Stationary measure for three-state quantum walk

TL;DR

This paper derives the stationary measure for a one-dimensional three-state quantum walk using a new method based on the eigenvalue problem, providing explicit solutions and examples.

Contribution

It introduces a novel recipe to obtain transfer matrices and solves the eigenvalue problem to find the stationary measure in general conditions.

Findings

Derived the stationary measure for three-state quantum walk

Provided explicit solutions for specific initial states

Presented examples illustrating the stationary measure

Abstract

We focus on the three-state quantum walk(QW) in one dimension. In this paper, we give the stationary measure in general condition, originated from the eigenvalue problem. Firstly, we get the transfer matrices by our new recipe, and solve the eigenvalue problem. Then we obtain the general form of the stationary measure for concrete initial state and eigenvalue. We also show some specific examples of the stationary measure for the three-state QW. One of the interesting and crucial future problems is to make clear the whole picture of the set of stationary measures.

Equations

$$\mu(x)=\left\{\begin{array}[]{ll}c&(\in l^{\infty}\!-\!space\;on\;\mathbb{Z}),\\ \\ \left(\dfrac{q}{p}\right)^{x}&(\notin l^{\infty}\!-\!space\;on\;\mathbb{Z}),\\ \end{array}\right.$$

$${\Psi }_n(x)=\left[\begin{array}{c}{\Psi }_n^L(x)\\ {\Psi }_n^O(x)\\ {\Psi }_n^R(x)\end{array}\right](x\in \mathbb{Z}).$$

$$U_x=\left[\begin{array}{ccc}{a}_x& b_x& c_x\\ d_x& e_x& f_x\\ g_x& h_x& i_x\end{array}\right],$$

$${\Psi }_{n+1}(x)=P_{x+1}{\Psi }_n(x+1)+R_x{\Psi }_n(x)+Q_{x-1}{\Psi }_n(x-1)(x\in \mathbb{Z}),$$

$$P_x=\left[\begin{array}{ccc}{a}_x& b_x& c_x\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],R_x=\left[\begin{array}{ccc}0& 0& 0\\ d_x& e_x& f_x\\ 0& 0& 0\end{array}\right],Q_x=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ g_x& h_x& i_x\end{array}\right],$$

$${\Psi }_n=\left[\cdots ,\left[\begin{array}{c}{\Psi }_n^L(-1)\\ {\Psi }_n^O(-1)\\ {\Psi }_n^R(-1)\end{array}\right],\left[\begin{array}{c}{\Psi }_n^L(0)\\ {\Psi }_n^O(0)\\ {\Psi }_n^R(0)\end{array}\right],\left[\begin{array}{c}{\Psi }_n^L(1)\\ {\Psi }_n^O(1)\\ {\Psi }_n^R(1)\end{array}\right],\cdots \right]{}^T\in ({\mathbb{C}}^3{)}^{\mathbb{Z}},$$

$$U^{(s)}=\left[\begin{array}{ccccccc}\ddots & ⋮& ⋮& ⋮& ⋮& ⋮& \text{iddots}\\ \cdots & R_{-2}& P_{-1}& O& O& O& \cdots \\ \cdots & Q_{-2}& R_{-1}& P_0& O& O& \cdots \\ \cdots & O& Q_{-1}& R_0& P_1& O& \cdots \\ \cdots & O& O& Q_0& R_1& P_2& \cdots \\ \cdots & O& O& O& Q_1& R_2& \cdots \\ \text{iddots}& ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right]withO=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],$$

$$\varphi (\Psi )(x)=|{\Psi }^L(x){|}^2+|{\Psi }^O(x){|}^2+|{\Psi }^R(x){|}^2(x\in \mathbb{Z}).$$

$$\mu (x)=\varphi (\Psi )(x).$$

$${\Sigma }_s=\{\varphi ({\Psi }_0)\in {\mathbb{R}}_{+}^{\mathbb{Z}}:\exists {\Psi }_0,s.t.\varphi ((U^{(s)}{)}^n{\Psi }_0)=\varphi ({\Psi }_0)\forall n\ge 0\},$$

