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

Abstract

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.

Appendix: Generating matrices under Hypothesis (H)

In this appendix, we give the explicit expression of the rank 1 matrices $B^*P_iB$ and $C^*P_iC$ for $i=1,2$ under the Hypothesis (H).

Let us denote the unitary matrices $U_B$ and $U_C$ appeared in the singular value decomposition of B and C in (3.5) by

$$\begin{aligned} U_B=\left( \begin{matrix} u_{11}&{}u_{12}\\ u_{21}&{}u_{22}\end{matrix}\right) , \quad U_C=\left( \begin{matrix} v_{11}&{}v_{12}\\ v_{21}&{}v_{22}\end{matrix}\right) . \end{aligned}$$

Lemma A.1

Let B and C have the singular value decomposition as in (3.5) and satisfy the hypothesis (H). Then the rank 1 operators $B^*P_iB$ and $C^*P_iC$, $i=1,2$, have the following forms:

(i) Case 1: $0<\lambda <1$, $0<\mu < 1$.

$$\begin{aligned} B^*P_1B=\lambda P_1,\,\,B^*P_2B=\mu P_2\text { or }B^*P_1B=\mu P_2,\,\,B^*P_2B=\lambda P_1, \end{aligned}$$

and similarly,

$$\begin{aligned}{} & {} C^*P_1C=(1-\lambda )P_1,\,\,C^*P_2C=(1-\mu )P_2\text { or}\\{} & {} C^*P_1C=(1-\mu )P_2,\,\,C^*P_2C=(1-\lambda )P_1. \end{aligned}$$

(ii) Case 2: $0<\lambda <1$, $\mu =0$.

$$\begin{aligned}{} & {} B^*P_1B=\lambda |u_{11}|^2P_1,\,\,B^*P_2B=\lambda |u_{21}|^2P_1,\text { and}\\{} & {} C^*P_1C= P_2,\,\,C^*P_2C=(1-\lambda ) P_1 \text { or } C^*P_1C= (1-\lambda )P_1,\,\,C^*P_2C= P_2. \end{aligned}$$

(iii) Case 3: $0<\lambda <1$, $\mu =1$.

$$\begin{aligned}{} & {} B^*P_1B= P_2,\,\,B^*P_2B=\lambda P_1\text { or } B^*P_1B= \lambda P_1,\,\,B^*P_2B= P_2,\text { and}\\{} & {} C^*P_1C= (1-\lambda )|v_{11}|^2P_1,\,\,C^*P_2C=(1-\lambda )|v_{21}|^2 P_1. \end{aligned}$$

(iv) Case 4: $\lambda =0$, $0<\mu <1$.

$$\begin{aligned}{} & {} B^*P_1B=\mu |u_{12}|^2P_2,\,\,B^*P_2B=\mu |u_{22}|^2P_2,\text { and}\\{} & {} C^*P_1C= (1-\mu )P_2,\,\,C^*P_2C= P_1 \text { or } C^*P_1C=P_1,\,\,C^*P_2C=(1-\mu ) P_2. \end{aligned}$$

(v) Case 5: $\lambda =1$, $0<\mu <1$.

$$\begin{aligned}{} & {} B^*P_1B= \mu P_2,\,\,B^*P_2B= P_1\text { or } B^*P_1B= P_1,\,\,B^*P_2B= \mu P_2,\text { and}\\{} & {} C^*P_1C= (1-\mu )|v_{12}|^2P_2,\,\,C^*P_2C=(1-\mu )|v_{22}|^2 P_2. \end{aligned}$$

(vi) Case 6: $\lambda =0$, $\mu =1$.

$$\begin{aligned} B^*P_1B=|u_{12}|^2 P_2,\,\,B^*P_2B=|u_{22}|^2 P_2\text { and } C^*P_1C= |v_{11}|^2P_1,\,\,C^*P_2C= |v_{21}|^2 P_1. \end{aligned}$$

(vii) Case 7: $\lambda =1$, $\mu =0$.

$$\begin{aligned} B^*P_1B=|u_{11}|^2 P_1,\,\,B^*P_2B=|u_{21}|^2 P_1\text { and } C^*P_1C= |v_{12}|^2P_2,\,\,C^*P_2C= |v_{22}|^2 P_2. \end{aligned}$$

(viii) Case 8: $\lambda =0$, $\mu =0$ or $\lambda =1$, $\mu =1$.

These cases result in $B=0$ or $C=0$, respectively, and are out of consideration in the model.

Proof

The proof follows easily by using the Hypothesis (H) and the fact that the operators $B^*P_iB$ and $C^*P_iC$ are of rank 1. $\square$