Accessible maps in a group of classical or quantum channels

The Schrödinger equation describes the time evolution of an isolated quantum system. However, for a system open to an environment, one needs a broader framework. Gorini, Kossakowski and Sudarshan GKS76 , and independently Lindblad Li76 , derived such an equation of motion governing the evolution of open quantum systems – see CP17 for historical remarks. The most general form of such a GKLS generator, acting on $N$ dimensional systems in Hilbert space $\mathcal{H}_{N}$ and implying Markovian dynamics, is given by

where $H$ is the effective Hamiltonian, $\Lambda$ is a completely positive map, and $*$ denotes the dual in the Heisenberg picture. The first term on the right-hand side is responsible for the unitary part of the evolution, whereas the remaining terms describe the dissipation. Any operation $L$ may be written in the GKLS form (1) if and only if it is (a) Hermiticity preserving, $L(X^{\dagger})=L(X)^{\dagger}$, (b) trace suppressing, $L^{*}({\rm 1l})=0$, and (c) conditionally completely positive, i.e. $L\otimes{\rm 1l}(|\psi_{+}\rangle\langle\psi_{+}|)$ is positive semidefinite in the subspace orthogonal to the maximally entangled state, $|\psi_{+}\rangle=\sum|ii\rangle/\sqrt{N}$ WECC08 . All terms in the above equation can generally be time-dependent. While a time-dependent Lindblad operation, $L_{t}$, governs a broader class of evolutions, the simple structure of a semigroup is predicated by a time-independent case, $L_{t}=L$.

On the other hand, one may adopt yet another approach, namely quantum channels, applicable in a more general situation. Indeed, this method works whenever the physical system and its interacting the environment are initially disjoint RH11 , and even correlated in some cases SL09 ; SL16 ; Betal13 ; B14 ; DSL16 . By a quantum channel, we mean a completely positive and trace-preserving map sending the convex set of $N$-dimensional quantum states into itself. Assuming that the time evolution of the system and environment is governed by the time-dependent unitary operator $U_{t}$, one has $\mathcal{E}_{t}(\rho)={\mathrm{Tr}}_{E}\left[U_{t}(\rho\otimes\sigma)U^{\dagger}_{t}\right]$. The Stinespring dilation S55 then guarantees that $\mathcal{E}_{t}$ is a quantum channel. Accordingly, quantum channels may cover a broader set of evolutions rather than Markovian ones. Note that since $U_{t}$ satisfies the Schrödinger equation, $\mathcal{E}_{t}$ is a one parameter quantum channel continuous on $t$.

The question of Markovianity is then brought up. It was introduced into quantum theory in 2008 WECC08 ; however, the corresponding classical problem has a long history K62 ; Ru62 ; C95 ; D10 . The question, if a given classical map described by a stochastic matrix is embeddable, so a continuous Markov process can generate it remains open, and it was recently generalized for the quantum case KL21 .

Markovianity asks for a given quantum channel $\mathcal{E}$ whether there exists a Lindblad generator of the form (1) whose resulting quantum map is equal to $\mathcal{E}$ at some time $t$. For a general time-dependent Lindblad generator, the answer to this question is positive if and only if $\mathcal{E}$ is an infinitesimal divisible channel D89 ; WC08 . By definition, an infinitesimal divisible channel is one that can be written as a concatenation of quantum maps arbitrarily close to identity D89 .

Here, we are more interested in the case with a time-independent Lindblad generator. Thus the central question for an assumed quantum channel $\mathcal{E}$ is to find whether there exists a dynamical semigroup starting from identity and equal to $\mathcal{E}$ at some time $t=T$, i.e. if $\mathcal{E}={\mathrm{exp}}\{TL\}$ where $L$ is a time-independent Lindblad generator in the form (1). Equivalently, one may ask if there exists a logarithm for a quantum channel $\mathcal{E}$ satisfying properties (a)-(c) mentioned above WECC08 . However, the non-uniqueness of logarithm for an assumed matrix leaves this a highly difficult question. Indeed, it has been proved to be an NP-hard problem from the computational perspective as well CEW12 .

