Log-Convex set of Lindblad semigroups acting on $N$-level system
Fereshte Shahbeigi David Amaro-Alcalá Zbigniew Puchała Karol Życzkowski
quant-ph math-ph math.MP



Department of Physics, Ferdowsi University of
Mashhad, Mashhad, Iran
Department of Physics, Sharif University of Technology, Tehran, Iran

Instituto de Física, Universidad Nacional
  Autónoma de México, México D.F. 01000, México

Institute of Theoretical and Applied Informatics, Polish Academy
of Sciences, ulica Bałtycka 5, 44-100 Gliwice, Poland
Faculty of Physics, Astronomy and Applied Computer Science,
Jagiellonian University, ul. Łojasiewicza 11,  30-348 Kraków, Poland

Faculty of Physics, Astronomy and Applied Computer Science,
Jagiellonian University, ul. Łojasiewicza 11,  30-348 Kraków, Poland
   Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników
32/46, 02-668 Warszawa, Poland
 National Quantum Information Centre (KCIK), University of Gdańsk,
  81-824 Sopot, Poland





We analyze the set A_N^Q of mixed unitary channels represented in the
Weyl basis and accessible by a Lindblad semigroup acting on an N-level quantum
system. General necessary and sufficient conditions for a mixed Weyl quantum
channel of an arbitrary dimension  to be accessible by a semigroup are
established. The set A_N^Q  is shown to be log–convex and star-shaped
with respect to the completely depolarizing  channel.
A decoherence supermap acting in the space of Lindblad operators
transforms them into the space of Kolmogorov generators of
classical semigroups. We show that for mixed Weyl channels
the hyper-decoherence commutes with the dynamics,
so that decohering a quantum accessible channel
we obtain a bistochastic matrix form the set A_N^C
of classical maps accessible by a semigroup.
Focusing on 3-level systems we investigate the geometry
of the sets of quantum accessible maps, its classical counterpart
and the support of their spectra.
We  demonstrate that the set  A_3^Q
is not included in the set U^Q_3
of quantum unistochastic channels,
although an analogous relation holds for N=2.
The set of transition matrices obtained
by hyper-decoherence of unistochastic channels of order N≥ 3
is shown to be larger than the set of unistochastic matrices of this order,
and yields a motivation to introduce the larger sets of k-unistochastic
matrices.



Log-Convex set of Lindblad semigroups acting on N-level system
    Karol Życzkowski
    March 26, 2020
==============================================================





§ INTRODUCTION

Open quantum systems have attracted a lot of interest from
different perspectives <cit.>. Two major
methods to describe the dynamics of a quantum system
interacting with an environment are frequently used.
The first one takes place in continuous time,
as one deals with differential equations of motion
governing the time evolution of the system.
The most general form of such a non-unitary quantum dynamics
was introduced by Gorini, Kossakowski, Sudarshan, and independently by
Lindblad <cit.>. For a historical account on these
fundamental results, see the recent review <cit.>.

In the second, stroboscopic approach, time changes in a discrete way
and the evolution of a quantum system is described by
quantum operations: linear  completely positive maps
which preserve the trace of the density matrix.
Let
Φ:M_N→M_N be a
quantum operation acting on an N-dimensional system
described in the complex Hilbert space ℋ_N.
The action of this map on an
N-level density matrix ρ in terms of Kraus operators, K_j,
reads,
 Φ(ρ)=∑ K_jρ K_j^†.  In an
alternative approach, one can represent  a channel by the corresponding
superoperator Φ acting on an extended Hilbert
space, ℋ_N⊗ℋ_N.
The superoperator, denoted by the same symbol Φ as the map it represents,
 can be determined  by its entries in a product basis,
  Φ_ m μnν = ⟨m μ|Φ|n  ν⟩=∑_j
(K_j)_mn(K_j)_μν.
It is known that a given quantum channel may have many corresponding
sets of Kraus operators, while the superoperator
associated with a channel is unique.

  Although the superoperator  Φ is not necessarily Hermitian,
  reshuffling its entries results in a positive
  semidefinite dynamical matrix <cit.>, also known as the Choi matrix  <cit.>
  of the channel,
   D_Φ = Φ^R, where
X_ m μnν^R
=X_ m nμν – see  <cit.>.
It is customary to refer to the N^2 eigenvalues of Φ as
the channel eigenvalues.
Due to Hermiticity and trace preserving conditions,
the eigenvalues of a channel
are either real or appear in complex conjugate pairs.
There exists at least one leading eigenvalue equal to unity,
which  corresponds to the invariant state <cit.>.
 It is interesting to study relations between both approaches <cit.>
 and investigate which discrete channel Φ can be accessed by a
 continuous dynamics in the Lindblad form.
 This general question, sometimes called embedding problem
 or Markovianity problem, is known to be hard for a large system size  <cit.>.
 A necessary condition is the divisibility property <cit.>,
 so that the map Φ can be represented as a composition of two
 other quantum maps, Φ=Ψ_2Ψ_1,
 which are completely positive and trace preserving
 such that one of them is not unitary.
 There exist non-divisible quantum maps,
 which correspond to non-Markovian dynamics <cit.>.

 In the simplest case of  N=2
 and one-qubit unital maps,
 which are unitary equivalent to Pauli channels,
  Φ_p=∑ p_i σ_i ⊗σ_i,
 necessary and  sufficient conditions for
 accessibility by a dynamical semigroup were
 established <cit.>.
 If all eigenvalues λ_i of Φ_p are real and positive,
 the map Φ
 is shown to be accessible by a semigroup if and only if
 the relation for subleading eigenvalues,  λ_i≥λ_jλ_k,
 holds for
 any choice of three different indices i, j and k from the set
  {1,2,3}, as the leading eigenvalue reads
 λ_0=1.
 These relations are equivalent to the following inequalities
 for the weights in the convex combination of Pauli matrices,
   p_0p_i ≥ p_jp_k,
 where the weight p_0 standing by σ_0=_2
 is assumed to be the largest.
 
 Furthermore, it was shown that the set 𝒜^Q_2
 of quantum channels accessible by a semigroup,
 forms a fourth part of the set U_2^Q
 of quantum unistochastic channels <cit.>,
 determined by unitary matrices of size  N^2=4
 which couple the principal system with the environment
 initially in the maximally mixed state <cit.>.
Further geometric properties of the set of one qubit Pauli maps
were recently investigated in <cit.>.

The aim of the present work is to generalize earlier results
on the structure of the set  A_2^Q
of one-qubit accessible channels <cit.>
for maps acting on  N-level systems.
In this work we concentrate on a class of
mixed unitary channels (also called random external fields <cit.>),
defined as convex combinations of rotations
by Weyl matrices, which can be considered as
unitary generalizations of the Pauli matrices.
The set W_N of Weyl mixed unitary channels of size N
was recently investigated  to study the
degree  of non-Markovianity  <cit.>.
Here we find necessary and sufficient conditions for
a quantum Weyl channel acting on an N dimensional system
to be  accessible by a Lindblad  semigroup.
Furthermore, we  investigate  the set
 A_N^Q of Weyl channels accessible by a semigroup
 and demonstrate that it is log-convex and star-shaped.
This property is also inherited by the
corresponding set  A_N^C
of circulant bistochastic matrices accessible by classical
semigroups.
As the relative volumes of different classes of quantum maps
were investigated <cit.>,
we analyze the ratio of the volume of the set A_N^Q
of accessible maps to the total volume of the set
 W_N of Weyl channels.

 This paper is organized as follows.
 In Section <ref> we fix the notation,
  introduce the basis of unitary Weyl  matrices
 and define channels accessible by a  Lindblad semigroup.
 Furthermore, we recall the definitions
 of stochastic, bistochastic and unistochastic quantum channels,
 and discuss the corresponding classical transition
 matrices.
 In Section <ref>, we establish a condition for
the Weyl  channels  to be
 accessible by a semigroup.
  Geometric properties of the set
 A_N^Q of accessible quantum channels are
  analyzed in Section <ref>.


 In  Section <ref> we investigate properties
of classical transition matrices generated
from the Weyl channels by  hyper-decoherence,
and demonstrate that these bistochastic matrices are circulant.
As decoherence acts in the space of quantum states <cit.>,
an analogous process acting in the larger  space of quantum maps and
described by a supermap is sometimes called hyper-decoherence.
We show that  the semigroup dynamics
commutes with hyper-decoherence for mixed Weyl channels,
so that any accessible quantum map subjected to decoherence leads to
a bistochastic matrix accessible by a classical semigroup.
In mathematical literature such Markov stochastic matrices are also
called embeddable <cit.>
and this property,  related to  classical analogue of divisibility of a channel  <cit.>,
was also studied in quantum context <cit.>.

Explicit conditions for a circulant bistochastic matrix,
obtained  by hyper-decoherence of mixed Weyl channels,
to be accessible by classical semigroup are derived
and geometry of the set  A̧_N^C of these matrices is analyzed.
 In Section <ref>,  we focus on 3-level systems,
  called qutrits, to emphasize
 prominent  difference with respect to the simpler
case of maps  acting on 2-level systems.
  Possible relations between the sets of
 accessible channels,  unistochastic channels,
 and quantum channels,
 such that the corresponding classical transition matrix
 is unistochastic  are analyzed
 in Section <ref>.
  In particular, we demonstrate that for N=3
 there exists a quantum map accessible by a semigroup,
  which is not unistochastic. The work is concluded in Section   <ref>,
 in which we summarize results achieved and present a list of open questions.
 Some properties of unistochastic matrices of order N=3 and unistochastic channels
 acting in this dimension are discussed in Appendices.



 

§ SETTING THE SCENE

 
   

 §.§ Unitary Weyl matrices

 

In this Section we review necessary results on discrete
quantum maps and continuous dynamics  of the Lindblad form.
We will be mainly concerned  with  mixed unitary channels,
defined by a convex combination of unitary transformations,

    Φ_p (ρ)=∑_n=1^M p_nV_nρ V_n^†.

Here {V_n}_n=1^M denotes an arbitrary collection of M unitary
matrices of order N, while
p⃗=(p_1,…, p_M)^T represents a probability vector.
By  definition, any mixed unitary channel is unital, i.e.
Φ()=.
In this work we focus our attention on the set  W_N  of
mixed unitary channels
in the form of Eq. (<ref>),
where matrices V_n form the basis of
Weyl unitary matrices,

    U_kl=X^kZ^l,   k,l=0,…,N-1.

Here X denotes the shift operator, X|i⟩=|i⊕1⟩,
and Z is a diagonal unitary matrix,
Z={1,ω__N,ω__N^2,…,ω__N^N-1}
with  ω__N=^i2π/N.
The notation i⊕ j, refers to addition i+j modulo N.
For convenience, we often  use a single index
  μ=0,…,N^2-1, instead of two, μ = Nk+l.

 This set of unitary matrices, applied in context of quantum
 theory  by Weyl <cit.>,
 and later popularized by Schwinger <cit.>,
was invented by Sylvester <cit.>  already in 1867.
These matrices form N-dimensional unitary representation
 of the elements of the Weyl-Heisenberg group
 and they are sometimes called clock–and–shift
matrices, which  refers to the operators Z and X, respectively.

With the exception of the identity matrix, U_00= _N,
all remaining Weyl matrices are traceless.
Full set of N^2 matrices U_μ  forms an orthogonal basis
composed of unitary matrices of size N,
which can be considered as a unitary generalization of Pauli
matrices of size N=2. It is known that
any one-qubit bistochastic channel Φ_B is unitarily equivalent to a
Pauli channel,
Φ_B(ρ)= U_2(Φ_p(U_1ρ U_1^†))U_2^†,
which is not the case for Weyl channels acting in higher dimensions.
Furthermore, for one-qubit systems all unital channels  are mixed
unitary, which is not true in higher dimensions  <cit.>.



It is easy to see that the Weyl unitary matrices of an arbitrary order N
enjoy the following property,

a) U_klU_k^' l^'=ω__N^lk^'
U_k⊕ k^',l⊕ l^',
 which will be crucial in the further analysis.
 In turn it implies,

b) Commutativity up to a phase,  U_klU_k^' l^'=
ω__N^lk^'- kl^'
U_k^' l^'U_kl,

c) Reflection symmetry, U_k,lU_-k⊕ N,-l⊕ N=
ω__N ^-lk _N. 


   

 §.§ Mixed unitary channels accessible by a Lindblad semigroup

  
Through celebrated GKLS theory <cit.>,
the time evolution of a quantum system experiencing
Markovian dynamics can be described by a Lindblad generator ℒ
 and represented
 as ρ(t)=^ℒt[ρ(0)]=Λ_t[ρ(0)].
The action of the generator ℒ on a quantum state ρ of size N
can be written in terms of at most N^2-1 jump operators L_j,

    ℒ(ρ)=∑_j=1^N^2-1( L_j ρ L_j^† -1/2 L_j^† L_j ρ -1/2ρ L_j^† L_j ).

The corresponding superoperator  ℒ
can be represented as a matrix acting in the extended space  H_N ⊗ H_N,

    ℒ=∑_j=1^N^2-1 L_j ⊗L_j	- 1/2∑_j L_j^† L_j ⊗_N- 1/2_N ⊗∑_j  L_j^T  L_j.


With these preliminaries in mind, let Φ_p⃗ be a mixed
unitary quantum channel written in terms of the Weyl unitary
matrices:

    Φ_p⃗=∑_k,l=0^N-1 p_klU_kl⊗U_kl=∑_μ=0^N^2-1p_μ U_μ⊗U_μ.

Probabilities in this convex combination form an (N^2-1)-dimensional simplex
 p⃗∈Δ_N^2-1⊂ℝ^N^2-1.
In this paper we find out,  for which p⃗
the map is semigroup accessible,
Φ_p⃗∈ A_N^Q,
so that there exists a Lindblad generator ℒ
and t>0 such that ^ℒt=Φ_p⃗.

A quantum channel Φ belongs to a dynamical semigroup if and only if there is
a Lindblad generator ℒ such that Φ=^ℒ, where ℒ fulfils the following three conditions <cit.>:

 
  * Hermiticity preserving, ℒ[Y^†]=ℒ[Y]^†
 for any operator Y.
 This in turn results in the following condition
  for entries  of the Lindblad generator,
  ℒ_ m μn ν=ℒ_μ mν n.
 
  * Trace preserving, (ℒ[Y])=0 for any Y, which implies,
 ∀ n,ν:  ∑_mℒ_ m mn ν=0.
 
  * Conditionally completely positive, (_N^2-ψ_+)(ℒ⊗_N)[ψ_+](_N^2-ψ_+)≥0,
  where |ψ_+⟩=1/√(N)∑|ii⟩ is the maximally entangled state acting on the composite space
H_A  ⊗ H_B.
 This condition is equivalent to (_N^2-ψ_+) ℒ^R(_N^2-ψ_+)≥0 in
 which ℒ^R is the reshuffled form of  ℒ.


However, as  logarithm of a matrix is usually not unique
<cit.>, it is not  straightforward to
verify the last condition for an arbitrary map Φ.

 

 §.§ Bistochastic quantum maps and classical bistochastic matrices

  

Any completely positive map Φ acting on a
quantum system of size N can be represented
by the corresponding dynamical matrix,
determined by the Choi–Jamiołkowski isomorphism,
D_Φ=N( Φ⊗𝕀_N)|ψ_+⟩⟨ψ_+|.
As discussed in the previous Section,
the dynamical matrix is related to the superoperator Φ
by a particular reordering of the entries of the matrix, called
reshuffling, D_Φ = Φ^R.

