Recurrence Criteria for Reducible Homogeneous Open Quantum Walks on the Line

TL;DR

This paper establishes multiple recurrence criteria for open quantum walks on the line and grid, covering various coin types and dimensions, advancing understanding of quantum walk recurrence behavior.

Contribution

It introduces new recurrence criteria for open quantum walks on the line and grid, including the first for lazy OQWs and a general criterion for finite-dimensional coins.

Findings

Three recurrence criteria for OQWs on 

Complete characterization for 2D lazy OQWs

General recurrence criterion for OQWs on ^2

Abstract

In this paper, we study the recurrence of Open Quantum Walks induced by finite-dimensional coins on the line ($\mathbb{Z}$) and on the grid ($\mathbb{Z}^2$). Two versions are considered: discrete-time open quantum walks (OQW) and continuous-time open quantum walks (CTOQW). We present three distinct recurrence criteria for OQWs on $\mathbb{Z}$, each adapted to different types of coins. The first criterion applies to coins whose auxiliary map has a unique invariant state, resulting in the first recurrence criterion for Lazy OQWs. The second one applies to Lazy OQWs of dimension 2, where we provide a complete characterization of the recurrence for this low-dimensional case. The third one presents a general criterion for finite-dimensional coins in the non-lazy case, which generalizes many of the previously known results for OQWs on $\mathbb{Z}$. Also, we present a general recurrence criterion for OQWs on $\mathbb{Z}^2$ via the open quantum jump chain, obtained from a recurrence criterion for CTOQWs on $\mathbb{Z}^2$.