On stopping rules for tree-indexed quantum Markov chains

Abstract

In the present paper, we introduce stopping rules and related notions for quantum Markov chains on trees (QMCT). We prove criteria for recurrence, accessibility and irreducibility for QMCT. This work extends to trees the notion of stopping times for quantum Markov chains (QMC) introduced by Accardi and Koroliuk, which plays a key role in the study of many properties of QMC. Moreover, we illustrate the obtained results for a concrete model of XY-Ising type.