A coarse-graining channel sends any density matrix ρ into its diagonal,
D(ρ)=ρ̃:= diag(ρ),
which can be interpreted as a probability vector.
Hence such a decoherence process <cit.>,
equivalent to stripping away all off-diagonal elements of the density matrix,
projects the set of quantum states Ω__N
into the classical probability  simplex Δ_N.

An analogous process applied to any
Choi matrix D_Φ produces a diagonal matrix D̃.
Its diagonal forms a  vector t⃗ of length N^2 which
 can be reshaped into a classical transition matrix
T(Φ)=T(D^R).
More formally, writing  t⃗ = (D_Φ),
we have T_ij=t⃗_k,
 where k=N(i-1)+j, with i,j =1,…, N.

The process of stripping of the off-diagonal elements of the Choi matrix
can thus be called  decoherence process
or also hyper-decoherence, to emphasize that it acts
on a larger space of generalized states or quantum maps <cit.>.
More precisely, we describe it in the formalism of super-maps,
which send the set of all quantum operations into itself, Φ'=Γ(Φ)
– see <cit.>. The hyper-decoherence  Ḑ_h
acting on a map Φ  corresponds to a coarse graining supermap
and technically it can be described by the procedure of reshuffling, D=Φ^R, decohering and then reshaping the diagonal matrix D̃,
 so one can put the above recipe in a nutshell as:
T(Φ)=Ḑ_h(Φ).
Alternatively, one may write equations for the discrete evolution of populations,
 T_ij =  Tr[|i⟩⟨ i|  Φ(|j⟩⟨ j|)].
In this way we represent the process of hyper-decoherence
acting in the space of quantum maps
by a projection from N^2 dimensions to N.
Let Π_N=∑_i=1^N|i,i⟩⟨ i,i|
denote a projection operator of dimension N.
Then the classical transition matrix T and its entries
can be written by
 
    T(Φ)=  Π_N ΦΠ_N ,    and   
     T_ij =Φ_ii
     jj ,

 where the four index notation introduced in the previous section
is used. Note that no sum  is performed over the repeated indices.

If a Choi matrix D is positive and satisfies the partial trace condition,
Tr_A D= 𝕀,  it represents a stochastic map Φ.
Then the corresponding transition matrix T(Φ) is stochastic,
so that T_ij≥ 0 and ∑_i T_ij=1.
Ş_N^Q and Ş_N^C represent the sets of
N-dimensional quantum and classical stochastic processes, respectively.
The condition of unitality,  Φ(𝕀)=𝕀,
 analogous to the trace preserving property,
 is equivalent to the dual partial trace condition,
Tr_B D = 𝕀.
Any completely positive,  trace preserving and unital map
Ψ_B is called bistochastic and  B_N^Q  will
denote the set of quantum bistochastic maps acting
 on N dimensional system.
Decohering any bistochastic map
one obtains a bistochastic transition matrix  B=T(Ψ_B),
such that B_ij≥ 0 and ∑_i B_ij=∑_j B_ij=1
– see <cit.>.
Due to the theorem of Birkhoff any bistochastic matrix
can be represented as a convex combination of
permutation matrices.
The set of bistochastic matrices, denoted by  B_N^C,
has (N-1)^2 dimensions and is called the Birkhoff polytope.

Consider any unitary matrix V of order N.
Unitarity condition, VV^†=𝕀, implies
that the matrix given by the Hadamard (element-wise) product,
B=V ⊙V̅, is bistochastic,
as its entries read B_ij=|V_ij|^2.
Thus one may ask, whether any bistochastic  matrix B
can be represented in such a way.
The answer is negative, so one introduces the set   U_N^C,
of unistochastic matrices, for which
there exist a unitary V such that B=V ⊙V̅.

It is easy to check that for N=2 any bistochastic matrix
is unistochastic, so both sets coincide, U_2^C = B_2^C.
However, already for N=3 there exist bistochastic matrices
which are not unistochastic, so that
U_3^C  ⊂ B_3^C.
As a simple example let us mention the bistochastic matrix considered by Schur,
B_S=1/2(X_3+X_3^2) ∉ U_3^C, where
X_3 is the three–element cycle permutation matrix.
In general, to verify whether a given bistochastic matrix
 B∈ B_3^C is unistochastic one can compute
 the quantity Q, related to the Jarlskog invariant  <cit.>,
 and check if it is positive – see Appendix <ref>.
For larger dimensions no explicit criteria necessary and sufficient
for unistochasticity are known <cit.>,
but in the case N=4 one can rely on an efficient
numerical procedure based on the algorithm of Haagerup <cit.>.

As unitary matrix V of size N determines a bistochastic matrix
B=V ⊙V̅∈ B_N^C,
a unitary matrix U of size N^2 determines a
certain quantum bistochastic channel.
Physically, such an operation
corresponds to a coupling of the principal system with
an d-dimensional ancillary subsystem E, initially in the maximally
mixed state, by a non-local unitary matrix U,
followed by the partial trace over the environment,

    Ψ_U = _E [U(ρ⊗𝕀_d/d)U^†],

By construction such maps are trace preserving and unital
for any dimension d of the environment.
The case d=N is distinguished,
as in this case the rank of the corresponding Choi matrix, r≤ d^2
can be full, so in analogy to the unistochastic matrices
such quantum maps were called unistochastic channels
<cit.>.
The set of unistochastic maps, denoted by
U_N^Q, forms a proper subset of
the set B_N^Q of bistochastic maps.
In the one-qubit case, the set of  bistochastic maps
forms the tetrahedron of Pauli channels Φ_p⃗,
while  U_2^Q, forms its non-convex subset,
bounded by ruled surfaces <cit.>
and called Steinhaus tetrahedron <cit.> – see Fig. <ref>
and ref. <cit.>.
The set of one-qubit accessible maps was analyzed recently
in <cit.> from a slightly different perspective.




The set of unistochastic channels was found
important by studying the maps accessible by a semigroup,
as it was shown <cit.>
that  A_2^Q ⊂ U_2^Q.
 In this work we show  that
 an analogous relation does not hold for N=3
 and analyze classical analogues of these sets.
  For any unitary dynamics, Ψ_V^ I= V ⊗V̅,
 the corresponding classical transition matrix is  unistochastic,
  T=V ⊙V̅. A stronger property  states <cit.>
 that a classical transition matrix T can be coherified to a unitary dynamics
 if and only if  it is unistochastic,  so the set I_N
 of isometries -  unitary quantum rotations -
 decoheres to the set of classically  unistochastic matrices.

Therefore, although the word unistochastic
in both cases refers to the underlying unitary matrix,
the sets of quantum unistochastic  operations U_N^Q
 and classical unistochastic matrices  U_N^C
 are not directly related by decoherence.
To explain this fact consider an arbitrary unistochastic
map Ψ_U,  determined  in (<ref>)
by a unitary matrix U of order N^2.
It is convenient to represent it in  the four-index notation,
U_αβ
γδ=
⟨α, β |U| γ , δ⟩.
It is known <cit.>
that the corresponding Choi matrix  reads,
  D_Ψ_U=(U^R)^†U^R/N.
Taking its diagonal and reshaping it we arrive at
the classical transition matrix T(Ψ_U) with entries

    T_ij= 1/N∑_b,c=1^N| U_ib
    jc|^2,   i,j=1,…, N.

Note that each entry of T can be written as an average of the entries of a
block of size N of a unistochastic matrix U⊙U̅ of order N^2.
The above observation provides a clear motivation
 to introduce a family of generalized unistochastic matrices.
 Define an auxiliary kN × N rectangular matrix L_N,k,
 being a suitable extension of identity,

    L_N,k =  𝕀_N⊗ |ϕ̅_k⟩ ,

 where the k-dimensional state of the uniform superposition reads,
|ϕ̅_k⟩=1/√(k)∑_j=1^k |j⟩.
For instance, for
N=2 and k=1 one has L_2,1= 𝕀_2
while for k=2 such a coarse graining matrix
reads L_2,2^T=1/√(2)[ 1 1 0 0; 0 0 1 1 ].

  Definition. A bistochastic matrix B or order N
will be called k–unistochastic if there exists an unitary matrix U_kN of size kN
such that B is given by the following matrix coarse graining <cit.>
of the larger unistochastic matrix of order kN,

    B  = L^T_N,k (U_kN⊙U̅_kN) L_N,k .


Since  L_N,1= 𝕀_N this
definition implies that any standard unistochastic matrix is 1-unistochastic,
while expression  (<ref>)
for the classical transition matrix is equivalent to
T  = L_N,N^T (U_N^2⊙U̅_N^2) L_N,N.
Thus a classical transition matrix T
obtained by decoherence of any unistochastic channel
is N–unistochastic.
By construction the set  U_N,N^C
 of N–unistochastic matrices of order N
includes the original set
U_N^C of unistochastic matrices.
For N=2 every bistochastic matrix  is unistochastic,
but already for N=3 there exits  a unistochastic operation
 Ψ_U
which decoheres to a non-unistochastic classical transition matrix T,
which is 3–unistochastic – see Appendix  <ref>.







A list of sets  of quantum operations discussed further in this work
and the corresponding sets of classical transition matrices
is presented in Table <ref>.
Note that the inclusion relations for quantum maps,
visible in the table,
correspond to analogous relations in the classical case.
Natural decoherence relations, like
D_h ( S_N^Q)= S_N^C
and
D_h ( B_N^Q)= B_N^C,
connect the sets of unistochastic channels and
N-unistochastic matrices,
D_h ( U_N^Q)= U_N,N^C.
The latter one contains the set
U_N^C of unistochastic matrices arising
  from isometric unitary quantum maps  by hyper-decoherence,
D_h ( I_N)= U_N^C,
which can be modeled by Schur superchannels <cit.>.
The set of mixed unitary Weyl channels
decoheres to the set of circulant bistochastic matrices,
D_h ( W_N)= C_N,
while quantum accessible Weyl channels
decohere to classical accessible bistochastic
matrices,
D_h ( A_N^Q)= A_N^C.

Any unitary rotation forms a unistochastic map, so that
U_N^Q ⊃ I_N^Q.
A classical analogue of unitary rotations is played by permutations matrices
and their set sits inside the set of unistochastic matrices,
P_N^C  ⊂ U_N^C ⊂ B_N^C.
In the case N=2 the following relations hold:
A_2^Q ⊂ U_2^Q
and U_2^C  =  B_2^C, which are not true for higher N.
The set U_N^Q of unistochastic channels and the set
W_N of the Weyl channels are incomparable.
So are the sets of Weyl channel and accessible channels,
so in this work we will restrict our attention to the Weyl channels
which are accessible by a semigroup.



§ GENERAL CASE: ACCESSIBILITY BY A SEMIGROUP FOR AN ARBITRARY DIMENSION N



  

 §.§ Channel eigenvalues and probabilities

In order to answer the question, whether a given map is
accessible by a semigroup we need to analyze further properties of
channels in the form  (<ref>).
The following lemma presents  a key
feature of mixed unitary channels written in the Weyl basis.


Weyl unitary operators satisfy  [U_ij⊗U_ij,U_kl⊗U_kl]=
U_ijU_kl⊗U_ijU_kl-
U_klU_ij⊗U_klU_ij=0
for any choice of the indices i,j,k,l=0,…, N-1.
 The equality holds as Weyl unitaries are commutative up to a phase,
  which is not relevant here as the expressions above contain
   U and its complex conjugate, U.

 Accordingly, all  combinations  (not necessarily convex) of
 tensor products of Weyl unitaries and their complex conjugates
 are compatible, so are all mixed unitary channels
 Φ_p⃗, i.e. ∀p⃗,p⃗'  [Φ_p⃗,Φ_p⃗']=0. For the sake of
 brevity, from now on we may drop subscript p⃗ and denote
 these channels by Φ.  The common eigenbasis of the
 channels defined in Eq. (<ref>) is given by the maximally
 entangled states proportional to the vectorized form of Weyl
 unitaries.
 For any matrix A=∑ A_ij ij let us use
 the shorthand  notation
 A=∑ A_ij|i⟩|j⟩ to
 represent its vectorized form. It  enjoys
 the following properties: (A⊗ C^T) B=ABC
 and B A⟩⟩=(B^† A),
 which  defines the Hilbert-Schmidt inner product in the space of matrices.
A matrix element of the superoperator Φ_p⃗
in the non-normalized basis {U_ij} reads,

    U_ijΦ_p⃗U_mn   =   ∑_kl p_klU_ijU_kl⊗U_klU_mn=∑_kl p_kl(U_ij^† U_klU_mnU_kl^†)
       =   ∑_kl
     p_klω__N^ml-kn(U_ij^† U_mn)=N∑_kl p_klω__N^ml-knδ_imδ_jn.

The above equalities holds due to
orthogonality of Weyl unitaries
 and their commutativity up to a phase mentioned above.
For any mixed unitary channel (<ref>)
represented by a superoperator in the form
Ψ_p⃗=∑ p_nV_n⊗V_n,
the dynamical matrix D, obtained by reshuffling its entries,

    D_Ψ= Ψ_p⃗^R=∑_np_n(V_n⊗V_n)^R=∑_np_nV_nV_n,

 corresponds to the map by the  Choi-Jamiołkowski isomorphism <cit.>.
Note that the last form yields  the spectral decomposition of the Choi matrix
 D_Ψ if and only if the set
{V_n V_n∈ U(N)}_n=1^N forms an orthogonal basis.
Observing that ⟨⟨ U_kl|U_ij⟩⟩=Nδ_ikδ_jl
and  Φ_p⃗^R=∑_klp_klU_klU_kl we infer
 that all the channels Φ_p⃗ are compatible
 and also that they commute with the set of the corresponding dynamical maps.
This fact implies  the following proposition clarifying the relation between
the eigenvalues  λ_kl of the superoperator Φ_p⃗
and the probabilities, p_kl,  which  are proportional to the
eigenvalues of the corresponding dynamical matrix D=Φ_p⃗^R.


There exists a complex Hadamard  matrix H
(unitary up to rescaling  with all unimodular entries),
 which  transforms probabilities p_kl  into eigenvalues  λ_mn
 of the superoperator Φ_p⃗ defined in Eq. (<ref>), namely
   λ⃗=Hp⃗ and  p⃗=1/N^2Hλ⃗.
   The matrix H of order N^2 is hermitian and its entries read <cit.>,

    H_ mnkl=ω__N^  ml-kn,
       k,l,m,n=0,…, N-1.


To demonstrate the above statement one needs to show that
H^2=HH^†=N^2, which means that H
is a complex Hadamard matrix <cit.>,
self inverse up to the constant N^2.
Hermiticity of H is obvious from definition,
and for unitarity we have

    ∑_k,l=0^N-1H_ mnkl
    H_ klpq=∑_kl ω__N^ml-knω__N^kq-lp=∑_lω__N^(m-p)l ∑_k ω__N^(q-n)k=N^2δ_mpδ_nq.



For any  quantum channel of the form Φ_p⃗=∑ p_klU_kl⊗U_kl=1/N∑λ_klU_klU_kl, the following facts can be verified
with the help of the matrix H,


  * For any superoperator Φ_p⃗ of this form its leading eigenvalue (corresponding to invariant state 1/√(N)) is equal to unity, λ_00=1.

  * For completely depolarizing channel Φ_∗ corresponding to
the flat probability vector, p_kl=1/N^2 for k,l=0,…, N-1,
 all subleading eigenvalues vanish, λ_kl=0 for k≠0 and l≠0.

