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.