$$\displaystyle\Psi(x)=\left\{\begin{array}[]{ll}\prod^{x}_{y=1}T^{(+)}_{y}\Psi(0)&(x\geq 1),\\ \ \Psi(0)&(x=0),\\ \ \prod^{x}_{y=-1}T^{(-)}_{y}\Psi(0)&(x\leq-1),\end{array}\right.$$

$$T_y^{(+)}$$

$$t_{11}^{(+)}$$

$$t_{13}^{(+)}$$

$$t_{22}^{(+)}$$

$$t_{31}^{(+)}$$

$$t_{11}^{(-)}$$

$$t_{21}^{(-)}$$

$$t_{23}^{(-)}$$

$$t_{32}^{(-)}$$

$$U_x=\frac{1}{3}\left[\begin{array}{ccc}-1& 2& 2\\ 2& -1& 2\\ 2& 2& -1\end{array}\right].$$

$$\displaystyle\Psi(x)=\left\{\begin{array}[]{ll}\prod^{x}_{y=1}T^{(+)}_{y}\Psi(0)&(x\geq 1),\\ \ \Psi(0)&(x=0),\\ \ \prod^{x}_{y=-1}T^{(-)}_{y}\Psi(0)&(x\leq-1),\end{array}\right.$$

$$T_y^{(+)}=\left[\begin{array}{ccc}1& 0& 0\\ & & \\ Konno-\frac{1}{3}& \frac{2}{3}& -\frac{1}{3}\\ & & \\ -\frac{2}{3}& -\frac{2}{3}& \frac{1}{3}\end{array}\right],T_y^{(-)}=\left[\begin{array}{ccc}\frac{1}{3}& -\frac{2}{3}& -\frac{2}{3}\\ & & \\ -\frac{1}{3}& \frac{2}{3}& -\frac{1}{3}\\ & & \\ 0& 0& 1\end{array}\right].$$

$$\Psi (x)=[\alpha ,0,-\alpha ]{}^T(x\in \mathbb{Z}),$$

$$\mu (x)=2|\alpha {|}^2(x\in \mathbb{Z}).$$

$$\displaystyle U_{x}=\left\{\begin{array}[]{ll}\dfrac{1}{3}\begin{bmatrix}-1&2&2\\ 2&-1&2\\ 2&2&-1\end{bmatrix}\quad(x\neq 0),\\ \\ \dfrac{\rho}{3}\begin{bmatrix}-1&2&2\\ 2&-1&2\\ 2&2&-1\end{bmatrix}\quad(x=0),\end{array}\right.$$

$$\displaystyle\Psi(x)=\left\{\begin{array}[]{ll}\prod^{x}_{y=1}T^{(+)}_{y}\Psi(0)&(x\geq 1),\\ \ \Psi(0)&(x=0),\\ \ \prod^{x}_{y=-1}T^{(-)}_{y}\Psi(0)&(x\leq-1),\end{array}\right.$$

$$T_1^{(+)}=\left[\begin{array}{ccc}1& 0& 0\\ & & \\ -\frac{5}{3}& -\frac{2}{3}& \frac{1}{3}\\ & & \\ \frac{2}{3}& \frac{2}{3}& -\frac{1}{3}\end{array}\right],T_{-1}^{(-)}=\left[\begin{array}{ccc}-\frac{1}{3}& \frac{2}{3}& \frac{2}{3}\\ & & \\ \frac{1}{3}& -\frac{2}{3}& -\frac{5}{3}\\ & & \\ 0& 0& 1\end{array}\right],$$

$$T_y^{(+)}=\left[\begin{array}{ccc}1& 0& 0\\ & & \\ -\frac{1}{3}& \frac{2}{3}& -\frac{1}{3}\\ & & \\ -\frac{2}{3}& -\frac{2}{3}& \frac{1}{3}\end{array}\right],T_y^{(-)}=\left[\begin{array}{ccc}\frac{1}{3}& -\frac{2}{3}& -\frac{2}{3}\\ & & \\ -\frac{1}{3}& \frac{2}{3}& -\frac{1}{3}\\ & & \\ 0& 0& 1\end{array}\right](|y|\ge 2).$$

