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

TL;DR

This paper explores open quantum random walks, analyzing hitting times and classical probability analogs, revealing nonclassical features and applying these concepts to quantum versions of gambler's ruin and potential theory.

Contribution

It introduces a quantum trajectory approach to analyze hitting times in open quantum walks, extending classical results to quantum systems with nonclassical properties.

Findings

Hitting time concepts extend to quantum systems with nonclassical features.

Quantum versions of gambler's ruin and birth-and-death chains are developed.

Probability expressions for walks induced by normal commuting contractions are derived.

Abstract

We consider a model of open quantum random walk and together with a quantum trajectory approach we are able to examine a notion of hitting time. We see that many constructions, such as minimal solutions to hitting time problems, are variations of well-known classical probability results, but the density matrix degree of freedom on each site gives rise to systems which are seen to be nonclassical. As a more specific application we study the collection of walks induced by normal commuting contractions, for which the corresponding probability expressions are obtained. We examine quantum versions of the gambler's ruin, birth-and-death chain and a basic theorem on potential theory.