Recurrence criteria for reducible homogeneous open quantum walks on the line and on the grid

Abstract In this paper, we study the recurrence of Open Quantum Walks induced by finite-dimensional coins on the line (Z Z) and on the grid (Z² 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 Z 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 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 several of the previously known results for OQWs on Z Z. Also, we present a general recurrence criterion for OQWs on Z² Z 2 via the open quantum jump chain, obtained from a recurrence criterion for CTOQWs on Z² Z 2.