As a consequence, while for qubit channels more facts are revealed DZP18 ; PRZ19 ; CC19 ; JSP20 , less is known about Markovianity of quantum channels in higher dimensions WECC08 ; SCh17 ; SAPZ20 ; Si21 . For instance, in the case of Pauli channels, $\mathcal{E}(\rho)=\sum_{i=0}^{3}p_{i}\sigma_{i}\rho\sigma_{i}^{\dagger}$ where $\sigma_{0}={\rm 1l}_{2}$ and $\sigma_{i}$ for $i\in\{1,2,3\}$ is one of Pauli matrices, the subset of Markovian channels can be divided into two classes DZP18 . The first class contains three measure-zero subsets of the channels with two negative degenerated eigenvalues $\lambda_{-}$ satisfying $\lambda_{-}^{2}\leq\lambda_{+}$ where $\lambda_{+}$ is the other nontrivial positive eigenvalue of the channel. While the second class is a subset of channels with positive eigenvalues such that eigenvalues satisfy $\lambda_{i}\lambda_{j}\leq\lambda_{k}$ for all combinations of different $i,j,k$. The latter set occupies $3/32$ of the whole set of Pauli channels SAPZ20 . Interestingly, in the case of Pauli channels by taking a convex combination of three Lindblad generators of the form $L_{i}(\rho)=\sigma_{i}\rho\sigma_{i}^{\dagger}-\rho$ for $i=1,2,3$ as the generator of the semigroup, one can exhaust the entire volume of Markovian maps PRZ19 . Motivated by this fact, a slightly different and simplified version of the Markovianity problem called accessibility has been introduced SAPZ20 . Given a set of quantum channels $S=\{\mathcal{E}_{i}\}$ with a not necessarily finite number of elements, in the accessibility problem one asks which quantum channels can be generated by a Lindblad generator of the form $L(\rho)=\sum q_{i}\mathcal{E}_{i}(\rho)-\rho$ for some probabilities $q_{i}$.

The accessibility problem was studied for mixed unitary channels given by a convex combination of Weyl unitary operators W27 – a unitary generalization of Pauli matrices to higher dimensions – which form the set $S$ of extremal channels SAPZ20 . Thus accessible channels form a subset $\cal A$ of the polytope of all convex combinations of the elements of the generating set $S$. The set $\cal A$ can be obtained by proper Lindblad generators related to the elements of the set $S$ of Weyl channels itself. Although the set $\cal A$ of accessible channels is a proper subset of Markovian ones by definition, like the Pauli channels, the accessible Weyl channels recover the full measure of Markovian maps. This volume is reduced by an increase in the dimension of the system SAPZ20 . In the general case, the set $\cal A$ of accessible maps occupies a positive volume of the set of channels.

Another relevant fact about Markovian and accessible Pauli channels concerns the rank of these channels. It is known that Pauli channels of Choi rank $3$ are neither accessible nor Markovian DZP18 ; PRZ19 . In higher dimensions, to our best knowledge, the rank problem has been solved only for a mixture of qutrit Weyl channels accessible by a Lindblad semigroup confirming the existence of accessible channels only of rank $1,3,9$ SAPZ20 . The accessibility rank has been an open problem for Weyl channels in other dimensions as well as other quantum channels.

The aim of this work is to study the problem of accessibility for a large class of quantum channels admitting a semigroup structure. Such a set contains Weyl channels as a special subset. We show that the group properties play an essential role in the accessibility problem. Applying such an approach, we can solve the accessibility rank for Weyl channels of any dimension and show which channels with admissible accessible ranks are Markovian. Furthermore, we obtain analytic results for the relative volume of the set $\cal A$ of accessible channels in some cases.