$$\displaystyle\Psi(x)=\left\{\begin{array}[]{ll}\left[\alpha,0,-\alpha\right]{}^{T}\!&(|x|\neq 1),\\ \\ \left[\alpha,-2\alpha,\alpha\right]{}^{T}\!&(x=1),\\ \\ \left[-\alpha,2\alpha,-\alpha\right]{}^{T}\!&(x=-1).\end{array}\right.$$

$$\displaystyle\mu(x)=2|\alpha|^{2}\left\{\begin{array}[]{ll}1&(|x|\neq 1),\\ \\ 3&(|x|=1).\\ \end{array}\right.$$

$$U_x=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& \omega & {\omega }^2\\ 1& {\omega }^2& \omega \end{array}\right]$$

$$\displaystyle\Psi(x)=\left\{\begin{array}[]{ll}\prod^{x}_{y=1}T^{(+)}_{y}\Psi(0)&(x\geq 1),\\ \ \Psi(0)&(x=0),\\ \ \prod^{x}_{y=-1}T^{(-)}_{y}\Psi(0)&(x\leq-1),\end{array}\right.$$

$$T_y^{(+)}=\left[\begin{array}{ccc}\omega & 0& 0\\ & & \\ \frac{-\omega -5}{3\omega }& \frac{1-\omega }{3}& \frac{1-\omega }{3\omega }\\ & & \\ \frac{1}{\omega (1-\omega )}& \frac{\omega }{1-\omega }& \frac{1}{1-\omega }\end{array}\right],T_y^{(-)}=\left[\begin{array}{ccc}\frac{1}{\omega (1-\omega )}& \frac{1}{\omega (1-\omega )}& \frac{1}{\omega (1-\omega )}\\ & & \\ \frac{-1-2\omega }{3(1+\omega )}& \frac{-1-2\omega }{3(1+\omega )}& \frac{-4-5\omega }{3(1+\omega )}\\ & & \\ 0& 0& 1\end{array}\right](|y|\ge 1).$$

Full text

1 Introduction

Owing to their specific properties, quantum walks(QWs) have attracted much attention in various fields, such as quantum algorithms [3, 28], and topological insulators [17]. For the rich application of QWs, it is important to further study both analytically and numerically. Indeed, over the past decade many researchers have investigated the asymptotic behaviors of QWs from various viewpoints [5, 14, 18, 22, 23, 30].

So far, localization and the ballistic spreading have been known as the characteristic properties of QWs [2, 12, 13, 14]. They are described mathematically by two kinds of limit theorems, which are composed of measures, i.e., the time-averaged limit measure corresponding to localization, and the weak limit measures describing the ballistic spreading [22]. We should note that the weak limit measure is consisted by the Dirac measure part corresponding to localization and absolutely continuous part, expressing the ballistic spreading. The weak convergence theorem for various types of QWs in one dimension, such as Hadamard walk [18], Grover walk [19], the two-phase QWs [9, 11] were derived.

Recently, the stationary measure of the QW has received peculiar interests as another key measure for the distribution of QW. The stationary measure provides the stationary distribution, for instance. Here we briefly review the past studies of the stationary measure. Comparing to the study of the stationary distributions of Markov chains, the corresponding study of QWs has not been done sufficiently. Most of the present papers deal with stationary measures of the discrete-time case mainly on $\mathbb{Z}$, where $\mathbb{Z}$ is the set of integers. As for the stationary measure of the two-state QW, Konno et al. [22] treated QW with a single defect at the origin and showed that a stationary measure with exponential decay for the position is identical to the time-averaged limit measure for the QW. [24] investigated stationary measures for various types of QWs. Endo et al. [6] got a stationary measure of the QW with a single defect whose quantum coins are defined by the Hadamard matrix at $x\neq 0$ and the rotation matrix at $x=0$. Endo and Konno [7] calculated a stationary measure of QW with a single defect which was introduced and studied by Wojcik et al . [30]. Furthermore, Endo et al. [11] and Endo et al. [8] obtained stationary measures of the two-phase QWs without defect and with a single defect, respectively, and investigated the relation to localization and the topological insulator. Konno and Takei [25] considered stationary measures of QWs and provided non-uniform stationary measures of non-diagonal QWs. They also showed that the set of the stationary measures generally contains uniform measure. As for the stationary measure of the three-state QW, Konno [24] obtained the stationary measures of the three-state Grover walk. Moreover, Wang et al. [29] investigated stationary measures of the three-state Grover walk with a single defect at the origin. Kawai et al. [15] constructed stationary measures for some types of the three-state QWs by using the reduction matrices. As for higher dimensional case, Komatsu and Konno [16] gave the stationary measures of the Grover walk on $\mathbb{Z}^{d}(d\in\mathbb{N})$, where $\mathbb{N}$ is the set of natural numbers. In this paper, we consider stationary measures for the three-state QWs.

