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In this paper, we study the recurrence of homogeneous Open Quantum Walks(HOQWs) induced by finite-dimensional coins $(L,B,R)$. We present threedistinct recurrence criteria, each adapted to different types of coins. Thefirst criterion applies to coins whose auxiliary map has a unique invariantstate, resulting in the first recurrence criterion for Lazy HOQWs. The secondone applies to Lazy HOQWs of dimension 2, where we provide a completecharacterization of the recurrence for this low-dimensional case. Finally, wepresent a general criterion for finite-dimensional coins in the non-lazy case$(B=0)$, which generalizes many of the previously known results. This newcriterion holds for irreducible and reducible HOQWs.

Recurrence Criteria for Reducible Homogeneous Open Quantum Walks on the Line