The structure of the paper is as follows. In Section II the necessary notions of Lindblad dynamics and accessible channels are introduced. Subsequent Section III provides a detailed description of the set $\cal A$ of accessible maps generated by a given group $G$ of quantum channels. Some key results of this work are presented in Section IV, in which we show that the set $\cal A$ is non-convex, but it has the star-shape property with respect to the uniform mixture of all elements of the group $G$. The following section discusses some further results on quantum channels, while the case of stochastic matrices and classical maps is described in Sec. VI. The final Section VII concludes the work and presents a list of open problems. Derivation of the relative volume of the set $\cal A$ for cyclic and non-cyclic groups of order $g=4$ is provided in Appendix A.

Consider a quantum channel $\mathcal{E}$ acting on $N$ dimensional states and described by a set $\{K_{a}\}$ of Kraus operators. The superoperator is then represented by a matrix of order $N^{2}$, which reads $\Phi_{\mathcal{E}}=\sum_{a}K_{a}\otimes\overline{K}_{a}$, where overline denotes complex conjugation.

Let $B_{\alpha}$ with $\alpha\in\{0,\dots,N^{2}-1\}$ form a basis of $N^{2}$ matrices of order $N$ such that ${\mathrm{Tr}}(B_{\alpha}B^{\dagger}_{\beta})=\delta_{\alpha\beta}$ for any $\alpha,\beta$. Hence any operator $X$ of dimension $N$ can be expanded as $X=\sum_{\alpha}x_{\alpha}B_{\alpha}$. If a quantum channel $\mathcal{E}$ acts on $X$ as the input, the output reads $\mathcal{E}(X)=\sum(\Phi_{\mathcal{E}}\bm{x})_{\alpha}B_{\alpha}$. Here the $N^{2}$ dimensional matrix $(\Phi_{\mathcal{E}})_{\alpha,\beta}={\mathrm{Tr}}\left(B^{\dagger}_{\alpha}\ \mathcal{E}(B_{\beta})\right)$ introduces the channel effects to the vector $\bm{x}$ of same dimension formed by coefficients $x_{\alpha}$ of the input. Selecting $B_{\alpha}=B_{ij}=|i\rangle\langle j|$ as the basis, the superoperator takes the standard form $\sum K_{a}\otimes\overline{K}_{a}$. Using the Hermitian basis of generalized Gell-Mann matrices, with $B_{0}=\frac{1}{\sqrt{N}}{\rm 1l}_{N}$, the affine parameterization of quantum channels in terms of the generalized Bloch vector is obtained,

Here $M$ denotes a real matrix of dimension $N^{2}-1$ called the distortion matrix, while $\mathbf{t}$ is a real translation vector of the same dimension presenting how the channel $\mathcal{E}$ shifts the identity. The same approach gives the superoperator assigned to a Lindblad generator $L$ given in (1), as $(\mathcal{L}_{L})_{\alpha,\beta}={\mathrm{Tr}}\left(B^{\dagger}_{\alpha}L(B_{\beta})\right)$. Hereafter, quantum channels and Lindblad generators are denoted by their corresponding superoperators, $\Phi$ and $\mathcal{L}$, and we will drop the subscripts $\mathcal{E}$ and $L$ for the sake of brevity.

If $\Phi$ denotes a quantum channel, then $\mathcal{L}_{\Phi}=\Phi-{\rm 1l}$ is a Hermiticity preserving and trace suppressing map. Moreover, complete positivity of $\Phi$ imposes conditional complete positivity on $\mathcal{L}_{\Phi}$ resulting in a valid Lindblad generator assigned to any quantum channel WC08 . It is possible to show this fact through Eq. (1) by choosing $H={\rm 1l}_{N}$ and $\Lambda=\mathcal{E}$. Being trace preserving, $\mathcal{E}$ admits $\Lambda^{\ast}({\rm 1l})=\mathcal{E}^{\ast}({\rm 1l})={\rm 1l}$ which implies $L_{\mathcal{E}}(\cdot)=\mathcal{E}(\cdot)-\frac{1}{2}\{{\rm 1l}_{N},\cdot\}$. However, the inverse is not valid, i.e. not all Lindblad generators can be obtained by subtracting the identity from a quantum channel. The operation $\mathcal{L}_{\Phi}=\Phi-{\rm 1l}$ is a generator of a dynamical semigroup, ${\mathrm{e}}^{t\mathcal{L}_{\Phi}}$, which provides a completely positive and trace-preserving map at each moment, $t\geq 0$, starting from ${\mathrm{e}}^{0\mathcal{L}_{\Phi}}={\rm 1l}$ and tending to a point in the set of quantum channels at $t\to\infty$. The quantum channel $\Phi$, based on which $\mathcal{L}_{\Phi}$ is defined, does not necessarily belong to the trajectory of ${\mathrm{e}}^{t\mathcal{L}_{\Phi}}$.

