Eigenvalues of quantum walk induced by recurrence properties of the underlying birth and death process: application to computation of an edge state

Abstract

In this paper, we consider an extended coined Szegedy model and discuss the existence of the point spectrum of induced quantum walks in terms of recurrence properties of the underlying birth and death process. We obtain that if the underlying random walk is not null recurrent, then the point spectrum exists in the induced quantum walks. As an application, we provide a simple computational way of the dispersion relation of the edge state part for the topological phase model driven by quantum walk using the recurrence properties of underlying birth and death process.

Appendix A: computation of \(\ker (C_j+ie^{i\phi })\)

The $(-ie^{i\phi })$-eigenvector of $C_j$ is expressed by

$$\begin{aligned} \ker (C_j+ie^{i\phi })={\mathbb {C}}\begin{bmatrix} \rho _j \\ \eta _j-ie^{i\phi } \end{bmatrix}. \end{aligned}$$

Recall that $\cos \phi =\kappa =\mathrm {Im}(\eta _j)$ and $e^{i\phi }=\kappa +i\sqrt{1-\kappa ^2}$. Then, each element of the eigenvector can be deformed by

$$\begin{aligned} \rho _j=\sqrt{ (1-\kappa ^2)-\mathrm {Re}(\eta _j)^2 }=A_+A_-, \; \eta _j-ie^{i\phi } =\sqrt{ 1-\kappa ^2 }+\mathrm {Re}(\eta _j) =A_+^2, \end{aligned}$$

respectively. Here we put

$$\begin{aligned} A_\pm :=\sqrt{\sqrt{1-\kappa ^2}\pm \mathrm {Re}(\eta _j)}. \end{aligned}$$

Therefore the normalized eigenvector is expressed by

$$\begin{aligned} \frac{1}{\sqrt{A_+^2+A_-^2}}\; {}^T[A_-\; A_+]={}^T[\sqrt{p_j}\; \sqrt{q_j}], \end{aligned}$$

where

$$\begin{aligned} p_j=\frac{1}{2}\left( 1-\frac{\mathrm {Re}(\eta _j)}{\sqrt{1-\kappa ^2}} \right) ,\;q_j=\frac{1}{2}\left( 1+\frac{\mathrm {Re}(\eta _j)}{\sqrt{1-\kappa ^2}} \right) . \end{aligned}$$