Localization in quantum walks with periodically arranged coin matrices

The study of quantum walks, which began in the early 2000s [1, 2], has spread and attracted much attention, especially for its applications in quantum information [3, 4, 5, 6, 7, 8]. One of the most characteristic properties of the quantum walk is localization, an essential property for manipulating particles. Numerical and theoretical analyses have been actively conducted, and we are particularly interested in the mathematical analysis of localization [9, 10, 11, 8, 12, 13]. It has been known that localization occurs in various quantum walk models. This study focuses on one-dimensional two-state quantum walks, which are considered the fundamental discrete-time quantum walks.

From a mathematical point of view, the investigation of localization can be regarded as an eigenvalue problem. This is because the occurrence of localization is equivalent to the existence of eigenvalues of the time evolution operator, and the corresponding eigenvectors are related to how likely the walker localizes [12, 14]. In a previous study [15], an eigenvalue analysis method using the transfer matrix was proposed. The eigenvalue analysis was performed for two-phase quantum walks with one defect, including a one-defect model where the time evolution differs at the origin and a two-phase model where the time evolution differs in the negative and positive parts, respectively. The localization phenomenon in the one-defect model has been used in quantum search algorithms [16, 3, 17], and the relationship between topological insulators and localization in the two-phase quantum walk has attracted much attention [18, 19]. Several other studies also used transfer matrices for deriving stationary measures [20, 21]. Furthermore, a previous study [14, 22] showed that the method could be applied to a more general model with a finite number of defects, which satisfies the following conditions:

$$C_{x}=\begin{cases}C_{+\infty},\quad&x\in[x_{+},\infty),\\ C_{-\infty},\quad&x\in(-\infty,x_{-}],\end{cases}$$

where $x$ and $x_{+}>0,x_{-}<0$ are integers, and $C_{x}$ denotes the coin matrix determining the time evolution in the position $x$. Models with periodic coin matrices have also been actively studied [23, 24, 25]. In particular, the model with self-duality studied in [25] is inspired by the well-studied Aubry-André model [26], and the Fourier transform method was applied. However, our approach can extend the discussion to models that cannot be handled by the Fourier transform method and simplify the eigenvalue problem since we only need to deal with 2 $\times$ 2 (transfer) matrices instead of the large matrix. In this study, we consider a two-phase periodic model with a finite number of defects that includes all of the above important models; the model has $n_{-}$ and $n_{+}$ different coin matrices arranged periodically in positions $x<x_{-}$ and $x\geq x_{+}$, respectively.

This paper is organized as follows. In Section 2, we define our model with periodically arranged coin matrices and its transfer matrix. This section also shows that the eigenvalue analysis via transfer matrix is applicable for the model. The necessary and sufficient condition for the eigenvalue problem is given in Theorem 2.3. Also, further discussion about the time-averaged limit distribution using eigenvalues and eigenvectors is provided. With the main theorem, Section 3 focuses on analyzing concrete eigenvalues for more specific models, which can be seen as natural generalizations of homogeneous, one-defect, and two-phase models to periodic models.

Firstly, we introduce two-state quantum walks on the integer lattice $\mathbb{Z}$. The Hilbert space is given as $\mathcal{H}$ defined by

$$\mathcal{H}=\ell^{2}(\mathbb{Z};\mathbb{C}^{2})=\left\{\Psi:\mathbb{Z}\to\mathbb{C}^{2}\ \middle|\ \sum_{x\in\mathbb{Z}}\|\Psi(x)\|_{\mathbb{C}^{2}}^{2}<\infty\right\},$$

where $\mathbb{C}$ denotes the set of complex numbers. For $\Psi\in\mathcal{H}$, we write $\Psi(x)=\begin{bmatrix}\Psi_{L}(x)&\Psi_{R}(x)\end{bmatrix}^{T}\in\mathbb{C}^{2}$, where $T$ is the transpose operator. The time evolution operator $U$ is defined by the product of a coin operator $C$ on $\mathcal{H}$ and a shift operator $S$ on $\mathcal{H}$. Here, $C$ and $S$ are defined by