  * λ_k,l=λ_-k⊕ N,-l⊕ N,
so apart from  λ_00, for an even N there exist three other eigenvalues,
λ_N/2,0, λ_0,N/2, and
λ_N/2,N/2 which are always real.

  * As the entries of H are unimodular and the eigenvalues of the superoperator
Φ_p⃗ are their convex combinations, they belong to the unit disk,
 |λ_k,l|≤1 for any k,l.




  

 §.§ Lindblad generators

Knowing all we need about the spectral form of quantum
channels and their dynamical maps, consider now a special
family of dynamical semigroups
Λ_t_μ=^t_μℒ_μ, defined by the
non-negative interaction time  t_μ and the Lindblad
generator ℒ_μ specified by a single jump
operator L_i=U_μδ_iμ.
Due to Eq. (<ref>),
the generator Ļ_μ is

    ℒ_μ=U_μ⊗U_μ-_N^2.

Note that for the case μ=0 Lindblad generator is trivial,
Ļ_0=0.
By the definition of ℒ_μ
and Lemma <ref>,
the Lindblad generators do commute,
[ℒ_μ ,ℒ_ν]=0.
The commutativity of Lindblad
generators implies commutativity on their respective
dynamical semigroups,
Λ_t_μΛ_t_ν=
Λ_t_νΛ_t_μ.
So it is convenient to write:

    Λ_t_0Λ_t_1…Λ_t_N^2-1   =   ^t_0Ļ_0^t_1ℒ_1…^t_N^2-1ℒ_N^2-1=
    ^∑_μ t_μℒ_μ=^
    tℒ=Λ_t.

Here t=∑_μ t_μ and

    ℒ=∑_μ p'_μℒ_μ,  where:   p'_μ=t_μ/t.

Regarding that Ļ_0=0, t_0 has no physical meaning and
one can set it to zero, t_0=0, which means the first component of the
N^2 dimensional probability vector p'
 is zero, p'_0=0.
Above representation of the semigroup Λ_t implies
the following statement.


For any dimension N  the set A_N^Q
of quantum channels accessible by a  dynamical semigroup is
log convex: any generator Ļ in the exponent can be represented by  a
convex combination of N^2-1 generators Ļ_μ.


Moreover, it is known that
any concatenation of quantum channels is a quantum channel itself.
However, for the family of semigroups in Eq. (<ref>), the stronger fact is
that quantum channels gained by an arbitrary concatenation
of dynamical semigroups, ^t_μℒ_μ, are mixed unitary channels in
the sense of Eq. (<ref>), i.e.
Λ_t=Φ,  and the order
of dynamical semigroups is not important due to
commutativity. To prove the claim,
by applying Eq. (<ref>), we obtain ℒ=∑_μ
p'_μ U_μ⊗U_μ-_N^2,
so the spectral form of the Lindblad generator is:

    ℒ=1/N∑_ν=0^N^2-1(λ_ν^'-1)U_νU_ν,

where λ_ν^'
 are the eigenvalues of the operator ∑ p'_μ U_μ⊗U_μ
obtained  by acting with the matrix H defined in Eq. (<ref>) on the
 probability vector  p'_μ of rescaled interaction times.
In conclusion, the dynamical semigroup associated with ℒ has the form:

    Λ_t=1/N∑_ν=0^N^2-1^t(λ_ν^'-1)U_νU_ν.


Now we will show that the semigroup  (<ref>) can be written in the form (<ref>)
as a convex combination of Weyl unitary matrices.
   The probabilities p_μ in the latter equation
   are obtained by applying H to the eigenvalues of
   Λ_t, and they are proportional to the eigenvalues of
 Λ_t^R, the reshuffled form of  Λ_t.
 This implies that every entry of the probability vector is non negative, as required.
Accordingly, quantum
channels accessible by a dynamical semigroup Λ_t
form the subset 𝒜_N^Q within the simplex Δ_N^2-1
of mixed unitary channels in the Weyl basis.
Note that if λ_ν' is real, then  ^t(λ_ν^'-1), the
  corresponding eigenvalue of Λ_t,
  is  positive. Also, if λ_ν' and
  λ'_ν  form a complex conjugate pair,
   the corresponding eigenvalues of Λ_t
   are again a  complex conjugate pair.
In this case, at
t_+=2nπ/|Im(λ'_ν)|
and t_-=(2n-1)π/|Im(λ'_ν)| for any
natural number n∈ℕ, the equality
^t(λ_ν^'-1)=
^t(λ'_ν-1) holds. So
 Λ_t becomes a map with a positive degenerate or a negative
 degenerate eigenvalue, respectively, at t_+ and t_- .
Finally,
since |λ_ν^'|≤1 (see Remark <ref>),
the moduli of the eigenvalues of Λ_t decay
exponentially with respect to time t or
remain unchanged and equal to unity.


  Consider a mixed unitary channel represented in the Weyl basis,
    written in the vectorized form |U_μ⟩⟩,
  
    Φ = 1/N∑_μ=0^N^2-1λ_μU_μU_μ.

  In general the eigenvalues λ_μ of the superoperator are complex,
  and a
  suitable choice of   logλ_μ is discussed below.
   The necessary and sufficient
condition for Φ to be accessible by a dynamical
semigroup (<ref>) is that
the sum on the right hand side of Eq. (<ref>) is real and positive,

    t_μ=1/N^2∑_ν=1^N^2-1(
        H)_μνlogλ_ν≥0 ,

for  μ=1,2,…,N^2-1.
Here H denotes the complex Hadamard matrix defined in (<ref>),
while  =_N^2-00 is a projector operator onto (N^2-1)-dimensional space
and 00 written as a product basis projects λ⃗ to its leading element λ_0=1.


Before proceeding to the proof note that if there exists a single real
negative eigenvalue of Φ, its logarithm is complex, so the above
condition  cannot be satisfied, hence the map is not accessible.
On the other hand, if negative eigenvalue is degenerated
(an even number of times) then the imaginary parts of logλ_ν
can cancel and the sum  (<ref>) can be real and positive.
The fact that a one-qubit map with a negative degenerated
eigenvalue of the superoperator can be accessible by a semigroup
was already reported in  <cit.>.



To prove the if part, we will assume Φ is accessible by a
dynamical semigroup (<ref>), Φ=Λ_t for some interaction
time t≥0, and apply Eq. (<ref>):

    logλ_μ=(λ_μ^'-1)t=
    (∑_ν H_μνp'_ν-1)t=
    ∑_ν(H_μν-1)t_ν=
    ∑_ν H_μν^' t_ν,

where H'=H-W with W_μν=1. Recalling the definition
of matrix H, see Eq. (<ref>), it is clear that
the entries of its first row and column are equal to 1, so
the corresponding
entries of H^' vanish. Consequently,  H^' has
the same structure of a projector onto a
(N^2-1)-dimensional subspace,
in other words, H^'=
H^'. This means that H' is not invertible,
however, one
can define its inverse on the (N^2-1)-dimensional subspace  Π.
We proceed to show that this inverse exists and is equal to H, up
to the constant N^2,

    (Π HΠ)(Π H^'Π)   =   ∑_ m n k l m'n'k'l'ω__N^ml-nk(ω__N^m'l'-n'k' 
    -1)δ_km'δ_ln'mnk'l'
       -   ∑_ m n  m'n'k'l'(ω__N^m'l'-n'k' 
    -1)(δ_m'0δ_n'0mnk'l'+δ_mm'δ_nn'00k'l')
       +   ∑_ m' n'  k'l'(ω__N^m'l'-n'k'
    -1)δ_m'0δ_n'000k'l'=N^2(∑_mnmn-00).

By multiplying both sides of  Eq. (<ref>)
by Π HΠ,
one gets the equality in Eq. (<ref>). Because
t_μ≥ 0 for μ=1,…,N^2-1, we completely recover
Eq (<ref>). Note that Eq. (<ref>) implies
logλ_μ=logλ_μ.

To prove then part, we should  show that if inequalities (<ref>)
 between eigenvalues hold, the channel is accessible by  dynamical
semigroups (<ref>). This fact is actually obvious as we have explicitly
 introduced the interaction time and this completes the proof.


Note that the above result allows us not only to
check, whether a given channel is accessible,
but if the answer is positive, one can also construct the desired semigroup.
For any probability vector  p⃗,
determining the mixed unitary channel  (<ref>),
we find the complex vector, λ⃗=Hp⃗,
check if conditions (<ref>) are satisfied,
and if it is the case, we use the non-negative vector t⃗
of interaction times to write the semigroup (<ref>).

Moreover, as it should be, Eq. (<ref>) is equivalent of existence
of a proper generator for the quantum channel Φ
 that satisfies properties mentioned in the section <ref>.
For accessible channels from A̧^Q_N, the (non-real) logarithm of Φ
providing such a generator is equal to:

    logΦ=1/N∑_μlogλ_μU_μU_μ.

The above equation fulfils trace preserving condition and by considering
logλ_μ=logλ_μ,
 Hermiticity preserving property  also holds.
Now, we  need to demonstrate that  logΦ is
conditionally completely positive.
Note that
(logΦ)^R=∑_μ
C_μU_μU_μ, where C_μ=1/N^2∑_ν
H_μνlogλ_ν, and that
the maximally entangled state
|ψ_+⟩ is equal to 1/√(N)U_0 where
U_0=_N. So conditionally completely positivity for
logΦ defined by Eq. (<ref>)  means that all of
 eigenvalues of its reshuffled form are positive
besides the one associated with U_0, i.e.
C_μ≥0 for  μ=1,…,N^2-1.
Observing that logλ_0 =0, we get
 ∑_ν (Π HΠ)_μνlogλ_ν≥0 which is satisfied
 as we assumed Φ∈A̧^Q_N.

Thus far,  we have shown which generator for channels in A̧_N^Q
is suitable, however, for logarithm of λ_ν one has
 logλ_ν=log
r_ν+iθ_ν+i2π M_ν with
M_ν∈ℤ and θ_ν∈[-π,π].
Note that both π and -π are included in the interval,
so for a negative degenerate eigenvalue one can assign π
to one of them and -π to another one.
 Recalling the condition imposed on the logarithm of
  eigenvalues, one can  verify (with two indices)
 M_-k⊕ N,-l⊕ N=-M_k,l. This implies:

    t_m,n=1/N^2∑_k,lH_
     mnkl(logr_kl+iθ_kl+i2π M_k,l)≥ 0,

 for m,n∈{0,⋯,N-1} which are not simultaneously zero,
 and so for m'=-m⊕ N and n'=-n⊕ N:

    t_-m⊕ N,-n⊕ N=
     1/N^2∑_k,lH_
     mnkl(logr_kl-iθ_kl
     -i2π M_k,l)≥ 0.

 These two relations result in the following bounds:
 
    -∑_k,lH_
     mnkl(logr_kl+iθ_kl)≤
     i2π∑_k,lH_
     mnkl M_k,l≤∑_k,lH_
     mnkl(logr_kl-iθ_kl).



In the next Remark we summarize further properties of the accessible
channels fulfilling relation (<ref>) and properties of M_k,l.


Regarding this fact that λ'_νs themselves
are eigenvalues  of a quantum Weyl
channel–see Eq. (<ref>),
so they have the properties mentioned in Remark <ref>,
one can achieve the following results for accessible maps,
 Φ=Λ_t:


  * As it is mentioned earlier in this section for a real λ_ν'
dynamical semigroup
Λ_t has a positive eigenvalue. Due to Remark <ref>
we know for an even N  there are three real subleading eigenvalues
λ'_ν.
 Consequently, for an even N the essentially real subleading eigenvalues of
 any accessible map Φ∈A̧_N^Q   are necessarily positive,
  i.e. 0≤λ_mn≤1 for
 {(m,n)}={(0,N/2),(N/2,0),(N/2,N/2)}.



  * Adding Eq. (<ref>) to Eq. (<ref>), we get the following
necessary condition for eigenvalues of an accessible map:
 
    ∑_k,lH_
     mnkllogr_kl≥0,

 where r_kl is the modulus of λ_kl, i.e.
 λ_kl=r_kl^iθ_kl.

 
  * If ∀ m,n:  ∑_k,lH_
 mnkl(logr_kl+iθ_kl)≥0, we do not need to
  check any M_k,l to see if the channel is accessible.
  However, if there exist M_k,l for which inequality
  (<ref>) holds for any m,n, then the associated quantum channel
  is accessible not by a unique Lindblad generator.

    
  * If ∑_k,lH_
 mnkl(logr_kl+iθ_kl)≤0 for
 both (m,n)=(m',n') and (m,n)=(-m'⊕ N,-n'⊕ N) the
  associated channel is not accessible.

  
  * Applying Eq. (<ref>) for an even N,
   we can show M_k,l=0 for
  {(k,l)}=
  {(0,N/2),(N/2,0),(N/2,N/2)}.




Eq. (<ref>) shows the eigenvalues of a quantum
channel belonging to a dynamical semigroup as an explicit
function of time. By applying matrix H, we can transform
eigenvalues into probabilities and gain the following
relation representing the probabilities as an explicit
function of time.

    p_μ=1/N^2∑_ν H_μν^∑_η H'_νη t_η.


Rewriting the previous equation using two indices kl rather than
μ, one has
p_kl=1/N^2∑_mnH_ kl
 mn^∑_rsH'_ mn 
rst_rs.
Computing the partial derivative with respect to t_ij,
we obtain the following useful relations

    ∂ p_kl/∂ t_ij   =   1/N^2∑_mnH_ kl  mn
    H'_ mn  ij^∑_rsH'_ mn  rst_rs=
    1/N^2∑_mnH_ kl  mn(H_ mn  ij-1)
    ^∑_rsH'_ mn  rst_rs
       =   1/N^2∑_mn
     H_ kl  mn
     H_ mn  ij^∑_rsH'_ mn  rst_rs-
     1/N^2∑_mn
     H_ kl  mn^∑_rsH'_ mn  rst_rs
       =   1/N^2∑_mnH_
               kl  mnH_ mn 
           ij^∑_rsH'_ mn 
       rst_rs-p_kl.

In the next step we get

    p_kl+∂ p_kl/∂ t_ij   =   1/N^2∑_mnω__N^(kn-ml)ω__N^(mj-ni)^∑_rsH'_ mn  rst_rs
       =   1/N^2∑_mnω__N^n(k-i)-m(l-j)^∑_rsH'_ mn  rst_rs
     = ∑_mnH_ -i⊕ k,
     -j⊕ lm , n^∑_rsH'_ mn 
     rst_rs,

which implies the relation

    p_kl+∂ p_kl/∂ t_ij=p_-i⊕ k, -j⊕ l.

This equality  shows that it is enough to know only one
component of the probability vector, say p_00, to
compute the rest of the probability vector,
p_00+∂ p_00/∂
t_ij=p_-i⊕ N, -j⊕ N.
By a proper substitution, this can be applied to
eigenvalues.

 To demonstrate this approach in action consider first the case N=2,
 in which the superoperator is expressed by the  Pauli matrices,
  Φ=∑_μ
 p_μσ_μ⊗σ_μ. In this case,
  the Hadamard matrix H of order N^2 defined in (<ref>) reads,
 
    H=[  1  1  1  1;  1  1 -1 -1;  1 -1  1 -1;  1 -1 -1  1 ].