There is no limitation on choosing the set $S$ of channels; for example, it can be the set of all channels, all unital channels, or a subset of channels determined based on what one can apply in a lab. Here we demand that the set $S$ be the semigroup formed by the convex hull of a group of channels. Hence the set $S$ here consists of mixed unitary channels; therefore, all elements are unital.

Let $G$ be a group consisting of $g=|G|$ quantum channels acting on states in $\mathcal{H}_{N}$, which means that all $\Phi\in G$ are unitary quantum channels, i.e. $\Phi=U\otimes\overline{U}$. As a convention, let us take $\Phi_{0}={\rm 1l}$ as the identity (neutral) element of the group. We define the corresponding set of Lindblad generators as $F_{\mathcal{L}}(G):=\{\mathcal{L}_{\Phi}=\Phi-{\rm 1l}\}$ where $\Phi\in\,G-\{{\rm 1l}\}$. Note that we have excluded by hand the useless generator corresponding to the identity from the set. Now, we can investigate the accessibility problem, i.e. ask which quantum channels obtained by a convex combination of the elements of $F_{\mathcal{L}}(G)$ belong to the set $\cal A$ of accessible channels. As the following theorem shows, the set $\cal A$ is a subset of the convex hull of the group elements $\sc C(G):=\{\Phi=\sum p_{\mu}\Phi_{\mu}\}$ with $\sum_{\mu}p_{\mu}=1$, which is a semigroup itself.

Additionally, as the group, $G_{\Phi}$ is assumed to be finite with unitary elements $\Phi_{\mu}$, a convex polytope with exactly $g$ extreme points in the set of all quantum channels can be constructed from them. Theorem 4 tells us the set of accessible maps also belongs to this polytope. The question concerning the position of the trajectory ${\mathrm{e}}^{t\mathcal{L}}$ in the polytope of the group is highly related to the group structure, its subgroups, and the non-zero interaction times (non-zero $q_{\mu}$). However, the trajectory is in the interior of the polytope once all $q_{\mu}$ are non-zero. Moreover, it tends to the centre of the polytope as $t\to\infty$.

In the case with some vanishing weights, $q_{\mu}$, the situation will be completely different. However, one gets the salient result from Theorem 4 that the trajectory remains in the subset characterized by the smallest subgroup containing all non-vanishing quantum channels appearing in the Lindblad generator. Such a curve ends in the centre of the polytope formed by the smallest subgroup as well. As an example, let us remind readers the group properties imply that every element $\Phi_{\mu}\in G_{\Phi}$ has order equal to the smallest integer and the positive number $h_{\mu}$ such that $\Phi_{\mu}^{h_{\mu}}={\rm 1l}$. The trajectory generated by $\mathcal{L}_{\Phi_{\mu}}=\Phi_{\mu}-{\rm 1l}$ is defined by