$\displaystyle(C\Psi)(x)=C_{x}\Psi(x),\qquad(S\Psi)(x)=\begin{bmatrix}\Psi_{L}(x+1)\\ \Psi_{R}(x-1)\end{bmatrix},$

where $\{C_{x}\}_{x\in\mathbb{Z}}$ is a sequence of $2\times 2$ unitary matrices called coin matrices. We define $x_{+},x_{-}\in\mathbb{Z}$ where $x_{+}\geq 0,\ x_{-}\leq 0$, then the periodicities of the model are defined by $n_{+}\in\mathbb{Z}_{>0}$ for positions $x\geq x_{+}$ and $n_{-}\in\mathbb{Z}_{>0}$ for positions $x<x_{-}$, where $\mathbb{Z}_{>0}$ is the set of positive integers. For $x\in\mathbb{Z}$, we define $r_{x}^{\pm}\in\{0,\dots,n_{\pm}-1\}$ by remainder of $x-x_{\pm}$ divided by $n_{\pm}\in\mathbb{Z}_{>0}$, that is,

$$r_{x}^{\pm}=x-x_{\pm}\ (\bmod\ n_{\pm}).$$

We treat the model whose coin matrices satisfy the following:

$$C_{x}=\begin{cases}C_{r_{x}^{+}}^{+},&x\in[x_{+},\infty),\\ C_{x}&x\in[x_{-},x_{+}),\\ C_{r_{x}^{-}}^{-},&x\in(-\infty,x_{-}).\end{cases}$$

The model has a finite number of defects in $x\in[x_{-},x_{+})$ and periodic coin matrices for each of $x\geq x_{+}$ and $x<x_{-}$. Here, we write $C_{x},C^{\pm}_{k}$ as follows:

$\displaystyle C_{x}=e^{i\Delta_{x}}\left[\begin{array}[]{ c c }\alpha_{x}&\beta_{x}\\ -\overline{\beta_{x}}&\overline{\alpha_{x}}\end{array}\right],\ \ C_{k}^{\pm}=e^{i\Delta_{k}^{\pm}}\left[\begin{array}[]{ c c }\alpha_{k}^{\pm}&\beta_{k}^{\pm}\\ -\overline{\beta_{k}^{\pm}}&\overline{\alpha_{k}^{\pm}}\end{array}\right],$

with $\alpha_{x},\beta_{x},\alpha_{k}^{\pm},\beta_{k}^{\pm}\in\mathbb{C},\ \Delta_{x},\Delta_{k}^{\pm}\in[0,2\pi),\ |\alpha_{x}|^{2}+|\beta_{x}|^{2}=\left|\alpha_{k}^{\pm}\right|^{2}+\left|\beta_{k}^{\pm}\right|^{2}=1,$ and $\alpha_{x},\alpha_{k}^{\pm}\neq 0$ for $x\in\mathbb{Z},\ k\in\{0,\dotsc,n_{\pm}-1\}$.

We let $\Psi_{0}\in\mathcal{H}\ (\|\Psi_{0}\|_{\mathcal{H}}^{2}=1)$ denote the initial state of the model. Then, the probability distribution at time $t\in\mathbb{Z}_{\geq 0}$ is defined by

$$\mu_{t}^{(\Psi_{0})}(x)=\|(U^{t}\Psi_{0})(x)\|_{\mathbb{C}^{2}}^{2},$$

where $\mathbb{Z}_{\geq 0}$ is the set of non-negative integers. Here, we say that the quantum walk model exhibits localization if there exists an initial state $\Psi_{0}\in\mathcal{H}$ and a position $x_{0}\in\mathbb{Z}$ satisfying

$$\limsup_{t\to\infty}\mu^{(\Psi_{0})}_{t}(x_{0})>0.$$

As a well-known fact that the quantum walk model exhibits localization if and only if the time evolution operator $U$ has an eigenvalue [12], which means that there exists $\lambda\in[0,2\pi)$ and $\Psi\in\mathcal{H}\setminus\{\mathbf{0}\}$ such that

$\displaystyle U\Psi=e^{i\lambda}\Psi.$

Let $\sigma(U)$ denotes the set of eigenvalues of $U$ henceforward. Subsequently, let $J$ be a unitary operator on $\mathcal{H}$ defined by

$\displaystyle(J\Psi)(x)=\begin{bmatrix}\Psi_{L}(x-1)\\ \Psi_{R}(x)\end{bmatrix}$

for $x\in\mathbb{Z}$. Here the inverse of $J$ is given as

$\displaystyle(J^{-1}\Psi)(x)=\begin{bmatrix}\Psi_{L}(x+1)\\ \Psi_{R}(x)\end{bmatrix}.$