 By acting with H on the probability vector p⃗=(p_0,p_1,p_2,p_3)^T, one
 obtains eigenvalues as a function of the probabilities <cit.>.
 Furthermore, by applying Eq. (<ref>) in qubit case, one obtains relation

    [ t_1; t_2; t_3 ]=1/4[  1 -1 -1; -1  1 -1; -1 -1  1 ][ logλ_1; logλ_2; logλ_3 ]=1/4[ logλ_1/λ_2λ_3; logλ_2/λ_1λ_3; logλ_3/λ_1λ_2 ],

 in which  only the `core' of size N^2-1  <cit.> of the complex Hadamard matrix H appears.
Non-negativity of all components of the vector t⃗
=(t_1,t_2,t_3) implies the desired set of three inequalities
 
    λ_i≥λ_jλ_k,
        for i≠ j≠ k.

  These relations
   form the necessary and
    sufficient condition for a quantum channel to be a seed
    for Pauli semigroups <cit.>. By
    applying Eq. (<ref>), one can easily
    find the time evolution of the probability
    vector,
 
    p_0   =   1/4(1+^-2(t_2+t_3)+
       ^-2(t_1+t_3)+^-2(t_1+t_2)),
       
     p_1   =   1/4(1+^-2(t_2+t_3)-
       ^-2(t_1+t_3)-^-2(t_1+t_2)),
       
     p_2   =   1/4(1-^-2(t_2+t_3)+
       ^-2(t_1+t_3)-^-2(t_1+t_2)),
       
    
       p_3   =   1/4(1-^-2(t_2+t_3)-
       ^-2(t_1+t_3)+^-2(t_1+t_2)),

where the total interaction time  reads, t=∑_μ=1^3 t_μ.
  



§ GEOMETRY OF THE SET  𝒜_N^Q OF MAPS ACCESSIBLE BY A SEMIGROUP




The set 𝒜_N^Q of  mixed unitary Weyl quantum channels
acting on N-dimensional states and accessible by a
dynamical semigroup is not convex <cit.>.
In Corollary <ref> we have shown that
for any dimension N  the set A_N^Q⊂ W_N
forms a log–convex subset of the probability
simplex  Δ_N^2-1 representing  all Weyl mixed unitary channels.
 Another important property of this set is established below.

  

 §.§ The set 𝒜_N^Q of accessible maps is star-shaped


  We show that the log-convex set
𝒜_N^Q of quantum Weyl channels accessible by a
dynamical semigroup is  star-shaped.



The set 𝒜_N^Q∈Δ_N^2-1 of mixed unitary
Weyl channels accessible by a semigroup has the
star-shape property with respect to completely depolarizing
channel Φ_*.


The aim is to show for a quantum channel
Φ∈𝒜_N^Q any convex combination with
Φ_∗ belongs to 𝒜_N^Q,
Φ'=mΦ+(1-m)Φ_∗∈𝒜_N^Q, where 0≤
m≤1. For any t_μ≥0  associated with
Φ∈𝒜_N^Q through Eq. (<ref>), one can
define the non-negative parameter t_μ^'=t_μ-N^-2logm  for which:

    Φ'   =   ^∑_μ t^'_μℒ_μ=
    ^∑_μ t_μℒ_μ^-log(m)/N^2∑_μℒ_μ=
    Φ_t ^-log(m)/N^2∑(U_μ⊗U_μ-_N^2)
       =   Φ_t^log(m)_N^2-log(m)Φ_∗=
    mΦ_t^-log(m)Φ_∗ = 
    mΦ_t((-1+1/m)Φ_∗+_N^2)=mΦ_t+(1-m)Φ_∗.

Here we use the fact that all subleading eigenvalues of
Φ_∗ are zero, and that the composition of any unital
channel with the completely depolarizing channel Φ_∗ is equal to Φ_∗.


Note that the boundaries  of the set 𝒜_N^Q
consist of channels accessible by a semigroup with
at least one of the non-trivial (μ≠0)
 parameters t_μ  in Eq. (<ref>) vanishing.
This means that all quantum channels of the form
exp(∑_μ=1^N^2-2Ļ_μ  t_μ)
belong to  ∂ A_N^Q,
for any choice of N^2-2 out of
N^2-1 non-trivial Lindblad generators and
those channels using N^2-1
Lindblad generators,
 Φ_p⃗=exp(∑_μ=1^N^2-1Ļ_μ t_μ),
lie inside A_N^Q. To see that,
note that if  Φ_p⃗ belongs to 𝒜_N^Q, it is
connected to _N^2 through a trajectory indicated by
non-negative interaction time and each point of this
trajectory corresponds to a channel belonging to
𝒜_N^Q∈Δ_N^2-1. This in turn leads us to
the fact that for any t≥0 corresponding to a point on
the trajectory connecting _N^2 to  Φ_p⃗,
the dynamical map is positive, also for small time t.
Expanding the dynamical map around t=0,
positivity condition of the corresponding Choi matrix, D_Φ
= Φ^R, implies:

    (_N^2+ϵdΦdt|_t=0)^R≥0,

which  holds for any ϵ>0 small enough
 for any Φ∈𝒜_N^Q. Applying Eq. (<ref>), it
 is easy to see that the above equation is equal to
 (_N^2+ϵℒ)^R=
(_N^2+ϵ(∑_μ p'_μ
U_μ⊗U_μ-_N^2))^R
which must remain positive
for small ϵ.
 Due to Proposition <ref>,  its eigenvalues belong to the set
{N(1-ϵ(1-p'_0)),ϵ Np'_1,…,ϵ Np'_N^2-1} and each of which should be non-negative.
 Noting that ϵ is small, the first eigenvalue is always
positive.  However,  if p'_μ=0 for any
μ∈{1,2,…,N^2-1}, the corresponding eigenvalue is
zero which means that  Eq. (<ref>) is saturated,
so that this subspace belongs to the boundary of 𝒜_N^Q.
Due to Eq. (<ref>) this fact implies
that  the boundary of 𝒜_N^Q is achieved
if we set to zero the interaction time t_μ  associated with
any selected generator ℒ_μ.
The above statement is valid for any N.
In particular, it implies Proposition (2) shown in <cit.> for the case of a single  qubit.



§ HYPER–DECOHERENCE OF QUANTUM WEYL CHANNELS





 §.§ Corresponding circulant bistochastic transition matrices


 As discussed in Sec. <ref>, one can assign a classical stochastic
  transition matrix T to any quantum channel by decohering and reshaping
  the dynamical matrix, D_Φ=Φ^R,
 so that T forms  a minor of order N of the superoperator Φ of order N^2.
  If {K_n} represents the set of Kraus operators related to a channel,
 it is straightforward to see the associated stochastic map is gained by
 T=∑_nK_n⊙K_n. Thus for the Weyl channels defined by
 Eq. (<ref>) the associated bistochastic transition
 T is given by
 
    T(Φ_p⃗)=∑_k,l=0^N-1p_klU_kl⊙U_kl=
     ∑_k,l=0^N-1p_klX^kZ^l⊙ X^kZ^l=
     ∑_k,l=0^N-1p_klX^k.

 To prove above equation we used the fact that for two arbitrary matrices
A and C and two diagonal matrices B and D, the equality
AB⊙ C D=(A⊙ C)(B⊙ D) holds.
Since the operators Z^l do not influence the terms
contributing to the transition matrix T,
 it is convenient to introduce an
N-point probability vector q,
given by the marginal  of the vector p of length N^2,

    q_k=∑_l p_kl.

Note that   q can also be considered as a reduction of
p,  analogous to partial trace.
Thus  the transition matrix T_q= D_h(Φ_p)
corresponding the Weyl channel  (<ref>)
has the following form
  
    T_q(Φ)=∑_k=0^N-1q_kX^k ∈ C_N.


The set C_N of classical transitions defined by Eq. (<ref>)
consists of circulant bistochastic matrices of order
N, i.e., bistochastic matrices whose rows (columns)
are cyclic permutations of its first row (column).
Hence the set C_N  forms an  (N-1)-dimensional
simplex Δ_N-1=  {_N, X, X^2, ..., X^N-1} embedded
inside the Birkhoff polytope B_N^C of dimension (N-1)^2
which contains all permutation matrices of order N.
Note that the shift operator X, also written X_N,
is actually the N-element cycle permutation,
thus hyper-decocherence of the Weyl channels
gives the set of circulant bistochastic matrices,
D_h( W_N)= C_N.



Arbitrary powers of cycle permutation do commute,  [X^k,X^k']=0.
This implies that all (not necessarily
convex) combinations of X^k are compatible, so the transitions matrices
in Eq.(<ref>) satisfy relation
T(Φ_p⃗)T(Φ_p⃗')
=T(Φ_p⃗')T(Φ_p⃗).





 §.§ Spectral properties of classical and quantum maps 


The Fourier matrix of order N, with entries
 F_mn=exp(-i 2 mn π/N),
diagonalizes any combination of
powers X_N^k of the cyclic permutation matrix X_N.
The common eigenbasis is given by
|x_j⟩=1/√(N) F^†|j⟩. Also eigenvalues and
coefficients of expansion can be transformed to each other applying F
and F^†. For instance,  for a transition matrix T  in Eq. (<ref>),
its eigenvalues ξ_i and the probabilities q_k satisfy relations,
ξ⃗=Fq⃗ and
q⃗=1/NF^†ξ⃗.
To see this let us rewrite T in the basis {|x_j⟩}
and use the notation, ω_N=exp(2 π i/N),

    ⟨x_m|T|x_n⟩   =   1/N∑_kq_k∑_x∑_i∑_j
    ω__N^-imω__N^jn⟨i| x⊕ k⟩⟨x| j⟩
       =   1/N∑_kq_kω__N^-mk∑_x
    ω__N^x(n-m)=∑_kω__N^-mkq_kδ_mn.



As any circulant bistochastic transition matrix T∈ C
contains non-negative real entries and
can be diagonalized by the Fourier matrix F_N,
the following spectral properties hold:


  * Any circulant bistochastic matrix T has the leading (Frobenius–Perron) eigenvalue
 ξ_0=1.


  * For the matrix  T_∗, located at the center of the
   Birkhoff polytope B_N of bistochastic matrices and corresponding to the flat probability vector,
q_k=1/N  for k=0,…,N-1,
all subleading eigenvalues vanish, ξ_i=0 for i≠ 0.


  * For a cyclic permutation matrix  X_N its spectrum belongs to the unit circle
  and forms the N–sided regular polygon.


  * Eigenvalues of an arbitrary T are either real or appear in complex conjugate pairs.


  * ξ_-i⊕ N=ξ_i,
so for an even N apart from the leading eigenvalue ξ_0,
 there exists another real eigenvalue ξ_N/2.


  * As the modulus of the entries of F is unity and eigenvalues
 of T are just a convex combination of these elements, so
 the spectrum belongs to the unit disk,
|ξ_i|≤1 for any i. Furthermore, as the entries of F_N
are given by powers of the N-th root of unity, ω_N=e^i 2π/N,
then all eigenvalues ξ_i belong to the regular N–sided polygon
containing  ξ_0=1.


The above properties of the circulant bistochastic matrices – classical maps –
can be extended for certain quantum maps.



Consider a Weyl channel Φ_p⃗ from the simplex
Δ(Φ_𝕀, Φ_ X, …, Φ_X^N-1).
The set of eigenvalues of the superoperator Φ_p⃗ is then equal (up to degeneracy)  to the spectrum of the corresponding circulant bistochastic matrix  T(Φ_p⃗).
Moreover, the spectrum of Φ_p⃗ is N-fold degenerate.


Let p_kl denote the probability vector of size N^2  defining  the Weyl
channel Φ_p⃗. Due to Proposition <ref>
the eigenvalues of the superoperator Φ_p⃗ read

    λ_mn = ∑_klω_N^ml-nk p_kl .

Since hyper-decoherence transforms this channel into
the classical map,  T(Φ_p⃗) = ∑ X^k q_k,
with q_k = ∑_l p_kl
the spectrum of the circulant bistochastic matrix
T(Φ_p⃗) can be expressed
by the Fourier matrix F of size N,

    ξ_n = ∑_k F_nk q_k,


Thus Proposition <ref> is equivalent to showing that ∀ m, λ_mn = ξ_n.
We start with  expression (<ref>)
 for eigenvalues λ_mn
 and notice that since  Φ_p⃗ belongs to the
 the simplex
Δ(Φ_𝕀, Φ_ X, …, Φ_X^N-1)
and p_kl≠ 0 only for l = 0, then

    λ_mn   = ∑_klω_N^ml-nk p_kl = ∑_k ω_N^-nk p_k0
       = ∑_k ω_N^-nk q_k = ∑_k F_nk q_k = ξ_n.

In the second row we used that q_k = ∑_l p_kl = p_k0 which is due to the fact that
Φ_p⃗ belongs to the simplex Δ.
This shows that both set of eigenvalues are equal.


  Consider an arbitrary Weyl  channel Φ_p⃗.
  There exists a change of  basis  such that  Φ_p⃗ is
  block diagonal, each block of order N forms a circulant
  matrix.  Thus the spectrum of the superoperator Φ_p⃗
  is contained in the regular N-polygon, with its rightmost vertex being 1.
  In addition, one of these blocks forms matrix T(Φ_p⃗),
  so the spectrum of  Φ_p⃗ contains the spectrum of the
  corresponding transition matrix  T.


  Consider the basis given by {|j⊕ k,j⟩}
  for  k, j∈{0,…, N-1},
  corresponding to a permutation of the original basis.
  The next step is to compute an entry of Φ_p⃗ in
  the new basis:
  
    ⟨j⊕ k, j|Φ_p⃗|j'⊕ k', j'⟩   =
        ⟨j⊕ k, j|(
          ∑_mn p_mn U_mn⊗U_mn)
        |j'⊕ k', j'⟩
       =
        ∑_mn
        p_mn⟨j⊕ k|
        U_mn|j'⊕ k'⟩⟨j|U_mn|j'⟩
       =
          ∑_mn p_mn⟨j⊕ k|
        X^mZ^n|j'⊕ k'⟩⟨j|X^mZ^n| j'⟩
       =
        ∑_n p_-j'⊕ j, nω_N^nkδ_k,k'.



  Thus the following decomposition holds
  
    Φ_p⃗ =
        ∑_k=0^N-1∑_j,j'=0^N-1(
        ∑_l p_-j'⊕ j, lω_N^kl)
          |j⊕ k, j⟩⟨j'⊕ k, j'|.

  Fixing k, the remaining sum on j and j' corresponds
  to a block Φ_p⃗^(k) of size N, given by
   
    Φ_p⃗^(k) =
       ∑_k'^N-1(
         ∑_l=0^N-1
         p_k' lω_N^kl)
         X^k'
         =
         ∑_k'
         X^k'(
           ∑_ml (Z^k)_ml p_k' l) =
         ∑_k'
         X^k'
         q_k'^Z^k,

  where we introduced a shorthand notation,
  q^Z^k_m  ≡∑_lk' (Z^k)_k'l p_ml.
Therefore, we can write

    Φ_p⃗
      = ⊕_kΦ_p⃗^(k).

Given that each block Φ_p⃗^(k) is circulant and can be
diagonalized by the N dimensional Fourier matrix,
 the spectrum of  Φ_p⃗ is contained in the regular
N-polygon.
Finally, observing that q^Z^0_m = q_m,
we realize that the first block of the superoperator is
equal to its classical analogue,
Φ_p⃗^(0) = T(Φ_p⃗).




