Recurrence Criteria for Reducible Homogeneous Open Quantum Walks on the Line

Abstract: In this paper, we study the recurrence of homogeneous Open Quantum Walks (HOQWs) induced by finite-dimensional coins $(L,B,R)$. We present three distinct recurrence criteria, 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 HOQWs. The second one applies to Lazy HOQWs of dimension 2, where we provide a complete characterization of the recurrence for this low-dimensional case. Finally, we present a general criterion for finite-dimensional coins in the non-lazy case $(B=0)$, which generalizes many of the previously known results. This new criterion holds for irreducible and reducible HOQWs.