Site recurrence of open and unitary quantum walks on the line

Abstract

We study the problem of site recurrence of discrete time nearest neighbor open quantum random walks (OQWs) on the integer line, proving basic properties and some of its relations with the corresponding problem for unitary (coined) quantum walks (UQWs). For both kinds of walks our discussion concerns two notions of recurrence, one given by a monitoring procedure, another in terms of Pólya numbers, and we study their similarities and differences. In particular, by considering UQWs and OQWs induced by the same pair of matrices, we discuss the fact that recurrence of these walks are related by an additive interference term in a simple way. Based on a previous result of positive recurrence we describe an open quantum version of Kac's lemma for the expected return time to a site.

References in corpus (12)

• Quantum walks: a comprehensive review
• Open Quantum Random Walks
• Recurrence for discrete time unitary evolutions
• Recurrence and Pólya number of quantum walks
• Quantum recurrence of a subspace and operator-valued Schur functions
• Limit theorems for open quantum random walks
• Homogeneous Open Quantum Random Walks on a lattice
• On a class of quantum channels, open random walks and recurrence
• Generalized Kac's Lemma for Recurrence Time in Iterated Open Quantum Systems
• Central limit theorem for reducible and irreducible open quantum walks
• Open quantum random walks: ergodicity, hitting times, gambler's ruin and potential theory
• Open quantum random walks with decoherence on coins with $n$ degrees of freedom

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• Open quantum random walks on the half-line: the Karlin-McGregor formula, path counting and Foster's Theorem
• Quantum Markov chains on the line: matrix orthogonal polynomials, spectral measures and their statistics
• Mean hitting times of quantum Markov chains in terms of generalized inverses
• Monitored Recurrence of a One-parameter Family of Three-state Quantum Walks