Here $M^{\mu}_{r}$ is the set of all non-negative numbers congruent modulo $h_{\mu}$, whose remainder, when divided by $h_{\mu}$, is equal to $r$. Note that

where $\omega_{\mu}={\mathrm{exp}}[\frac{2\pi i}{h_{\mu}}]$. Provided that $t_{\mu}>0$, these coefficients are all positive numbers adding up to $1$, i.e. they provide a probability vector of length $h_{\mu}$ and none of its elements vanishes. Thus we can write

which means at each moment of time we get a mixture of all $h_{\mu}$ different powers of $\Phi_{\mu}$. This implies ${\mathrm{e}}^{t_{\mu}\mathcal{L}_{\Phi_{\mu}}}$ (with no summation on $\mu$), for any $t_{\mu}>0$, is located in the interior of the polytope formed by taking the convex hull of $h_{\mu}$ extreme points each of which is assigned to different powers of $\Phi_{\mu}$. Moreover, at $t_{\mu}\to\infty$ the trajectory tends to the center of the polytope as discussed in Proposition 6, i.e. the uniform mixture of all different powers of $\Phi_{\mu}$. To see it explicitly, we should note that Eq. (10) can also be written as

This equation now shows that at large time scale $\lim_{t_{\mu}\to\infty}p_{r}(t_{\mu})=\frac{1}{h_{\mu}}$ holds for any $r$.

We emphasise that Eq. (10) is independent of the dimension and the individual channels that form the group. It only depends on the order of the element under investigation. Moreover, note that the set $H_{\Phi_{\mu}}:=\{\Phi_{\mu}^{r}\}$ for $r\in\{0,\dots,h_{\mu}-1\}$ is actually a cyclic subgroup of $G_{\Phi}$ of order $h_{\mu}$. For an arbitrary convex combination of Lindblad generators associated with different elements of $H_{\Phi_{\mu}},$ the resultant trajectory cannot leave the polytope formed by the elements of $H_{\Phi_{\mu}}$.

In other words, if all quantum channels appearing in ${\mathrm{exp}}[t(\sum q_{\mu}\Phi^{\prime}_{\mu}-{\rm 1l})]$ belong to a cyclic subgroup, then at each moment of time, the trajectory can be convexly expanded in terms of all members of the smallest cyclic subgroup containing all $\Phi^{\prime}_{\mu}$ in this summation. Due to commutativity of the elements of a cyclic group, we get ${\mathrm{exp}}{(t\mathcal{L})}={\mathrm{exp}}{\left(\sum t_{\mu}\mathcal{L}_{\Phi^{\prime}_{\mu}}\right)}=\prod{\mathrm{exp}}({t_{\mu}\mathcal{L}_{\Phi^{\prime}_{\mu}}})$. Each term in the last formula can be expanded based on Eq. (11) in which probabilities are given by Eq. (10). To see this fact explicitly, let $\Phi_{\mu}$ be the generator of $H_{\Phi_{\mu}}$, i.e. $H_{\Phi_{\mu}}=\{\Phi_{\mu}^{i}\}_{i=0}^{h_{\mu}-1}=\{\Phi^{\prime}_{\mu}\}_{\mu=0}^{h_{\mu}-1}$, then

where $h_{\mu}$, and $h_{\mu_{i}}$ are the order of $\Phi_{\mu}$, and $\Phi_{\mu}^{i}$, respectively. In the last equation $l=(ij\ {\rm mod}\ h_{\mu})$ and

Hence, we can find the time-dependent probabilities in the convex combination. However, this is only one of the possibilities since, in a general case, the polytope formed by the set elements $G_{\Phi}$ can differ form the simplex.

Consider now another quantum channel $\Phi^{\prime\prime}_{\alpha}$ from the set $G_{\Phi}$ such that there is no cyclic subgroup of $G_{\Phi}$ including both $\Phi_{\mu}$ and $\Phi^{\prime\prime}_{\alpha}$. To emphasize this fact, we will denote $\Phi^{\prime\prime}_{\alpha}$ by $\Theta_{\alpha}$. If we want to add $\mathcal{L}_{\Theta_{\alpha}}$ to the set discussed in Eq. (13) two different cases are possible. First, if we assume $[\Phi_{\mu},\Theta_{\alpha}]=[\mathcal{L}_{\Phi_{\mu}},\mathcal{L}_{\Theta_{\alpha}}]=0$, then

