Open quantum random walks: ergodicity, hitting times, gambler's ruin and potential theory

TL;DR

This paper investigates ergodicity, hitting times, and classical probabilistic concepts in open quantum random walks, introducing spectral criteria and quantum trajectory methods to analyze nonclassical behaviors and potential theory in quantum systems.

Contribution

It introduces a spectral criterion for ergodicity of nonhomogeneous quantum Markov chains and explores quantum analogs of classical probability results using a trajectory approach.

Findings

Spectral criterion for ergodicity in quantum Markov chains

Quantum hitting times and their classical counterparts

Quantum versions of gambler's ruin and potential theory

Abstract

In this work we study certain aspects of Open Quantum Random Walks (OQRWs), a class of quantum channels described by S. Attal et al. \cite{attal}. As a first objective we consider processes which are nonhomogeneous in time, i.e., at each time step, a possibly distinct evolution kernel. Inspired by a spectral technique described by L. Saloff-Coste and J. Z\'u\~niga \cite{saloff}, we define a notion of ergodicity for finite nonhomogeneous quantum Markov chains and describe a criterion for ergodicity of such objects in terms of singular values. As a second objective, and based on a quantum trajectory approach, we study a notion of hitting time for OQRWs and we see that many constructions are variations of well-known classical probability results, with the density matrix degree of freedom on each site giving rise to systems which are seen to be nonclassical. In this way we are able to examine open quantum versions of the gambler's ruin, birth-and-death chain and a basic theorem on potential theory.