Site recurrence of open and unitary quantum walks on the line

TL;DR

This paper investigates the recurrence properties of open and unitary quantum walks on the line, comparing their behaviors and establishing a quantum analogue of Kac's lemma for expected return times.

Contribution

It introduces a detailed comparison of recurrence notions for open and unitary quantum walks and derives an open quantum Kac's lemma relating to return times.

Findings

Recurrence notions for open and unitary quantum walks are related by an interference term.

A quantum version of Kac's lemma for expected return time is established.

Recurrence properties depend on the underlying matrices inducing the walks.

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 P\'olya 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.