Ergodicity and limit distribution of open quantum walks on the periodic graphs

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.