 §.§ Kolmogorov generators and accessible bistochastic matrices


In analogy with the quantum case, it is interesting to find
 which transition matrices out of those described in (<ref>) are accessible
by a classical dynamical semigroup.
From a mathematical perspective such stochastic matrices are also
called embeddable and their properties were analyzed in <cit.>.
In our physics–oriented approach it will be convenient to make use of the notion of
strictly incoherent operations (SIO), defined as these
stochastic maps, which are not able
 to generate and to use quantum coherence <cit.>.
Therefore an arbitrary quantum operation is  strictly incoherent
if and only if  its Kraus operators are of  the form,
K_n=∑_i c_niP_n(i)i
where P_n(i) is a permutation matrix <cit.> and c_ni
are  arbitrary complex coefficients  satisfying the  normalization condition.

In the analyzed case of mixed Weyl channels
the corresponding Kraus operators read,
K_kl=√(p_kl)∑_m
ω__N^mlm⊕ km,
which proves  that these channels are strictly incoherent.
This property is  equivalent to  the following decoupling property,

    Φ_p⃗(Ḑ(ρ))=Ḑ(Φ_p⃗(ρ)).

It states that the decoherence map Ḑ, acting in the space of density matrices –
see Sec. <ref> –
 is not coupled with the dynamics generated by a Weyl channel
Φ_p⃗.
Thus coherence  of input states of  Weyl channels  plays
no role in the evolution of probability vector of their diagonal entries.
Applying  a mixed unitary Weyl channel (<ref>) on
 an input state ρ we see
 that the evolution of its diagonal entries
 is governed by the classical transition matrix  T given in  (<ref>)
 and related
 to the Weyl channel through Eq. (<ref>).
If d_Y denotes the vector of diagonal
elements of a matrix Y, written d_Y=(Y),
then the following equality holds,

    Td_ρ=d_Φ(ρ).

This relation, not working for an arbitrary quantum channel,
 shows that in the case of a Weyl channel, the
 corresponding transition matrix  T(Φ)
 is responsible for the evolution of the diagonal elements of ρ.
This fact can be restated in the following way.



The evolution of diagonal elements of input states for the channels
defined by Eq. (<ref>) that also are accessible by a dynamical
semigroup, i.e. those Weyl channels describing a Markovian evolution, is
governed by Markovian Pauli master equation:

    d/dtd_ρ=Ķd_ρ,

where Ķ is the Kolmogorov operator.

 So d_ρ(t) in the above equation is determined as
 d_ρ(t)=d_ρ(t)=^tĶd_ρ(0)=𝕋_td_ρ(0).
 For future references let us explicitly mention necessary properties
 of a Kolmogorov operator  represented by a real matrix of order N – see <cit.>,

 a)∀ i≠ j    Ķ_ij≥0,

 b)∀ j       ∑_i Ķ_ij=0, or equivalently:

 b')∀ j        Ķ_jj=-∑_i≠ jĶ_ij.



As discussed in Section <ref>,
the process of decoherence  Ḑ brings a quantum state
into a classical probability vector,
while the hyper-decoherence  Ḑ_h
sends a quantum channel Φ into a classical
 transition matrix T.
 In a similar manner a Lindblad operator Ļ governing the quantum dynamics
 is transformed by process of hyper-decoherence into a Kolmogorov operator Ķ
 generating a classical semigroup.
 Consider first a Lindblad generator determining a quantum semigroup
 (<ref>) related to accessible Weyl channels.
 
Decohering and then reshaping the diagonal elements of  Ļ^R,
such that
Ļ is defined in Eq. (<ref>),  gives a proper Kolmogorov
operator corresponding to the Lindblad generator.
 
 
  It is straightforward to see the aforementioned method results in the
  following operator,
 Ķ=∑_k,lp'_k,lU_kl⊙U_kl-_N.
 Thus based on the discussions right after Eq.(<ref>), we have:
 
    Ķ=∑_k=0^N-1q'_kX^k-_N=∑_kq'_kĶ_k

 where q'_k=∑_lp'_kl and Ķ_k=X^k-_N is the
 Kolmogorov operator associated to Ļ_kl and is independent of
 l. Note that Ķ_0=0 is the trivial case. It is easy to see Ķ and
 Ķ_k  satisfy properties (a), (b) and (c) of a
 valid Kolmogorov operator which completes the proof.
 

In general, the process of hyper-decoherence acting in the space of Lindblad operators
produces a valid Kolmogorov generator  <cit.>,
with entries Ķ_ij =  Tr[|i⟩⟨ i| Ļ(|j⟩⟨ j|)].
Thus any Lindblad operator produces,  by the N–dimensional  projection operator Π_N
used in  Eq. (<ref>),
a corresponding classical  Kolmogorov generator,
 
    Ķ(Ļ)  = Ḑ_h(Ļ) =  Π_N ĻΠ_N ,    and   Ķ_ij =Ļ_ii
     jj .

However, the analyzed case of Weyl channels and the corresponding Lindblad operators
is rather special, as all the Lindblad  generators
are commutative - see Eq. (<ref>).
This property is also inherited in the classical setup,
as due to Lemma <ref>
 all the generators  Ķ_k are commutative,
 so are all Ķ and Ķ' in the form of  Eq.
 (<ref>). The commutativity of Kolmogorov operators Ķ_k
 imposes commutativity on their respective dynamical semigroups,
 𝕋_t_k𝕋_t_k'=
 𝕋_t_k'𝕋_t_k where
 𝕋_t_k=^t_kĶ_k.

Moreover,  classical transition matrices  gained by any concatenation of classical
 dynamical semigroups, ^t_kĶ_k,
 belong to the set C_N of circulant bistochastic
 matrices  (<ref>), i.e.
 
    𝕋_t=
    𝕋_t_0𝕋_t_1…𝕋_t_N-1=
    ^t_0Ķ_0^t_1Ķ_1…^t_N-1Ķ_N-1=^∑_k=0^Nt_kĶ_k=^t∑ q'_k
    Ķ_k=^tĶ=T_q,

 where the total interaction time t=∑ t_k and q'_k=t_k/t.
 Observe that  the order of
 classical dynamical semigroups is not important because of their
 commutativity. To see that, one can apply the Fourier matrix F to diagonalize the
 Kolmogorov operator and to find eigenvalues, so 𝕋_t has
 the following form:
  
    𝕋_t=∑_j^t(ξ'_j-1)x_j,

 where ξ'_j are the eigenvalues of ∑ q'_kX^k
 obtained by applying F on the probabilities q'_k. Therefore,
 the above classical dynamical semigroup
  can be rewritten in the form (<ref>) in
 which probabilities q_k are given by,

    q_k=1/N∑_jF^†_kj^t(ξ'_j-1)=
    1/N∑_jω__N^kjexp(∑_m
    (ω__N^-jm-1)t_m).


 In Theorem <ref>  we will prove  that this equality provides
 a valid probability vector with non-negative entries.
Observe that according to Eq. (<ref>),
 for any real ξ'_j, the corresponding eigenvalue of
 𝕋_t is positive.
 For the complex conjugate pair ξ'_j and
 ξ'_-j⊕ N, the corresponding eigenvalues of
 𝕋_t are complex conjugates as well.
Moreover, at t_+=2nπ/|Im(ξ'_j)| and
t_-=(2n-1)π/|Im(ξ'_j)| for any n∈ℕ,
𝕋_t will possess positive degenerate and negative
degenerate eigenvalues respectively. Finally, as
 |ξ'_j|≤ 1, the moduli of the eigenvalues of
 𝕋_t are either unity or
 decay exponentially with respect to t.

 We denote by A̧_N^C⊂ C_N the set of circulant bistochastic transitions
 satisfying Eq. (<ref>) and forming the set of
 transition matrices accessible by a classical dynamical semigroups with
 the Kolmogorov operator in the form of Eq. (<ref>). The
 aforementioned equation also implies the following result,
 analogous to the quantum counterpart
formulated in Corollary <ref>,

 
 The set A̧_N^C of classical transition matrices
 acting on N-point probability vectors
 and  accessible by a classical dynamical
 semigroup generated by Ķ in Eq. (<ref>) is
 log convex.
 


Assume now a circulant bistochastic transition in the form of

    T=∑_nξ_nx_n.


Again, we choose logarithm such that
logξ_n=logξ_n=logξ_-n⊕ N.
The necessary and sufficient
condition for T to be accessible by a classical dynamical
semigroup  (<ref>),  T∈A̧_N^C,
is that  the sum on the right hand side of Eq. (<ref>) is real and positive,

    t_m=1/N∑_n(QF^† Q)_mnlogξ_n≥0,

for m=1,…, N-1.
Here F denotes  the Fourier matrix, Q=_N-0   is the
projector operator on (N-1)-dimensional subspace and
0 projects ξ⃗ to ξ_0=1.
Note that there is no condition on t_0 as it is the coefficient of
 Ķ_0=0.



Let us first show that if T is an accessible classical map, then
Eq. (<ref>) holds. So we  start by the dynamical
semigroup in Eq. (<ref>)
that implies the following relation for the
eigenvalues of an accessible map:

    logξ_n=t(ξ'_n-1)=t(∑_m=0^N-1ω__N^-mnq'_m-1)=
    ∑_m=0^N-1(ω__N^-mn-1)t_m=
    ∑_m=0^N-1F'_mnt_m,

where F'=F-V with V_mn=1. As the entries of the first row and
column of  F' are all zero, F' is not invertible and F'=QF'Q. It is
again possible to define the inverse on the (N-1)-dimensional
subspace Q.
In the following we show this inverse exists and equals to
QF^† Q, up to the constant N:

    (QF^† Q)(QF'Q)   =   ∑_m,n=1^N-1∑_i,j=0^N-1∑_k,l=0^N-1ω__N^ij(ω__N^-kl-1)
     mi⟩⟨ j n k⟩⟨l|
       =   ∑_m,n=1^N-1∑_l=0^N-1ω__N^mn(ω__N^-nl-1) ml
       =   N∑_m=1^N-1∑_l=0^N-1δ_ml ml
    =N∑_m=1^N-1 mm= N(_N-0).

So by acting (QF^† Q) on the both sides of Eq. (<ref>) the
condition in Eq. (<ref>) is achieved for interaction time t_m
which should be non-negative. Moreover, note that Eq. (<ref>)
implies logξ_n=logξ_n.
On the other hand, if the inequality (<ref>) holds between
eigenvalues of a classical map, as the interaction time
is explicitly presented, it is clear that the map belongs to A̧_N^C.


As in the quantum version,  condition (<ref>) for T implies
existence of a proper logarithm of T satisfying
 properties of a Kolmogorov operator mentioned
right after Prop. <ref>. In analogy to the quantum case
one can simply write log T as follows:

    log T= ∑_n=0^N-1logξ_nx_n=1/N∑_m=0^N-1∑_n=0^N-1∑_j=0^N-1ω__N^mnlogξ_nm⊕ j j.

For this generator   ∀ j: ∑_i(log T)_ij=0 is straightforwardly verified.
In addition,   ∀ i≠ j: (log T)_ij≥0   means

    ∀ m≠0:      1/N∑_n=0^N-1ω__N^mnlogξ_n≥0.



Interestingly, the interaction times  t_m entering Eq. (<ref>)
can be obtained by
summation over the second index of t_mn introduced in Eq. (<ref>)
when it is rewritten by two indices m,n rather than
μ. This observation suggests a closer relation between
quantum accessible channels and classical accessible transition matrices.
 The following theorem
shows that A̧_N^C is the image of A̧_N^Q in classical space,
 i.e. Ḑ_h(A̧_N^Q)=A̧_N^C,
 where hyper-decoherence Ḑ_h is defined by the projection (<ref>).


Let Φ_p⃗ be a Weyl channel accessible by a Lindblad generator in
Eq. (<ref>), i.e. Φ_p⃗ =^tĻ∈A̧_N^Q.
Then the corresponding
circulant bistochastic matrix T(Φ) belongs to the set of
transition matrices  accessible by a classical semigroup whose generator Ķ is
defined by  Eq. (<ref>) based on Ļ, i.e.
T=^t Ķ∈A̧_N^C and Ķ is related to Ļ
 through Lemma <ref>. Hence for a generator Ļ related to a Weyl channel the
 hyper-decoherence commutes with the time evolution,
 
    Ḑ_h(^tĻ)=^tḐ_h(Ļ)
      =e^t Ķ.



We start the proof by finding classical transition of Φ=^tĻ.
Assuming t_mn=tp'_mn, and rewriting Eq. (<ref>)
using two indices k,l one has:

    Φ_p⃗ =^tĻ=∑_k,lp_k,lU_kl⊗U_kl,

where

    p_kl=1/N^2∑_ijω__N^kj-liexp(∑_mn(ω__N^in-jm-1)t_mn).

Hence T can be gained by Eq. (<ref>), in which the probabilities q_k read,

    q_k   =   ∑_l p_kl=1/N^2∑_ijω__N^kj∑_l(ω__N^-li)
    exp(∑_mn(ω__N^in-jm-1)t_mn)
       =   1/N∑_jω__N^kjexp(∑_m(ω__N^-jm-1)t_m),

where t_m=∑_nt_mn=t∑_np'_mn=tq'_m,
while  q'_m is defined by Eq. (<ref>).
Equalities (<ref>) and  (<ref>) complete the proof.
Furthermore,  Eq. (<ref>) explicitly  represents a  valid probability vector q
with non-negative components, as it was claimed before.


Thus the classical image of a quantum accessible Weyl channel
 is a circulant bistochastic matrix accessible by a classical semigroup.
 An immediate consequence of this fact and Prop. <ref>  is the following Corollary.


 The set A̧_N^C of circulant bistochastic transition matrices,
   accessible by a semigroup  generated by Kolmogorov operators (<ref>), is
star-shaped with respect to the flat matrix, T_∗=1/N∑_kX^k,
in analogy to its quantum counterpart A̧_N^Q.

Already we have seen the set of classical accessible transitions are
log-convex and star-shaped. The following Proposition shows how
one can get the boundaries of the set A̧_N^C.



The boundaries of the set, ∂A̧_N^C, are accessible when at
least one of the probabilities q'_k related to non-trivial Kolmogorov
operators (k≠0) (<ref>) is zero. In other words, if one takes at most N-2
out of N-1 non-trivial Kolmogorov operators, the resultant transitions,
^tĶ,  form the boundaries ∂A̧_N^C.


To prove the claim, note that if T belongs to A̧_N^C it is
 connected to __N through a trajectory that at each
 t≥ 0 belongs to the set, so does for small time t=ϵ.
 Expanding ^tĶ around t=0, one has:
 
    ^ϵĶ   =   (__N+ϵdTdt|_t=0)=
     (__N+ϵĶ)=
     (1-ϵ(1-q'_0))
     __N+ϵ∑_k=1^N-1q'_kX^k,

that should describe a proper bistochastic transition for small
ϵ>0. This means all of its entries should be non-negative and
sum of them over each row and column equals to unity. The latter is
satisfied automatically. The first condition means
1-ϵ(1-q'_0)≥0 and ϵ q'_k≥0 for
k∈{1,…,N-1}. Regarding that ϵ is small
1-ϵ(1-q'_0) is always positive, however, if q'_k is taken
to be zero for k∈{1,…,N-1}, the second inequality is satisfied
with equality. So in that case we get the boundaries of the set.







 §.§ Spectra of accessible quantum Weyl channels and circular
bistochastic matrices 


In this subsection we analyze the boundary of certain cross-sections
of the set A_N^Q of accessible Weyl channels
and characterize the support of their spectra in the complex plane.
As Proposition <ref> relates the eigenvalues of a superoperator Φ_p,
corresponding to a Weyl channel, with eigenvalues of its classical action,
described by a circulant bistochastic matrix, our results concern also
support of spectra of transistion matrices accessible by a classical semigroup.
Circulant bistochastic matrices of order N
are spanned by the cycle permutation matrix X and its powers,
Δ(X,X^2,..., X^N), where X^N = 𝕀_N.

