We discuss the limit distribution of open quantum walks on the periodic graphs, particularly on the cycles. We show that under certain hypothesis, we can benefit from the theory of the classical Markov chains. Thereby we can show that under certain condition the stationary distribution is unique. For certain models, we show directly the stationary distribution. We also notice that the open quantum walks cannot be always modeled as classical Markov chains by showing that it can break some classical probability rule. By providing with some examples, we show that there can be multiple stationary states for the open quantum walks on the cycles.

中文翻译：

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周期图上开放量子游走的遍历性和极限分布

我们讨论周期图上的开放量子游走的极限分布，特别是在周期上。我们证明，在某些假设下，我们可以从经典马尔可夫链理论中受益。由此可以证明，在一定条件下，平稳分布是唯一的。对于某些模型，我们直接显示平稳分布。我们还注意到，开放量子游走不能总是通过证明它可以打破一些经典概率规则来建模为经典马尔可夫链。通过提供一些例子，我们表明循环上的开放量子行走可以存在多个稳态。