Divisibility of qubit channels and dynamical maps
David Davalos
1
, Mario Ziman
2,3
, and Carlos Pineda
1,4
1
Instituto de Física, Universidad Nacional Autónoma de México, México, D.F., México
2
Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, Bratislava 84511, Slovakia
3
Faculty of Informatics, Masaryk University, Botanická 68a, 60200 Brno, Czech Republic
4
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
d
of di-
mension d. In particular, we consider linear maps
on bounded operators, B(H
d
), 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
d
) → B(H
d
) 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
n
⊗ E are
positive maps for all n > 1, where id
n
is the iden-
tity channel on a n-dimensional quantum system. In
general a channel has the form E[X] =
P
i
K
i
XK
†
i
.
The minimum number of operators K
i
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
1
E
2
implies that either E
1
,
or E
2
, 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
(C
∞
) and the set of infinitesimal divisible channels
(C
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
A
n
such that E = (A
n
)
n
. 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
j
with ||id − ε
j
|| ≤ and E =
Q
j
ε
j
.
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
t
∈ C for all t belonging to an interval with min-
imum element 0 and satisfying the initial condition
E
0
= id. Let E
t,s
= E
−1
t
E
s
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 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
t
= e
tL
with
L(ρ) = i[ρ, H]+
X
α,β
G
αβ
F
α
ρF
†
β
−
1
2
{F
†
β
F
α
, ρ}
,
where H = H
†
∈ B(H
d
) is known as Hamilto-
nian, {F
α
} are hermitian and form an orthonor-
mal basis of the operator space B(H
d
), 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
E
t
=
ˆ
Te
R
t
0
L(τ)dτ
(
ˆ
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
L
, C
CP
, and C
P
of L-divisible,
CP-divisible and P-divisible channels, respectively. In
particular, we say E ∈ C
L
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
L
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
CP
(E ∈ C
P
)
Accepted in Quantum 2019-04-24, click title to verify 2
if there exists a CP-divisible (P-divisible) dynamical
map E
t
such that E = E
t
(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
L
[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
P
d
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
L
⊂ C
CP
⊂ C
P
. 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
L
⊂ C
∞
.
Indeed Denisov has shown in [4] that infinitely divis-
ible channels can be written as E = E
0
e
L
, with L a
Lindblad generator, and E
0
idempotent operator such
that E
0
LE
0
= E
0
L. Further, it was shown in Ref. [25]
that E ∈ C
inf
implies det E ≥ 0 and also that E can
be approximated by
Q
j
e
L
j
, i.e. C
CP
= 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
∞
⊂ C
Inf
⊂ C
div
⊆
=
C
L
⊂ C
CP
⊂ C
P
. (2)
The relation between C
P
and C
div
is unknown, al-
though it is clear that C
div
⊂ C
P
is not possible since
channels in C
P
are not necessarily divisible in CP
maps. The intersection of C
P
and C
div
is not empty
since C
CP
⊆ C
div
and C
CP
⊆ C
P
. Later on we will
investigate if C
P
⊆ C
div
or not.
C
C
L
C
div
C
P
C
CP
C
∞
C
P
∩ 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
√
2
{1, σ
x
, σ
y
, σ
z
}, and the stan-
dard Hilbert-Schmidt inner product, the real repre-
sentation for qubit channels is given by [10, 20]:
ˆ
E =
1
~
0
T
~
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 = U
1
DU
2
. (4)
This can be performed by decomposing ∆ in ro-
tation matrices, i.e. ∆ = R
1
DR
2
, where D =
diag(λ
1
, λ
2
, λ
3
) 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
by
ˆ
D =
1
~
0
T
~τ D
, (5)
where ∆ = R
1
DR
2
and ~τ = R
T
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
σ
x
~
λ = (−1, 1, −1) to σ
y
and
~
λ = (−1, −1, 1) to σ
z
(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
P
and C
CP
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
X
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 =
1
4
P
3
i,j=0
R
ij
σ
i
⊗σ
j
the matrix R can
be decomposed as R = L
1
ΣL
T
2
. Here L
1,2
are proper
orthochronous Lorentz transformations and Σ is ei-
ther Σ = diag (s
0
, s
1
, s
2
, s
3
) with s
0
≥ s
1
≥ s
2
≥ |s
3
|,
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
λ
1
λ
2
λ
3
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
i
> 0∀i, i.e.
channels of the positive octant belonging to C
P
. The whole
set C
P
includes three other bipyramids corresponding to the
other vertexes of tetrahedron, i.e. C
P
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
P
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
1
, T
2
such that T = T
1
ET
2
is propor-
tional to
1
~
0
T
~
0 Λ
, (7)
where Λ = diag(s
1
, s
2
, s
3
) with 1 ≥ s
1
≥ s
2
≥ |s
3
|.
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
P
and C
CP
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
L
, 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
P
([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
1
λ
2
λ
3
≥ 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
CP
if and only if it has
diagonal Lorentz normal form with
s
2
min
≥ s
1
s
2
s
3
> 0 (9)
where s
min
is the smallest of s
1
, s
2
and s
3
, 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
ˆ
E given a real matrix
ˆ
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
ˆ
E has a real generator (i.e. a log
ˆ
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
ˆ
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
ˆ
E, as finding an appropriate
parametrization is essential to test for the ccp con-
dition, see Eq. (1). If
ˆ
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.
ˆ
E, the Jordan form is given by
ˆ
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
L
,
unitary conjugations can bring it to C
Inf
\C
L
and vice
versa.
3.3 Unital channels
We shall start our study of unital qubit channels, by
considering Pauli channels, defined as convex combi-
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