We consider the spectrum of accessible Weyl channels
A̧_N^Q, i.e., for which
there exists a trajectory, z(t), such that z(t=0)=Φ_ 
 and belongs to the set of Weyl channels at all times t.
This is equivalent to the statement that the trajectory in the space  of transition
matrices corresponding   to z(t) by hyper-decoherence
belongs to the simplex Δ for all times t.
In order to obtain an analytical description, it suffices
to study the behavior of the curve at times t close to 0.
Using this observation, we get the boundary of spectrum A̧_N^Q by a parametric
expression,

    e^± i t - t tan(π/N)  for 
    t ∈[0,π].




Note that in the region described above, equal to the support of spectra
of accessible classical transition matrices from 𝒜_N^C, 
there is a line segment of negative values. 
More precisely, the intersection of 
of the spectra of accessible classical and quantum maps
with the real line is
𝒜_N ∩ℝ = [-e^-πtan(π/N), 1].
Thus, the set 𝒜_N^C of bistochastic matrices accessible by a classical semigroup
contains also matrices with degenerated eigenvalues which are real and negative.
The minimum real eigenvalue for a matrix T belonging to 𝒜^C_N,
or for a superoperator Φ from 𝒜^Q_N,
reads

    x_ min(N)= -e^-πtan(π / N) =
    -1+π ^2/N+𝒪(1/N^2).

For N=3 this number is close to zero, x_ min(3)=
-e^-√(3)π≃ -0.004333,
but for large dimension one has, x_ min(N) → -1,
and the support of the spectrum of accessible maps covers completely the unit disk.
In order to calculate the area of this set, we consider real and imaginary part
of the parametric expression, x(t) and y(t), respectively

    x(t)    =
    cos (t) e^-t  tan(π/N),
    
    
    y(t)    =
    sin (t) e^-t  tan(π/N).

Next, we calculate the area enclosed by this curve using standard analytical methods

    V_N=
    2 ∫_0^π x(t) y'(t)  t
    =
    1/2(1-e^-2 πtan(π/N)) (π/N)
    =
    π -π ^3/N+𝒪(1/N^2) ,

which for N=3 gives
 V_3=1-e^-2 √(3)π/2 √(3)≃ 0.28867.
Observe that for any z∈ℂ\ℝ,
with |z| ≤ 1, the curve z^t tends to 0 as t →∞. The trajectory is a logarithmic spiral (Spira mirabilis),
 and therefore, it crosses the negative real axis infinitely many times.

It should be mentioned here that in the mathematical
literature, stochastic matrices accessible by a classical semigroup are called
embeddable and an expression equivalent to Eq. (<ref>) appeared in the
literature <cit.>.
The characteristic heart-like set plotted in Fig. <ref>a
was analyzed recently <cit.> in the context of
Markovian evolution of quantum coherence for N=3 systems.
Note that the logaritmic spiral  (<ref>) plotted in the case N=3
in Fig. <ref>a, is relevant for our study in several ways:
 it characterizes
a) the boundary of the set of quantum accessible Weyl channels 𝒜^Q_3
at the triangular face Δ(Φ_𝕀, Φ_ X, Φ_X^2) – see
Fig. <ref>a; b)  the boundary of  the set 𝒜^C_3 of classical accessible matrices
in the triangle  Δ(X,X^2, 𝕀_3) of circular bistochastic matrices – see Fig. <ref>c;
 the boundary of the support of the spectrum of members of
  c)  quantum accessible maps from 𝒜^Q_3;  d) classical accessible transition matrices from 𝒜^C_3 – see Fig. <ref>c.
 The two latter statements concerning the boundary of the spectra
   in the complex plane are also valid for higher dimensions N
   – see Fig.   <ref>b-d.





§ LINDBLAD DYNAMICS FOR QUTRITS


  

 §.§ Explicit criterion for accessibility by a Lindblad semigroup


Let us now apply previous results for the special case N=3.
Although the explicit form of Weyl unitary matrices is not relevant here,
 we wish to explain the notation for future reference.
For the sake of brevity, we are going to use one
index μ∈{0,…,8} instead of two k,l∈{0,1,2}, related by
μ=3k+l.
This means,
up to a phase, U_1 is equal to  U_2^†,   U_3 is
U_6^†, U_4 equals to U_8^†, and  U_5 is
U_7^† which in turn implies  following relations on
the eigenvalues:  λ_2=λ_1,
λ_3=λ_6, λ_4=λ_8, and
λ_5=λ_7.



In the case N=3 the Hermitian matrix H of size N^2=9,
 defined in Eq. (<ref>)
can be represented in the element-wise logarithm notation
useful for complex Hadamard matrices <cit.>,

    log H   =   2 π  i/3(
    [ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙; ∙ ∙ ∙ 2 2 2 1 1 1; ∙ ∙ ∙ 1 1 1 2 2 2; ∙ 2 2 ∙ 1 2 ∙ 1 2; ∙ 1 2 2 ∙ 1 1 2 ∙; ∙ 1 2 1 2 ∙ 2 ∙ 1; ∙ 2 1 ∙ 2 1 ∙ 2 1; ∙ 2 1 2 1 ∙ 1 ∙ 2; ∙ 2 1 1 ∙ 2 2 1 ∙;   ]),

where
 ∙ represents zero,
 so that H_11=exp(0)=1, while H_24=exp(i 4π/3)=ω_3^2.
 
Applying H on the probability vector p_μ defining the
mixed unitary channel (<ref>)
 
yields eigenvalues λ_μ of the superoperator Φ_p⃗.
To describe a legitimate quantum channel
accessible by a semigroup, Φ_p⃗∈𝒜_3^Q,
the interaction times need to be non-negative, t_μ≥ 0,
which  due to Eq. (<ref>) yields
constraints for the complex eigenvalues λ_μ.


    log(|λ_1|^2/|λ_3||λ_4||λ_5|)±√(3)(θ_3+θ_4+θ_5
    +M_3+M_4+M_5)   ≥   0,
    log(|λ_3|^2/|λ_1||λ_4||λ_5|)±√(3)(θ_1+θ_4-θ_5
     +M_1+M_4-M_5)   ≥   0,
    log(|λ_4|^2/|λ_1||λ_3||λ_5|)±√(3)(θ_1-θ_3+θ_5
    +M_1-M_3+M_5)   ≥   0,
    log(|λ_5|^2/|λ_1||λ_3||λ_4|)±√(3)(θ_1+θ_3-θ_4
    +M_1+M_3-M_4)   ≥   0.

Here  θ_i denotes the phase of λ_i and M_i∈ℤ
is an integer.
Since the arguments of the logarithm have to be positive
the second line implies |λ_3|≥√(|λ_1| |λ_4||λ_5|).
Substituting this to an analogous equation
 from the first line, |λ_1|^2≥√(|λ_3| |λ_4||λ_5|),
we obtain an inequality,  |λ_1|^3/2≥  |λ_4|^3/2 |λ_5|^3/2.
Taking both sides to the power 2/3 and repeating these
steps with the other equations,
we arrive  at the following necessary condition on the eigenvalues:

    |λ_α|≥|λ_β||λ_γ|,

for any choice of indices but the leading one,
αβγ 0. These  conditions are analogous to
 relations (<ref>) defining the set A_2^Q
 of  qubit channels accessible by a semigroup <cit.>,
  except that for N=3 these conditions are necessary but not sufficient.

  

 §.§ Product probability vectors

 Applying Eq. (<ref>),
 we obtain eigenvalues of a dynamical semigroup as an explicit function of time:

    λ_0   =   1,
    λ_1,2   =   ^-3/2(t_3+t_4+t_5+t_6+t_7+t_8)^± i√(3)/2(-t_3-t_4-t_5+t_6+t_7+t_8),
    λ_3,6   =   ^-3/2(t_1+t_2+t_4+t_5+t_7+t_8)^± i√(3)/2(t_1-t_2+t_4-t_5+t_7-t_8),
    λ_4,8   =   ^-3/2(t_1+t_2+t_3+t_5+t_6+t_7)^± i√(3)/2(t_1-t_2-t_3+t_5+t_6-t_7),
    λ_5,7   =   ^-3/2(t_1+t_2+t_3+t_4+t_6+t_8)^± i√(3)/2(t_1-t_2+t_3-t_4-t_6+t_8),

For channels Φ_p⃗ accessible  by a semigroup, the probabilities
can be expressed by Eq. (<ref>) as explicit functions of time.
These functions are analogous to Eq. (<ref>)
valid for N=2,
but contain many more terms,
For instance, in the case N=3 the time dependence of p_0
has the form

    p_0=1/9(1   +   2^-3/2
    (t_1+t_2+t_3+t_5+t_6+t_7)cos[√(3)/2(t_1-t_2-t_3+t_5+t_6-t_7)]
       +   2^-3/2
    (t_3+t_4+t_5+t_6+t_7+t_8)cos[√(3)/2(t_3+t_4+t_5-t_6-t_7-t_8)]
       +   2^-3/2
    (t_1+t_2+t_4+t_5+t_7+t_8)cos[√(3)/2(t_1-t_2+t_4-t_5+t_7+t_8)]
       +   2^-3/2
    (t_1+t_2+t_3+t_4+t_6+t_8)cos[√(3)/2(t_1-t_2+t_3-t_4-t_6+t_8)]).




A particular result characteristic to the single qubit case is the structure of the boundary
∂ A_2^Q
of the set of accessible maps, which consists of the channels corresponding
to probability vectors with a product structure  <cit.>,

    p_0p_i=p_jp_k,

for any different choice of i,j,k from the set {1,2,3}.
In the case N=3 we are dealing with 8-dimensional simplex and
product probability vectors form at most a 4-dimensional
subset of the 7-dimensional boundary of 𝒜_3.
Therefore, there are other channels at the boundary
represented by a probability vector p without
the product structure.


Consider any four out of eight Lindblad generators ℒ_μ defined
 by Eq. (<ref>) whose corresponding unitaries U_μ are Hermitian
 conjugates up to a phase. Let us represent them by
 ℒ_a,ℒ_b,ℒ_α,ℒ_β
  for which, up to a phase, U_a equals to U_b^† and U_α
  equals to U_β^†. For this set the associated dynamical
  semigroup defined by Eq. (<ref>) belongs to the 4-dimensional boundaries
   of 𝒜_3^Q and the corresponding  probability vector possesses
   the product structure.


The corresponding 8–dimensional vector t_μ of interaction times has
four components  equal to zero and four non-zero components,
so the channel  belongs to the boundary  of 𝒜_3^Q.
 To see the product structure of the probability vector p_μ
  one can express probabilities as  explicit functions of time through Eq. (<ref>).
  
 For the probabilities appearing in Λ_t_aΛ_t_bΛ_t_αΛ_t_β=∑ p_μ U_μ⊗U_μ,
one has p⃗̃⃗=w⃗(a,b)×w⃗(α,β)
in which p⃗̃⃗ is a permutation of the probability  vector p⃗,
while w⃗(x,y)=(f_1(x,y),f_2(x,y),f_3(x,y)) is a
local probability vector of size three given by

    f_1(x,y)   =   1/3(1+2^-3/2(t_x+t_y)cos[√(3)/2(t_x-t_y)]),
    
    f_2(x,y)   =   1/3(1-^-3/2(t_x+t_y)cos[√(3)/2(t_x-t_y)]+√(3)^-3/2(t_x+t_y)sin[√(3)/2(t_x-t_y)]),
    
    f_3(x,y)   =   1/3(1-^-3/2(t_x+t_y)cos[√(3)/2(t_x-t_y)]-√(3)^-3/2(t_x+t_y)sin[√(3)/2(t_x-t_y)]).


A particular case of pro:classicalproduct corresponds
to the channels with real eigenvalues λ_μ.
Note that this case is not generic: sampling uniformly
vectors r⃗, s⃗∈Δ_3 generates
channels with complex eigenvalues with probability one.
This can be seen by decomposing the eigenvalues
of Φ_p⃗ into its real and imaginary
parts. Equating the imaginary part of each eigenvalue to 0 impose
a large list of restrictions on r⃗ and s⃗,
leaving-at most-one free parameter of the original four (r_1,
r_2, s_1, s_2). Thus, the region
in the space of mixed unitary channels,
for which all the eigenvalues λ_μ are positive has zero measure.
An interesting implication of pro:classicalproduct is that
 inside the set 𝒜_3^Q of accessible maps there are no channels with rank 2.
 In other words, the edges connecting identity map U_0 with the Weyl unitary rotations U_μ
contain no accessible maps.
This fact is visualized in Fig. <ref>
which presents several cross-sections of the 8D
simplex Δ_8 of mixed unitary Weyl channels and
its subset 𝒜_3^Q  of accessible maps.

Another difference  concerning the structure of the set of accessible maps
for N=2 and N=3 is that in the former case  p_0 is the largest
component for the probability vector of Φ∈𝒜_2^Q
for any t≥0 <cit.>.
This is not the case for N=3, even if we choose only one
interaction time t_μ to be nonzero – see Fig. <ref>a.


 

 §.§ Accessible maps for qubits and qutrits

  
  To describe properties of the set  A_N of maps accessible by a semigroup
    we shall compare the volumes of these sets for N=2 and N=3
    relative to the volumes of the simplex Δ_N^2-1 of Weyl mixed unitary channels.
   In the qubit case the set of mixed unitaries corresponds to the tetrahedron
   spanned by the identity operation, Φ_0=⊗,
       and unitary rotations associated to three Pauli matrices,
Φ_1=σ_x⊗σ_x,
Φ_2=σ_y⊗σ_y, and
Φ_3=σ_z⊗σ_z.
As the length of each edge of the regular tetrahedron can be set to unity,
its  volume reads, V_Δ_3=√(2)/12.

To calculate the volume of its subset A_2^Q we use
Eq. (<ref>)
and work with three-dimensional Cartesian coordinates.
Each point inside the tetrahedron Δ_3 can specified by its coordinates
X,Y,Z.
 For channels in Δ_3 which also belong to A_2^Q, the
 probabilities p_0, p_1, p_2, and p_3 are defined by Eq. (<ref>)
 suggesting X, Y, and Z as an explicit function of t_1, t_2, and t_3.
Let |J|=exp[-4(t_1+t_2+t_3)]/√(2) denotes the Jacobian determinant associated to such a change of variables.
The volume of the set  of one-qubit accessible maps reads then

    V_A̧_2=∭_A̧_2dXdYdZ=∭|J|dt_1dt_2dt_3=
    1/√(2)∫_0^∞^-4t_1dt_1∫_0^∞^-4t_2dt_2∫_0^∞^-4t_3dt_3=1/4^3√(2).

Hence the relative volume of the set of maps accessible by a semigroup is
V_A̧_2/V_Δ_3=3/32=0.09375,
as it forms a quarter of the volume of the set
U̧_2^Q of one-qubit unistochastic channels
investigated in <cit.>.
On the other hand,  for the qutrit channels
the analogous ratio gained numerically is much smaller,
V_/V_Δ_8= 9.02×10^-4±
1.8× 10^-5. This estimation was obtained by sampling
10^6 different random probability vectors p⃗∈Δ_8
and verifying whether the corresponding mixed unitary channel
satisfies all conditions  (<ref>) for accessibility.
As for qubit channels this ratio  is larger by two orders of magnitude, one
can expect a significant decrease of the relative volume of accessible channels
with their dimension N.