Since there are rich applications of space-inhomogeneous QWs, such as the applications for networks [27] and topological insulator [11], the research on the mathematical aspects of space-inhomogeneous QW to exactly grasp the asymptotic behavior is important topic in the theoretical study of QWs [1, 4, 20, 21]. However, comparing to the study on the two-state QW, the three-state QW has not been treated enough. As for the stationary measures, there are few results for the quantum systems in comparison with that of the classical systems. In particular, the stationary measures for the classical systems have been known to be constant or exponential increase:

[TABLE]

with $c,p,q\in\mathbb{R}_{\geq 0}$, and $0\neq p<q,\;p+q=1$. Here $\mathbb{R}_{\geq 0}$ is the set of the real numbers $r\geq 0$. Also, in the study of the generalized eigenfunctions in the Spectral scattering theory, the eigenfunctions in $l^{\infty}$-space have been actively discussed [26]. According to the background, it is very important to study the stationary measures that are not in $l^{2}$-space on $\mathbb{Z}$.

Our main result is the general form of the stationary measure obtained by the transfer matrices of the three-state QW.

The rest of this paper is organized as follows. In Section 2, we give the definition of the three-state QW which is the main target in this paper, and present our main result. We show specific examples of our results in Section 3. In Appendix, we give the proof of Theorem 1.

2 Stationary measure in general condition originated from the eigenvalue problem

First, we introduce discrete-time space-inhomogeneous QW with three-state on $\mathbb{Z}$, which is a quantum version of classical random walk with an additional coin state. The particle has a coin state at time $n$ and position $x$ described by a three dimensional vector in $\mathbb{C}^{3},$ where $\mathbb{C}$ is the set of complex numbers:

[TABLE]

The upper and lower elements express the left and right chiralities, respectively, and the middle element corresponds to the loop. The time evolution is determined by $3\times 3$ unitary matrices $U_{x}$:

[TABLE]

where $x\in\mathbb{Z},$ and $a_{x},b_{x},c_{x},d_{x},e_{x},f_{x},g_{x},h_{x},i_{x}\in\mathbb{C}$.

Here the time evolution is determined by the recurrence formula

[TABLE]

where

[TABLE]

with $U_{x}=P_{x}+R_{x}+Q_{x}$. Note that $P_{x}$ and $Q_{x}$ correspond to the left and right movements, respectively, and $R_{x}$ represents the loop. Here we remark that the walker steps dependent on position.

From now on, we introduce the stationary measure of the QW. Let

[TABLE]

and

[TABLE]

where $T$ means the transposed operation. Then the state of the QW at time $n$ is given by $\Psi_{n}=(U^{(s)})^{n}\Psi_{0}$ for any $n\geq 0$. Let $\mathbb{R}_{+}=[0,\infty)$. Here we introduce a map $\phi:(\mathbb{C}^{3})^{\mathbb{Z}}\rightarrow\mathbb{R}_{+}^{\mathbb{Z}}$ such that for $\Psi_{n},$ we put

[TABLE]

Then we define the measure $\mu:\mathbb{Z}\to\mathbb{R}_{+}$ by

[TABLE]

We should note that $\mu(x)$ is a measure of the QW at position $x$. Now put

[TABLE]

and we call the element of $\Sigma_{s}$, the stationary measure of the QW.

