Passage times, exit times and Dirichlet problems for open quantum walks

TL;DR

This paper investigates passage and exit times in open quantum walks on graphs, extending classical Markov chain results to quantum settings, and explores recurrence, Dirichlet problems, and harmonic measures.

Contribution

It introduces new analysis of passage and exit times for open quantum walks, extending classical Markov chain theory to quantum systems.

Findings

Finite passage time probabilities characterized

Expected passage and visit times computed

Dirichlet problems solved for quantum walks

Abstract

We consider open quantum walks on a graph, and consider the random variables defined as the passage time and number of visits to a given point of the graph. We study in particular the probability that the passage time is finite, the expectation of that passage time, and the expectation of the number of visits, and discuss the notion of recurrence for open quantum walks. We also study exit times and exit probabilities from a finite domain, and use them to solve Dirichlet problems and to determine harmonic measures. We consider in particular the case of irreducible open quantum walks. The results we obtain extend those for classical Markov chains.