Homogeneous open quantum walks on the line: criteria for site recurrence and absorption

TL;DR

This paper establishes criteria for site recurrence and absorption in homogeneous open quantum walks on the line, extending classical recurrence concepts to quantum systems with internal degrees of freedom.

Contribution

It provides the first complete criteria for site recurrence in 2-dimensional internal spaces and generalizes results to arbitrary finite dimensions.

Findings

Complete recurrence criteria for 2-dimensional internal spaces

Extension of recurrence criteria to arbitrary finite dimensions

Analysis of absorption phenomena on semi-infinite lines

Abstract

In this work, we study open quantum random walks, as described by S. Attal et al. These objects are given in terms of completely positive maps acting on trace-class operators, leading to one of the simplest open quantum versions of the recurrence problem for classical, discrete-time random walks. This work focuses on obtaining criteria for site recurrence of nearest-neighbor, homogeneous walks on the integer line, with the description presented here making use of recent results of the theory of open walks, most particularly regarding reducibility properties of the operators involved. This allows us to obtain a complete criterion for site recurrence in the case for which the internal degree of freedom of each site (coin space) is of dimension 2. We also present the analogous result for irreducible walks with an internal degree of arbitrary finite dimension and the absorption problem for walks on the semi-infinite line.