Here let $S^{1}=\{z\in\mathbb{C}:|z|=1\},$ and consider the eigenvalue problem $U^{(s)}\Psi=\lambda\Psi(\Psi\in{\rm Map}(\mathbb{Z},\mathbb{C}^{3}),\lambda\in S^{1})$. Since $U^{(s)}$ is unitary, we directly see $\phi(U^{(s)}\Psi)\in\Sigma_{s}$.

Now we solve the eigenvalue problem as follows. The proof is devoted to Appendix.

Theorem 1

Let $\{U_{y}\}_{y\in\mathbb{Z}}$ be the set of $y$-parameterized unitary matrices of the three-state inhomogeneous QW, and $\Psi(x)=[\Psi^{L}(x),\;\Psi^{O}(x),\;\Psi^{R}(x)]{}^{T}\!$ be the probability amplitude. Note that there is a restriction for the initial state $\Psi(0)$ [10]. Then the solutions for $U^{(s)}\Psi=\lambda\Psi(\Psi\in{\rm Map}(\mathbb{Z},\mathbb{C}^{3}),\lambda\in S^{1})$ are

[TABLE]

where $T^{(\pm)}_{y}$ are the transfer matrices defined by

[TABLE]

with

[TABLE]

and

[TABLE]

Furthermore, the stationary measure is given by $\mu(x)=\phi(\Psi)(x)=\|\Psi(x)\|^{2}\;(x\in\mathbb{Z})$.

We note that the initial state develops by the transfer matrices. We can obtain the stationary measure for given three-state QW and initial state by calculating the transfer matrices. Also, we should remark that we cannot apply Theorem 1 in the case that the elements of the transfer matrices diverge. Our result is more effective for the three-state QWs in general than the result obtained by Kawai et al. [15], since they use the reduced matrix, which restricts the models that can be analyzed.

3 Examples

In this section, we exhibit some concrete typical examples of our result, Theorem 1.

The Grover walk:

First, we consider the Grover walk defined by the unitary matrix

[TABLE]

We got the stationary amplitude originated from the eigenvalue problem $U^{(s)}\Psi=\lambda\Psi$ as follows.

Proposition 1

Let $\{M_{y}\}_{y\in\mathbb{Z}}$ be the set of $y\in\mathbb{Z}$-parameterized unitary matrices of the Grover walk, and $\Psi(x)=[\Psi^{L}(x),\;\Psi^{O}(x),\;\Psi^{R}(x)]{}^{T}$ be the probability amplitude. Put $\alpha=\Psi^{L}(0),\;\gamma=\Psi^{O}(0)$ and $\beta=\Psi^{R}(0)$. Then we have $(1+3\lambda)\Psi^{O}(0)=2\Psi^{L}(0)+2\Psi^{R}(0)$, and for every $\lambda\in S^{1}$, we can choose $\Psi(0)=[\alpha,0,-\alpha]{}^{T}$ as an initial state. Now we take $\lambda=-1$, and the solutions for $U^{(s)}\Psi=\lambda\Psi(\Psi\in Map(\mathbb{Z},\mathbb{C}^{2}),\lambda\in S^{1}),$ are

[TABLE]

where $T^{(\pm)}_{y}$ are the transfer matrices defined by

[TABLE]

For $\Psi(0)=[\alpha,0,-\alpha]{}^{T}$, we see

[TABLE]

and therefore, we obtain

[TABLE]

We see that the stationary measure is uniform, which is same with the result in Section $4$ of [15]. 2. 2.

The Grover walk+1-defect model:

Next, we focus on the QW defined by the unitary matrices

[TABLE]

with $\rho=e^{i\phi}\;(\phi\in[0,\infty))$.

The QW is given by putting the weight $\rho$ at $x=0$ to the Grover walk. We derived the stationary amplitude of the eigenvalue problem $U^{(s)}\Psi=\lambda\Psi$ as follows.

