Recurrence Criteria for Reducible Homogeneous Open Quantum Walks on the Line

Newton Loebens

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arXiv · 2501.01249

Recurrence Criteria for Reducible Homogeneous Open Quantum Walks on the Line

Newton Loebens

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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 $\mathbb{Z}^2$.

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Newton Loebens. 2026-03-19. Recurrence Criteria for Reducible Homogeneous Open Quantum Walks on the Line. https://arxiv.org/abs/2501.01249

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