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In this work we study the recurrence problem for quantum Markov chains, whichare quantum versions of classical Markov chains introduced by S. Gudder anddescribed in terms of completely positive maps. A notion of monitoredrecurrence for quantum Markov chains is examined in association with Schurfunctions, which codify information on the first return to some given state orsubspace. Such objects possess important factorization and decompositionproperties which allow us to obtain probabilistic results based solely on thoseparts of the graph where the dynamics takes place, the so-called splittingrules. These rules also yield an alternative to the folding trick to transforma doubly infinite system into a semi-infinite one which doubles the number ofinternal degrees of freedom. The generalization of Schur functions --so-calledFR-functions-- to the general context of closed operators in Banach spaces isthe key for the present applications to open quantum systems. An importantclass 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 generalcompletely positive trace preserving maps. We also take the opportunity todiscuss basic results on recurrence of finite dimensional iterated quantumchannels and quantum versions of Kac's Lemma, in close association with recentresults on the subject.

Quantum Markov chains: recurrence, Schur functions and splitting rules