To analyze the structure of the set A̧^Q_3 of accessible maps
we shall analyze cross-sections of the simplex   Δ_8
determined by fixing the weight p_0 of the identity component
of a mixed unitary channel (<ref>).
  Consider an 8-point random vector
   φ⃗∈Δ_7 from the 7 dimensional simplex
 and define a 9 dimensional vector,
 p⃗ = [p_0, (1-p_0)φ⃗].
The corresponding mixed unitary channel Φ_p⃗ is  thus given by Eq. (<ref>).

Generating auxiliary vectors  φ⃗ according to the flat measure in the simplex Δ_7
we obtain an ensemble of random vectors p⃗
and thus random unitary channels with a fixed size of the identity component p_0 and
 study the fraction of probability vectors corresponding to accessible channels.
 Numerical results presented in  Fig. <ref>
 show that  for  N=3  the relative volume of the analyzed set ,
grows monotonically to 1 as a function of the
weight p_0 of the probability vector defining the channel.
For comparison we plotted analogous data obtain for the case N=2
for which the relative volume grows faster with the weight p_0.
However, in this case the curve has an inflection point
which does not occur for N=3. The inset shows behavior for small values of p_0,
as the relative volume is positive already for p_0 <1/9.
This observation is consistent with Fig. <ref>a,
showing that in the case N=3
 there exist accessible maps Φ_p,
  for which the component p_0 is not the largest one.



 

 §.§ Bistochastic matrices accessible by N=3  classical semigroup


As discussed in  Section <ref>,
any mixed Weyl channel decoheres to a classical circulant bistochastic matrix.
We shall analyze the issue here  for N=3 in some more detail.
Related question of describing the subset of the Birkhoff polytope
containing these bistochastic matrices of order three,
so that their square root is bistochastic, was studied in <cit.>.

Consider a random Weyl channel Φ_p⃗
distributed uniformly from the triangle  Δ(Φ_,Φ_X, Φ_X^2)
shown in  Fig. <ref>a
– a 2D face of the 8D simplex Δ_8 of all Weyl channels.
Circulant bistochastic transition matrices, obtained by hyper-decoherence,
T  = Ḑ_h(Φ_p⃗) cover the triangle
Δ(_3, X_3, X_3^2) uniformly – see Fig.  <ref>a.
This is a consequence of the fact that in this  subspace the supermap
 Ḑ_h is linear and transforms a regular triangle into itself,

    Ḑ_h(∑_j=0^2 q_j  Φ_X^j) =
      ∑_j=0^2 q_j  X^j  = T_q.

 On the other hand, if one takes an average over the entire set
 B_3^Q of Weyl channels and samples the probability
 vector p⃗ uniformly in the  simplex Δ_8
 the distribution of classical matrices exhibits maximum at the
 flat matrix T_* with all entries equal to 1/3
 –  see Fig. <ref>b.
 If the sampling is restricted to the set of accessible quantum channels only,
 its image under decoherence is supported inside the characteristic
 `heart–like' shape presented in panel  Fig. <ref>c.



In Eq. (<ref>) we used the fact that a circulant matrix T
can be diagonalized by the Fourier matrix F.
If the transition matrix is obtained by the convex combination of permutation matrices,
T=∑_i=0^2 q_j X_3^j, then its spectrum consists of 1, z, z̅,
where z=∑_i=0^2 q_j ω_3^j.
Therefore, if the distribution of the probability vector q is uniform in the
triangle of bistochastic matrices, so is the level density of subleading eigenvalues
in the regular triangle  Δ(1, ω_3, ω_3^2)
inscribed in the unit disk —compare Fig. <ref>a.

To analyze the shape of the set A_3^C of  circulant
  bistochastic matrices  accessible by a classical semigroup
 we may use Proposition <ref>.
 Since the hyper-decoherence transformation, T  = Ḑ_h(Φ_p⃗),
  acts linearly  (<ref>)
  in the
  triangle of quantum Weyl channels  Δ(Φ_,Φ_X, Φ_X^2),
  the shape of the set  A_3^C of classically accessible matrices
  is identical with cross-section of the set
   A_3^Q of quantum accessible maps
   presented in Fig. <ref>d.
   This set is uniformly covered when the same region,
   corresponding to quantum channels, is uniformly sampled.
   The structure of this star-shaped set, bounded by fragments of two
   symmetric logarithmic spirals, is discussed in Section <ref>.
   This set appeared in the literature in context of
    analyzing Markovian evolution of quantum coherence
    and N=3 embeddable stochastic matrices <cit.>.








§ SET   OF UNISTOCHASTIC CHANNELS ACTING ON N=3 SYSTEMS



As for N=2 the set A_2^Q of quantum accessible channels
forms a quarter of the set  U_2^Q of unistochastic channels,
so  the relation
A_2^Q  ⊂ U_2^Q holds <cit.>.
Hence in this Section we  compare the corresponding sets for N=3.
A similar analysis
will be performed for the analogous classical sets inside the Birkhoff polytope
B_3^C of bistochastic matrices of this size.



 §.§ Unistochastic channels and channels accessible by a semigroup
 

We start demonstrating that for one-qutrit system
the set  of quantum accessible channels is not included
in the set  of unistochastic channels,
written A_3^Q  ⊈ U_3^Q.


 For 3-level systems,
there exists a channel accessible by a Lindblad semigroup
which is not unistochastic.


  To show a  counterexample we analyze Weyl channels
  being a convex combination of , Z, and Z^2,
  
    Φ_p = p_1 ⊗ + p_2 Z⊗Z +
        (1-p_1-p_2) Z^2⊗Z^2.

  We wish to find a vector p=(p_1,p_2,p_3)
  such that there is no unitary dilation matrix U∈ U(9)
  for which Eq. (<ref>) produces the analyzed channel Φ_p.
  Given that Z is diagonal, the dilation matrix U is block diagonal,
  with three unitary blocks W_1,W_2 and W_3, each of size 3,
  
    U = W_1⊕ W_2 ⊕ W_3
        =
      (1,0,0)⊗ W_1 +(0,1,0)⊗ W_2
    +(0,0,1)⊗ W_3.

  By unistochasticity of the channel Φ_p the
  corresponding Choi matrix D=Φ_p^R has to satisfy
  D=1/N(U^R)^†U^R.
  This leads to the following conditions for W_1, W_2 
  and W_3:

    W_2^†W_1 =
      s,   W_3^†W_1 =
    s^*,   W_3^†W_2 =
      s,

where s =  1/2(3p_1-1) +i√(3)/2(p_1+2p_2-1).

Consider a probability vector with three non zero entries
given by p⃗_c = (p_1,p_2,1-p_1-p_2) = (35,4,1)/40.
Using p⃗_c in Eq. (<ref>),
we can verify that Φ_p⃗_c satisfies the three properties mentioned at
the end of Subsection <ref>,
which implies that the channel is accessible, Φ_p ∈ A_3^Q.
Making use of the parametrization of the group of unitary matrices of size 3
one can show that there are no
 unitary matrices W_1, W_2, W_3∈ U(3)
which satisfy conditions  (<ref>)
for the vector p⃗_c selected above:
if any two of these equalities are satisfied, the third one is not.
Hence for this p⃗ there is no unitary dilation matrix U∈ U(9)
entering expression  (<ref>),
so the channel  Φ_p⃗ is not unistochastic.


Although for 2-level systems,
the following inclusion relations holds
A_2^Q  ⊂ U_2^Q <cit.>,
we have shown that the analogous property is not valid for N=3.




 §.§ Unistochastic and N-unistochastic matrices
 

Due to the process of hyper-decoherence a quantum stochastic
map undergos transition to a classical stochastic matrix.
We shall  analyze relations between
subsets of the set S_N  of stochastic matrices
introduced in Section  <ref>.



For any N-level system, the set of
unistochastic matrices  U_N^C is contained in the set
D_h( U_N^Q) of transition matrices
obtained by the hyper-decoherence <cit.> of a unistochastic channel.


It is enough to take a dilation matrix of the product form,
U=V⊗_N.
Then due to
 Eq. (<ref>)  one obtains T=V ⊙V̅,
where ⊙ denotes the Hadamard product,
so the classical transition matrix is unistochastic.
Thus the set of classical transition matrices obtained by hyper-decoherence
of unistochastic channels U_N^Q
containing N-unistochastic matrices,
includes the set  of unistochastic matrices,
D_h( U_N^Q)= U_N,N⊃ U^C_N
– compare table  <ref>.





 §.§ Quantum unistochastic maps and classical
unistochastic matrices


To explore relations between unistochastic maps and
matrices, we will restrict our attention to the face
of the set W_N of mixed Weyl channels determined by three points:
, U_X, and  U_X^2.
In this equilateral triangle we need to distinguish two regions:
i) the  region bounded by the 3-hypocycloid <cit.> denoted by
ℍ_3. For
p_1, p_2 ∈Δ_2, the set ℍ_3
includes points satisfying the inequality,

    4p_1^2(1-p_1-p_2)p_2-(p_1-p_1^2-(1-p_1-p_2)p_2)^2 ≥ 0.

The 3–hypocycloid
determines the boundary of the set of unistochastic matrices
at the cross-section of the N=3 Birkhoff polytope <cit.>,
 as the above expression implies positivity of the
 quantity (<ref>), which
 determines  the Jarlskog invariant of
the corresponding bistochastic matrix—see Appendix <ref>.

ii) the region resembling the Star of David, formed
by two equilateral triangles, _A and
_B, each one described by
three vertices on the plane:

    _A (1/3,0), (1/3, 1/√(3)), (5/6,
        1/2√(3)),     _B  (2/3,1/√(3)), (2/3,0), (1/6,1/2√(3)).

These subsets of the triangle Δ(,Φ_X,  Φ_X^2)
 are represented in Fig. <ref>.
 The 3–hypocycloid plays a key role <cit.>
 in characterizing the set of unistochastic matrices of order three.
 We will show now that it is also relevant to describe
 the set of unistochastic channels and its cross-sections.





Mixed unitary Weyl channel Φ_p⃗
belonging the the 3–hypocycloid,
 p⃗∈ℍ_3 ⊂Δ(Φ_,Φ_X, Φ_X^2),
 is unistochastic,  Φ_p⃗∈ U_3^Q,
and the corresponding transition matrix is unistochastic,
T(Φ_p⃗) ∈ U_3^C.



Consider a matrix U of order N^2=9,

    U =
        √(p_1)e^iϑ_1⊗+
        √(p_2)e^iϑ_2X ⊗ X +
        √(p_3)e^iϑ_3X^2 ⊗ X^2,

where {ϑ_i}_i=1^3 are real phases to be
determined.
Substituting U in Eq. (<ref>), we observe that it generates a
superoperator
Φ_p⃗ = p_1 𝕀⊗𝕀 + p_2 X ⊗X
+(1-p_1-p_2)X^2⊗X^2.
Thus we have constructed a channel of the desired form,
provided matrix U is unitary.
To analyze under what conditions it is the case
we will compute the scalar product between each pair of rows.
Let us introduce auxiliary real parameters,
 L_1 = √(1-p_1-p_2)√(p_1),
L_2 = √(1-p_1-p_2)√(p_2), and L_3 = √(p_1)√(p_2).
Then the  resulting expression takes the form,

    exp(i φ_1) L_1 +
      exp(i φ_2) L_2  +
      exp(i φ_3) L_3 = 0,

where {φ_i}_i=1^3 correspond to linear
combinations of phases {ϑ_i}.
Left-hand side of Eq. (<ref>), can be
pictured as three joined line segments on the complex plane,
of length L_1, L_2, and L_3, respectively.
Inequality holds if these three sides
form a triangle. Thus Eq. (<ref>)
is equivalent to the triangle inequality:

    |L_1-L_2| ≤ L_3 ≤ L_1 + L_2,

This inequality is satisfied for any point p in the interior of the hypocycloid,
p⃗∈ℍ_3.
In such  a case the columns of U form an orthonormal set,
hence U is unitary  <cit.>.
This implies that if p⃗∈ℍ_3
then due to Eq. (<ref>)
unitary U leads to a unistochastic Φ_p⃗,
while T(Φ_p⃗) is a unistochastic matrix.


Further analysis shows that the cross-section of the set  U_3^Q
of unistochastic channels by the plane containing the triangle
Δ(Φ_,Φ_X, Φ_X^2) is larger than the
hypocycloid p⃗∈ℍ_3.
Numerical results allow us to formulate the  following conjecture.


In the face of the simplex W_3 of the Weyl channels formed by
Δ_Q=Δ(Φ_𝕀, Φ_X, Φ_X^2),
the set  of unistochastic channels corresponds
to the region formed by the union ℍ_3∪_A∪_B.



The triangle Δ_Q of quantum Weyl channels
shown in Fig. <ref>a
corresponds to classical triangle of bistochastic matrices
Δ_C=Δ(𝕀, X, X^2),
which forms a cross-section of the Birkhoff polytope  <cit.>.
Its subset A_3^C of matrices accessible by a classical semigroup
has the same structure as the set
A_3^Q of accessible quantum maps  shown in Fig. <ref>c.
The region union ℍ_3∪_A∪_B
corresponds to the set U^C_3,3 of 3-unistochastic matrices
while ℍ_3 denotes its subset
U^C_3 of unistochastic matrices – see Fig. <ref>b.


To conclude note that not every unitary matrix U∈ U(9)
 produces by  Eq. (<ref>)
 a unistochastic channel of order three,
 which decoheres to a unistochastic transition matrix.
 An exemplary  channel  Φ∈ U_3^Q
  such that T(Φ) ∉ U_3^C
 is described  in Appendix <ref>.



  

§ CONCLUDING REMARKS

  
  In this work we investigated properties of mixed unitary Weyl  channels
  and circulant bistochastic matrices
  which describe corresponding classical dynamics.
  Extending one-qubit results presented in <cit.>
  we characterized the set A_N^Q
  of Weyl channels accessible by a quantum semigroup.
  We showed that this set is log-convex and star-shaped with respect to
  the completely depolarizing channel.

  The standard process of decoherence, occurring due to inevitable
  interaction of the system analyzed with an environment,
  transforms any quantum state ρ into the classical probability vector,
  p= D(ρ)= diag(ρ), and destroys all quantum effects.
  In a similar way one analyzes analogous transitions
  between a quantum operation Φ and
  a classical transition matrix, T= D_h(Φ).
  Such an effect, sometimes called hyper-decoherence,
  can also act on a Lindblad operator L,
  which generates  a quantum semigroup.
  The image gives a legitimate Kolmogorov operator,
   K= D_h( L),
  related to a classical semigroup <cit.>.

  We showed that any Weyl channel Φ_p⃗
  subjected to hyper-decoherence yields a circulant
   transition bistochastic matrix T_q⃗, where
   the N-point probability vector  q⃗
   arises as the marginal of the N^2-point initial probability vector
   p⃗. Furthermore, for mixed Weyl channels the dynamics
   commutes with decoherence, so that
   Ḑ_h(^tĻ)=^tḐ_h(Ļ).
   Hence   the set A_N^C
   of transition matrices accessible by a classical semigroup,
   arises by hyper-decoherence of its quantum counterpart,
   A_N^C=  D_h(  A_N^Q).
   This implies that this classical set inherits key properties of the
   its quantum relative  A_N^Q and is also log-convex and star-shaped.

