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 A

NOT

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

→ C

div

→ C

div

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

δ[E

], p

(t), χ[E

]

div

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 |αi. The blue curve shows the probability

of ﬁnding the two-level atom in its excited state, p

(t). The

ﬁgure shows that the fast oscillations in δ occur roughly at

the same frequency as the ones of p

(t), see the inset. Notice

that there are fast transitions between C

\ C

and C

\ C

occurring in the region of revivals, with a few transitions

between C

\ C

and C

in the second revival. The function

χ shows that during revivals channels are not entanglement

breaking, but we ﬁnd that channels belonging to C

div

are

always entanglement breaking, in agreement with theorem 6.

The particular chosen set of parameters are α = 6, g = 10,

= 5, and ω

= 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[E

] and χ[E

], together with

the probability of ﬁnding the atom in its excited state

(t), to study and compare the divisibility properties

with the features of the collapses and revivals. The

probability p

(t) is calculated choosing the ground

state of the free Hamiltonian ω

/2σ

of the qubit,

and it is given by [13]:

(t) =

hσ

(t)i + 1

, (27)

where

hσ

(t)i = −

∞

n=0



∆

4Ω



1 −

∆

4Ω



cos (2Ω



with P

= e

−|α|

|α|

/n!, Ω

∆

/4 + g

n and

∆ = ω

− ω

the detuning.

The divisibility indicator function δ exhibits an os-

cillating behavior, roughly at the same frequency of

(t), see inset in Fig. 11. The ﬁgure shows fast peri-

odic transitions between C

\ C

and C

\ C

oc-

curring in the region of revivals. There are also few

transitions among C

and C

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 C

div

\ C

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

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 C

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 R

matrix, both written in the Pauli basis:

E =







1 0 0 0

0 −

0 0

0 0 −

0 0







, (28)

and







1 0 0 0

0 −

0 0







. (29)

Using the algorithm introduced in Ref. [24] to calcu-

late R

’s Lorentz decomposition into orthochronous

proper Lorentz transformations we obtain







4 0 0 1

0 −γ

0 0

0 0 −γ

1 0 0 4







, (30)







89 + 9

√

97 0 0 −8

0 −γ

0 0

0 0 −γ

−8 0 0 89 + 9

√







and







11 +

109

√

0 0 −

√

97+1

√

97+873

0 −

0 0

1 +

√

0 0

−1 +

√







with γ

√

15, γ

= 3

178

√

97 + 1746, and γ

√

30. Although the central matrix Σ

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 L

1,2

being proper and orthochronous, we can bring Σ

to the desired form by conjugating Σ

with G =

diag (1, 1, 1, −1):

−1

G =







11 +

109

√

0 0

√

97+1

√

97+873

0 −

0 0

−

1 +

√

0 0

−1 +

√







In both cases (taking Σ

or G

−1

G as the normal

form of R

), 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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