Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules

Abstract

In this work we study the recurrence problem for quantum Markov chains, which are quantum versions of classical Markov chains introduced by S. Gudder and described in terms of completely positive maps. A notion of monitored recurrence for quantum Markov chains is examined in association with Schur functions, which codify information on the first return to some given state or subspace. Such objects possess important factorization and decomposition properties which allow us to obtain probabilistic results based solely on those parts of the graph where the dynamics takes place, the so-called splitting rules. These rules also yield an alternative to the folding trick to transform a doubly infinite system into a semi-infinite one which doubles the number of internal degrees of freedom. The generalization of Schur functions -so-called FR-functions-to the general context of closed operators in Banach spaces is the key for the present applications to open quantum systems. An important class of examples included in this setting are the open quantum random walks, as described by S. Attal et al., but we will state results in terms of general completely positive trace preserving maps. We also take the opportunity to discuss basic results on recurrence of finite dimensional iterated quantum channels and quantum versions of Kac's Lemma, in close association with recent results on the subject.walks on graphs, see [32] for a recent survey on the subject. Regarding hitting probabilities and recurrence in the setting of OQWs, see [5,14,26,30].This work provides a detailed study of the consequences of the splitting properties for FR-functions regarding recurrence in quantum Markov chains. Two kinds of FR-function splittings, related to factorizations and decompositions into sums of the underlying operator, yield two types of splitting rules for quantum Markov chains and, thus, for the particular case of classical Markov chains. As a consequence of these splitting rules we will see that, similarly to UQWs, the return probability is also invariant under certain local perturbations.Other novel contributions of this paper deal with generalizations of known results on recurrence in classical Markov chains and UQWs. Different quantum generalizations of Kac's lemma for classical Markov chains are commented in Subsect. 2.1. Besides, Theorem 2.8 constitutes the quantum version of a well known result for classical Markov chains, namely, that finiteness and irreducibility imply the positive recurrence of every state, i.e. every state returns to itself with probability one and in a finite expected return time. A similar result holds for finite-dimensional unitary evolutions, which in addition present the striking particularity of exhibiting integer valued expected times for the return to any state [19]. These results has been generalized to the return to a subspace in [9], while [33] proves that they hold for the return to a state in iterated quantum channels whenever they are unital. We extend this property of unital quantum channels to the r...

Homogeneous open quantum walks on the line: criteria for site recurrence and absorption

“…We can also study recurrence in terms of generating functions as follows (a formal discussion on the analytic behavior of such series can be seen in [11]). If we define…”

“…We note that site recurrence refers to the probability that, given an initial density concentrated on site i, say, η ⊗ i⟩⟨i , the walk will eventually return to site i, landing on such site with any density matrix. This is in contrast to the state recurrence problem, which consists of determining the probability that the walk will return to i with the same initial density η, see [11] for more on this.…”

“…The authors of [1] have formally defined OQWs and also stated a question on recurrence, which can be phrased as: given an OQW acting on some graph, how can we determine if a site is recurrent by looking at the transition matrices? At this point, one immediately needs to provide a proper definition of quantum recurrence and note that, in principle, more than one is available, see [4,11] for more on this. Generally speaking, recurrence problems are topics of interest in both open and closed (unitary) quantum dynamics and we seek a basic understanding of this kind of problem having in mind applications to quantum information and computation.…”

“…In addition, the authors of [4] provide a systematic study of first passage times of general OQWs, discussing in particular the irreducible case, giving further insight into the recurrence problem. More recently, in [11], a discussion on quantum Markov chains (QMCs) and associated subspace recurrence problems is presented, methods for determining site recurrence in terms of Schur functions are developed and, as OQWs are particular cases of QMCs, one obtains an useful understanding of the problem at hand from an analytic perspective.…”

Induced on-demand revival in coined quantum walks on infinite $d$-dimensional lattices

“…Moreover, it can be extended to the study of recurrences in QWs [29][30][31], even in more general scenarios that include monitored walks [32,33]. In fact, the latter approaches allow the study of recurrences in classical random walks, standard QWs, open QWs, and quantum Markov chains in general within the same mathematical framework [34,35]. Moreover, concerning recurrence in QWs, the first detection time or the time it takes for a quantum walker to return to its origin is studied in Ref.…”