   For concreteness, we set the dimension N=3
   and in this case investigated geometry of the above sets.
   In particular, we studied various cross-sections
   of 8D set A_3^Q of accessible quantum channels
   and described in details the corresponding set
   A_3^C of  embeddable bistochastic matrices <cit.>,
   which are accessible by a classical semigroup.
  Motivated by the relation between maps accessible by a quantum semigroup
  and  unistochastic channels established in the single qubit case,
  A_2^Q  ⊂ U_2^Q <cit.>,
  we studied the unistochastic operations.
  A constructive example of a qutrit channel is presented,
  which is accessible by a quantum semigroup
  but it cannot be obtained by the partial trace over the environment of the
  same size, initially prepared in the maximally mixed state.
  Such a counterexample  allows us to conclude that
  for higher dimensions such a relation
  between unistochastic and accessible channels breaks down,
  as the analogous inclusion relation does not hold already for N= 3.

A notion of higher order, k–unistochastic matrices of size N has been introduced.
We have shown that unistochastic channels acting on N-level systems
 decohere to N–unistochastic matrices,
   D_h( U_N^Q)= U^C_N,N.
  It contains the set of unistochastic matrices,
  which arise by hyper-decoherence of isometric unitary transformations,
   U^C_N= D_h( I_N^Q).

  
  Let us conclude the paper presenting a list of some open problems.
  
    
  * Results of this work are obtained for mixed unitary channels
      written in the Weyl basis (<ref>).
        Analyze which of them can be generalized for a) mixed unitary channels
          written in other orthogonal basis, b) any bistochastic maps, or
          c) any stochastic maps.
    
  * For one-qubit case, channels belonging to Pauli
      semigroups are the only ones connected to the identity map
      through a trajectory formed by mixed unitary channels.
        Is an analogous property true for higher dimensions?
    
  * One-qubit Pauli maps belonging to 𝒜_2^Q
       (accessible by a semigroup) are of rank 1,2 or 4.
       For qutrits, we identified only  accessible channels of rank 1, 3,
      and 9, so existence of N=3 accessible channels with rank 2,4,…,8
        remains open.
    
  * Check if  the rank of the operation
        does not change (or does not decrease)
          during the time evolution  induced by a quantum
           semigroup L specified in Eq. (<ref>).
    
  * Find necessary or sufficient criteria for a N=3 bistochastic  map Φ
        to be unistochastic,
         so there exists a unitary matrix U∈ U(9)
          leading to the representation (<ref>).
  

It is a pleasure to thank Fabio Benatti, Adam Burchardt,
Dariusz Chuściński, David Davalos,
Kamil Korzekwa, Carlos Pineda, Łukasz Rudnicki
and Konrad Szymański
for numerous discussions and helpful correspondence.
This research was support by  National Science Center in Poland
under the Maestro grant number DEC-2015/18/A/ST2/00274
and by Foundation for Polish Science under the grant Team-Net NTQC
number  17C1/18-00. Z.P. acknowledges the support by the Polish National 
Science Center under the Project Number 2016/22/E/ST6/00062.
DAA acknowledges support from projects CONACyT 285754,
UNAM-PAPIIT IG100518.



§ JARLSKOG INVARIANT FOR N=3 BISTOCHASTIC MATRICES
 


Consider the Birkhoff polytope B_3^C
of  bistochastic matrix of order three.
Any matrix from this four-dimensional set
can be parameterized by
four non negative numbers b_1,b_2,b_3,b_4,

    B =
        [               b_1               b_2         1-b_1-b_2;               b_3               b_4         1-b_3-b_4;         1-b_1-b_3         1-b_2-b_4 b_1+b_2+b_3+b_4-1;                   ].

Note that these parameters do satisfy some constraints,
as any entry of B has to be positive and not larger than one.
A bistochastic matrix B is called unistochastic
if  there  exists a unitary matrix V
such that B can be represented by the Hadamard (entrywise) product,
B=V ⊙V̅,
so its entries read  B_ij = | V_ij|^2.

For any matrix B ∈ B_3^C  one introduces the quantity

    𝒬(B) ≡
        4b_1b_2b_3b_4 - (b_1+b_2+b_3+b_4-1-b_1b_4-b_2b_3)^2,

which serves as an effective tool to detect unistochasticity
of bistochastic matrices of size N=3.
If  Q(B)≥ 0, the matrix B  is unistochastic <cit.>,
as the triangle inequality (<ref>) is satisfied
and there exists a corresponding unitary matrix  V
such that B=V ⊙V̅.
Conversely, if  Q (B) is negative,
the unitarity triangle cannot be constructed and B is not unistochastic.
This is a consequence of the fact that
the triangle conditions necessary to find two orthogonal vectors
forming first two columns of V
are not satisfied <cit.>.

Quantity (<ref>)
can be written as Q= 4𝒥^2,
where 𝒥(B)=𝒥(B(V)),
 is called the Jarlskog invariant,
originally invented for the corresponding  unitary V  <cit.>.
The name of this quantity is related to the following invariance property:
For two unitary matrices V and V',
equivalent with respect to permutations and multiplication
 from left and right by two diagonal unitary matrices,
the value of  𝒥 computed for the
associated bistochastic matrices B and B' is constant.

Interestingly, the maximal value of Q, equal to 1/27,
is attained by the flat bistochastic matrix, B_ij=1/3,
which forms the center of the Birkhoff polytope B_3^C.
The minimal value reads, 𝒬_ min=-1/16,
 and corresponds to the combination
 of two permutation matrices  B_S=(X_3+X_3^2)/2,
  introduced in Section  <ref>,
 which is most distant from the set   U_3^C
 of unistochastic matrices  <cit.>.



§ NOT EVERY TRANSITION MATRIX CORRESPONDING TO UNISTOCHASTIC CHANNEL
 IS UNISTOCHASTIC

 
Consider the unitary matrix U∈ U(9) defined by

    U = 𝕀_2⊕ H_2⊕(1,-1,1)⊕𝕀_2,

where H_2 is the Hadamard matrix representing the Hadamard
gate, and (1,-1,1) is the matrix with non-zero
entries in the antidiagonal.
Such a two-qutrit unitary matrix U defines by Eq.   (<ref>)
a single-qutrit unistochastic channel Ψ_U.
The corresponding classical transition matrix
generated by hyper-decoherence, T= D_h(Φ_U), reads

    T =
    1/6[ 5 1 0; 1 3 2; 0 2 4 ].


Expression (<ref>) implies that its squared Jarlskog
invariant is negative,   Q(T)= -1/324.
Therefore, the matrix U∈ U(9) determines
a single-qutrit unistochastic channel Ψ_U,
such that the corresponding classical transition matrix T(U) of size N=3
is not unistochastic. This example shows that the set U_3^Q of
unistochastic channels (yellow region in Fig. <ref>a)
is not included inside the blue set of
isometric unitary channels I_3^Q
 which decohere to unistochastic transition matrices  <cit.>.
 


§ UNISTOCHASTIC CHANNELS AT THE CORNERS OF THE STAR OF DAVID.


In this appendix we present  six unitary
matrices of order N^2=9
which produce unistochastic channels
represented in  Fig. <ref>a
by points at six corners of the Star of
David for the 2-face of Δ_8.
Additionally, we will show the relation between the
unistochastic channels corresponding to faces spanned
by the triads  (Φ_𝕀, Φ_Z,Φ_Z^2)
and (Φ_𝕀, Φ_X,Φ_X^2).
First, let us introduce a diagonal unitary matrix ∈ U(9),

    =
    diag(1,1,1,1,1, 1,ω,1,ω^2),

where ω=e^2π i /3.
Using definition Eq. (<ref>), we obtain
the Choi matrix for the corresponding unistochastic channel,

    D_ =
      1/3
      (^R)^†^R
      =
        [         1         0         0         0 (2+ω^2)/3         0         0         0   (2+ω)/3;         0         0         0         0         0         0         0         0         0;         0         0         0         0         0         0         0         0         0;         0         0         0         0         0         0         0         0         0;   (2+ω)/3         0         0         0         1         0         0         0 (2+ω^2)/3;         0         0         0         0         0         0         0         0         0;         0         0         0         0         0         0         0         0         0;         0         0         0         0         0         0         0         0         0; (2+ω^2)/3         0         0         0   (2+ω)/3         0         0         0         1;           ],

which can be represented as
D_=(2/3Φ_𝕀 +
1/3Φ_Z)^R.
This implies that the unistochastic channel
 Φ_U_Z=2/3Φ_𝕀 + 1/3Φ_Z,
 so that it is represented by a point at  one of the six
corners of the star of David inscribed in the triangle
Δ(Φ_𝕀, Φ_Z, Φ_Z^2).
In a similar way one can construct unistochastic
channels corresponding to the remaining corners of the star.

Let us now consider the other face of the simplex of Weyl channels
 determined by (Φ_𝕀, Φ_X,Φ_X^2).
The channel
1/3Φ_𝕀+2/3Φ_X
at a corner of the star of David inscribed into this triangle
and plotted in  Fig. <ref>a,
corresponds to the unitary matrix
𝔽𝔽.
we use here  the tensor product of two Fourier matrices, 𝔽 = F_3⊗ F_3,
while (·) denotes the complex conjugation.
Combining the tensor product  of matrix Z of order three with Fourier matrices and
matrix  introduced above, we obtain the remaining channels
at six corners of the star of David at this face,
listed in
Tab. <ref>.



As demonstrated in Table <ref>
all the corners of the star correspond to unistochastic channels.
Numerical results support the conjecture that all points inside the
star also represent unistochastic channels.
Not being able to prove this statement so far, we can show
that the problem of finding the boundary of the
sets of unistochastic channels in the triangles
Δ(Φ_𝕀, Φ_Z, Φ_Z^2) and
 Δ(Φ_𝕀, Φ_X, Φ_X^2)
are equivalent.


Consider a unistochastic channel
Φ_p =
  p_1
  Φ_𝕀
  +
  p_2
  Φ_Z+
  p_3
  Φ_Z^2,
associated with an unitary matrix U of order nine.
Then the local transformation of U'=(F_3⊗
F_3)U(F_3⊗F_3), corresponds to
the channel
Φ'_p= p_1  Φ_𝕀+  p_2  Φ_X+ p_3  Φ_X^2
belonging to the triangle
Δ(Φ_𝕀, Φ_X, Φ_X^2).



We consider a unistochastic channel
 Φ_p =  p_1    Φ_𝕀+
  p_2   Φ_Z+  p_3   Φ_Z^2,
with associated unitary matrix  V.
 Our aim is to show that there exists a
local transformation that transforms it into Φ'_p.
In other words, we show that
a unistochastic channel in the cross-section (𝕀, Z, Z^2)
can be transformed into a unistochastic channel in the cross-section
(𝕀, X, X^2) by means of a local transformation.
Firstly, we find the expression for the channel entries
in four-index notation:

    ⟨e_mf_μ|Φ|e_n f_ν⟩   =
    δ_mnδ_μν
    (
      p_1  + p_2 ω^n-ν
      + p_3 ω^2(n-ν)
    ) =
    Φ_mν
     nν

Then the corresponding reshuffling is

    (Φ^R_mn
    μν) =
      δ_mμδ_nν
    ( p_1  + p_2 ω^μ-ν
      + p_3 ω^2(μ-ν)).

On the other hand, Choi matrix entries corresponding to a
unitary matrix V are

    (Φ_V)_sσ
    tτ =
      1/3∑_lλ
      V_sl
    σλV_tl
    τλ.

Thus the relation between V entries and p⃗ =
(p_1,p_2,p_3) is

    1/3∑_lλ
      V_ml
    nλV_μ l
    νλ
      =
      δ_mμδ_nν
      ( p_1  + p_2 ω^μ-ν
        + p_3 ω^2(μ-ν)).

Now we proceed to find the entries of the matrix
(F_3⊗ F_3)
  U
  (F_3⊗F_3),

    ⟨e_mf_μ|
      (F_3⊗ F_3)
      U
      (F_3⊗F_3)
      |e_nf_ν⟩
     
     
     
     
     
     
         =
      ∑_lλ
      F_ml F_μλ⟨e_lf_λ|
      U
      (F_3⊗F_3)
      |e_nf_ν⟩
    
     
     
     
     
      
     
     
     
         =
      ∑_lλ
    tτ
      F_ml F_μλ
      U_lλ
    tτF_tnF_τν
       =
      1/3^2∑_lλ
    tτω_3^mlω_3^μλ
      U_lλ
    tτω_3^-tnω_3^-τν
      = V_mμ
    nν.

What follows now is to substitute Eq. (<ref>)—V =
(F_3⊗ F_3)U(F_3⊗F_3)—in the
Choi matrix definition of a
unistochastic channel, Eq. (<ref>),

    D_st
    στ
    =  (Φ_V)_sσ
    tτ
     =
      1/3∑_lλ1/3∑_lλ
      V_sl
    σλV_tl
    τλ   =
      1/31/3^4∑_lλ(
        ∑_d_1d_2
    d_3d_4ω_3^sd_1ω_3^ld_2ω_3^-σ d_3ω_3^-λ d_4
        U_d_1d_2
    d_3d_4)
                                        
           ×(
        ∑_c_1c_2
    c_3c_4ω_3^-c_1tω_3^-c_2lω_3^c_3τω_3^c_4λU_c_1c_2
    c_3c_4)  
       =
                        1/3^5∑_d_1d_2
    d_3d_4∑_c_1c_2
    c_3c_4ω_3^d_1s-d_3σ-c_1t+c_3τ
        U_d_1d_2
    tτU_c_1c_2
    c_3c_4∑_lλω_3^d_2l-d_4λ -c_2l+c_4λ.

Making use of the following identity,
∑_lλω_3^d_2l-d_4λ -c_2l+c_4λ
= 3^2 δ_d_2c_2δ_d_4c_4,
the above formulae
 can be reduced,

    D_st
    στ   =
      
                         1/3^5∑_lλ(
        ∑_d_1d_2
    d_3d_4ω_3^sd_1ω_3^ld_2ω_3^-σ d_3ω_3^-λ d_4
        U_d_1d_2
    d_3d_4)
                                        
           ×(
        ∑_c_1c_2
    c_3c_4ω_3^-c_1tω_3^-c_2lω_3^c_3τω_3^c_4λU_c_1c_2
    c_3c_4)  
       
      
     
     
     
     
     
     
     
                                        
       =
     
        1/3^3∑_d_1d_2
    d_3d_4∑_c_1
    c_3ω_3^d_1s-d_3σ-c_1t+c_3τ
        U_d_1d_2
    d_3d_4U_c_1d_2
    c_3d_4,

Now using Eq. (<ref>) we can rewrite the entries of
the Choi matrix D corresponding to the map Φ_V

    D_st
    στ   =
    1/3^2∑_d_1c_1
    d_3c_3ω_3^d_1s-d_3σ-c_1t+c_3τδ_d_1c_1δ_d_3c_3(p_1+p_2ω_3^c_1-c_3+p_3ω_3^2(c_1-c_3))
       =
      1/3^2∑_x
    yω_3^xs-yσ-xt+yτ(p_1+p_2ω^x-y+p_3ω^2(x-y)).

The last equation corresponds—after reshuffling—to the
desired channel
Φ_V=Φ'_p=  p_1 Φ_𝕀 +
  p_2
Φ_X+
  p_3
  Φ_X^2.
 

In general,  for any pair of triangles,
Δ(Φ_𝕀,  Φ_μ, Φ_μ̅)
and Δ(Φ_𝕀,  Φ_ν, Φ_ν̅)
with  ν≠μ,
there exists a similar relationship
determined by unitary matrices belonging to the Clifford group <cit.>.
The proof of this is similar to the one given above.

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