where $h_{\alpha}$ and $h_{\mu}$ are the order of $\Theta_{\alpha}$ and $\Phi_{\mu}$, and $p_{k}(t_{\alpha})$ and $\mathzapf{P}_{l}(t_{\mu})$ are introduced in Eqs. (10) and (14), respectively. Note that commutativity of $\Theta_{\alpha}$ and $\Phi_{\mu}$ imposes compatibility on any power of them, i.e. all elements of the cyclic subgroups $H_{\Theta_{\alpha}}$ and $H_{\Phi_{\mu}}$ commute. Moreover, as we assumed that there is no cyclic subgroup including both $\Theta_{\alpha}$ and $\Phi_{\mu}$, thus $H_{\Theta_{\alpha}}$ and $H_{\Phi_{\mu}}$ have no element in common but the identity. This implies that the product of $H_{\Theta_{\alpha}}$ and $H_{\Phi_{\mu}}$ is a subgroup of $G_{\Phi}$ of order $h_{\alpha}h_{\mu}$, i.e. $\forall k,l:\ \ \Theta_{\alpha}^{k}\Phi_{\mu}^{l}$ belongs to a subgroup of $G_{\Phi}$ formed by multiplication of its cyclic and commutative subgroups. Applying the above arguments, we can generalize these results as follows.

This Corollary can be proved through the fundamental theorem of abelian groups and the above discussion.

The condition on commutativity in Corollary 7 may be relaxed in a general scenario, i.e. $[\Theta,\Phi]\neq 0$. In that case, $H_{\Theta}\times H_{\Phi}$ is not necessarily a subgroup. Therefore, we need to add some other quantum channels from $G_{\Phi}$ to the set $H_{\Theta}\times H_{\Phi}$ so it satisfies closure. The smallest subgroup $H$ of $G_{\Phi}$ that includes all members of the set $H_{\Theta}\times H_{\Phi}$ expands convexly the dynamical semigroups generated by Lindblad generators of our interest. This is a direct consequence of the closure property, as the group structure plays a crucial role for Theorem 4.

In this section we will apply the aforementioned theorems and relations for certain constructive examples. For some groups associated with quantum channels acting on $N$-dimensional systems and represented by matrices of size $N^{2}$ we demonstrate how to find the quantum channels, their polytopes, and the subset of accessible maps inside the polytope.

Before proceeding with the examples let us mention that if $G_{\Phi}=\{\Phi_{\mu}=U_{\mu}\otimes\overline{U}_{\mu}\}_{\mu=0}^{g-1}$ is a group of $g$ unitary channels, then the set of the corresponding Kraus operators denoted by $\tilde{G}_{U}=\{U_{\mu}\}_{\mu=0}^{g-1}$ is a group up to a phase, which is also called a projective representation of group $G$. Explicitly, in the set $\tilde{G}_{U}$: (i) there are no two elements which are equal up to a phase, i.e. if $U\in\tilde{G}_{U}$, then ${\mathrm{e}}^{i\phi}U\notin\tilde{G}_{U}$, (ii) the set is closed up to a phase, i.e. if $U_{1},U_{2}\in\tilde{G}_{U}$, then ${\mathrm{e}}^{i\phi}U_{1}U_{2}\in\tilde{G}_{U}$ for some $\phi$, (iii) and it contains an inverse up to a phase for each element, i.e. for any $U_{1}\in\tilde{G}_{U}$ there exists an element $U_{2}\in\tilde{G}_{U}$ and a phase $\phi$ such that $U_{1}U_{2}={\mathrm{e}}^{i\phi}{\rm 1l}$. Clearly, the last two conditions imply that (iii’) the set possesses a neutral element up to a phase. Moreover, the set $G_{\Phi}$ is abelian if and only if $\tilde{G}_{U}$ is abelian up to a phase, i.e. $\forall U_{1},U_{2}\in\tilde{G}_{U}$ one has $U_{1}U_{2}={\mathrm{e}}^{i\phi}U_{2}U_{1}$ for some phase $\phi$.