Proposition 2

Let $\{M_{y}\}_{y\in\mathbb{Z}}$ be the set of $y\in\mathbb{Z}$-parameterized unitary matrices of the Grover walk+1-defect model, and $\Psi(x)=[\Psi^{L}(x),\;\Psi^{O}(x),\;\Psi^{R}(x)]{}^{T}$ be the probability amplitude. Put $\alpha=\Psi^{L}(0),\;\gamma=\Psi^{O}(0)$ and $\beta=\Psi^{R}(0)$. Then we see $(1-3\lambda)\Psi^{O}(0)=2\Psi^{L}(0)+2\Psi^{R}(0)$, and for every $\lambda\in S^{1}$, we can take $\Psi(0)=[\alpha,0,-\alpha]{}^{T}$ as an initial state. Now we take $\lambda=-1$, and the solutions for $U^{(s)}\Psi=\lambda\Psi(\Psi\in Map(\mathbb{Z},\mathbb{C}^{2}),\lambda\in S^{1}),$ are

[TABLE]

where $T^{(\pm)}_{y}$ are the transfer matrices defined by

[TABLE]

For $\Psi(0)=[\alpha,0,-\alpha]{}^{T},$ we obtain

[TABLE]

Hence, we have

[TABLE]

Thereby, the stationary measure is not uniform, and the result does not coincide with the result in Section $4$ of [10]. The defect at the origin seems to influence the value of the stationary measure not at the origin, but at both sides of the origin. 3. 3.

The Fourier QW: Here we treat the Fourier QW defined by the unitary matrices

[TABLE]

with $\omega=e^{\frac{2}{3}i\pi}$.

By putting $\lambda=(1+2\omega)/\sqrt{3}$, we got the stationary amplitude of the eigenvalue problem $U^{(s)}\Psi=\lambda\Psi$ as follows.

Proposition 3

Let $\{M_{y}\}_{y\in\mathbb{Z}}$ be the set of $y\in\mathbb{Z}$-parameterized unitary matrices of the Fourier QW, and $\Psi(x)={}^{T}\![\Psi^{L}(x),\;\Psi^{O}(x),\;\Psi^{R}(x)]$ be the probability amplitude. Then we see $(\sqrt{3}\lambda-\omega)\Psi^{O}(0)=\Psi^{L}(0)+\omega^{2}\Psi^{R}(0)$, and for every $\lambda\in S^{1}$, we can choose $\Psi(0)=[\alpha,0,-\alpha\omega^{-2}]{}^{T}$ as an initial state. Now we take $\lambda=i$, and the solutions for $U^{(s)}\Psi=\lambda\Psi(\Psi\in Map(\mathbb{Z},\mathbb{C}^{2}),\lambda\in S^{1}),$ are

[TABLE]

where $T^{(\pm)}_{y}$ are the transfer matrices defined by

[TABLE]

For $\Psi(0)={}^{T}\![\alpha,0,-\alpha\omega^{-2}]$, we get

[TABLE]

where $m\in\mathbb{Z}_{\geq 0}$ with $\mathbb{Z}_{\geq 0}=\{0,1,2,\cdots\}$. Hence, we obtain

[TABLE]

We notice that the stationary measure has period $3$, which is same with the result in Section $4$ of [15].

4 Summary

In this paper, we obtained for the three-state QW, the general form of the stationary measure originated from the eigenvalue problem. Our method is mainly based on the transfer matrices, and is more effective than the reduction method developed by Kawai et al. [15], since we do not need to reduct to two-state model in our recipe. By using our result, we can derive various types of stationary measures for many types of the three-state QWs, which contributes to clear the whole picture of the set of stationary measures. One of our future problems is to examine the eigenvalues and eigenspaces, which directly connects to Spectral theory. Also, to investigate the stationary measure which does not come from the eigenvalue problem is fundamental for the study of the stationary measure of the QW. On the other hand, to explore the relation between the three-state QWs and physical phenomenon, such as the topological insulator, leads to the application of the three-state QWs to quantum information sciences.

Acknowledgments

TE is supported by financial support of Postdoctoral Fellowship from Japan Society for the Promotion of Science.

References