Furthermore, for $\lambda\in[0,2\pi)$, $x\in\mathbb{Z}$ and $k\in\{0,\dotsc,n_{\pm}-1\}$, we introduce the transfer matrix $T_{x}(\lambda),\ T^{\pm}_{k}(\lambda)$ as followings:

$$T_{x}(\lambda)=\frac{1}{\alpha_{x}}\left[\begin{array}[]{ c c }e^{i(\lambda-\Delta_{x})}&-\beta_{x}\\ -\overline{\beta_{x}}&e^{-i(\lambda-\Delta_{x})}\end{array}\right],\ \ T_{k}^{\pm}(\lambda)=\frac{1}{\alpha_{k}^{\pm}}\left[\begin{array}[]{ c c }e^{i\left(\lambda-\Delta_{k}^{\pm}\right)}&-\beta_{k}^{\pm}\\ -\overline{\beta_{k}^{\pm}}&e^{-i\left(\lambda-\Delta_{k}^{\pm}\right)}\end{array}\right].$$

We abbreviate the transfer matrix $T_{x}(\lambda)$ as $T_{x}$ and $T^{\pm}_{k}(\lambda)$ as $T^{\pm}_{k}$ henceforward. Here, $\Psi\in\mathcal{H}$ satisfies $(U-e^{i\lambda})\Psi=0$ if and only if $\Psi$ satisfies the following equation for all $x\in\mathbb{Z}$ :

$\displaystyle(J\Psi)(x+1)=T_{x}(J\Psi)(x).$

(1)

For more details, see [15]. In this paper, we define $\prod$ notation for the products of matrices by the following:

$$\prod_{i=k}^{n}A_{i}=\begin{cases}A_{n}\cdots A_{k+1}A_{k},&n\geq k,\\ 1,&n<k.\end{cases}$$

For $\lambda\in[0,2\pi)$ and $\varphi\in\mathbb{C}^{2}$, we define $\tilde{\Psi}:\mathbb{Z}\to\mathbb{C}^{2}$ as follows:

	$\displaystyle\tilde{\Psi}(x)$	$\displaystyle=\left\{\begin{array}[]{ l l }T_{x-1}T_{x-2}\cdots T_{1}T_{0}\varphi,&x>0,\\ \varphi,&x=0,\\ T_{x}^{-1}T_{x+1}^{-1}\cdots T_{-2}^{-1}T_{-1}^{-1}\varphi,&x<0.\end{array}\right.$
		$\displaystyle=\begin{cases}\displaystyle\prod_{i=0}^{r_{x}^{+}-1}T_{i}^{+}\left(\prod_{i=0}^{n_{+}-1}T_{i}^{+}\right)^{m_{x}^{+}}T_{+}\varphi,&x>x_{+},\\ \displaystyle\prod_{i=0}^{x-1}T_{i}\varphi,&0<x\leq x_{+},\\ \varphi,&x=0,\\ \displaystyle\prod_{i=1}^{|x|}T_{-i}^{-1}\varphi,&x_{-}\leq x<0,\\ \displaystyle\left(\prod_{i=r_{x}^{-}}^{n_{-}-1}T_{i}^{-}\right)^{-1}\ \left(\prod_{i=0}^{n_{-}-1}T_{i}^{-}\right)^{-m_{x}^{-}}T_{-}\varphi,&x<x_{-},\end{cases}$

where $T_{+}=\prod_{i=0}^{x_{+}-1}T_{i},\ T_{-}=\prod_{i=1}^{|x_{-}|}T_{-i}^{-1}$ and $m_{x}^{+}=\frac{x-x_{+}-r_{x}^{+}}{n_{+}}\in\mathbb{Z}_{\geq 0},\ \ m_{x}^{-}=\frac{\left|x-x_{-}-r_{x}^{-}+n_{-}\right|}{n_{-}}\in\mathbb{Z}_{\geq 0}.$ This can be rewritten as

$\displaystyle\tilde{\Psi}(x)=\begin{cases}\displaystyle\left(\tilde{T}_{r_{x}^{+}}^{+}\right)^{m_{x}^{+}}\prod_{i=0}^{r_{x}^{+}-1}T_{i}^{+}T_{+}\varphi,&x>x_{+},\\ \displaystyle\prod_{i=0}^{x-1}T_{i}\varphi,&0<x\leq x_{+},\\ \varphi,&x=0,\\ \displaystyle\prod_{i=1}^{|x|}T_{-i}^{-1}\varphi,&x_{-}\leq x<0,\\ \displaystyle\left(\tilde{T}_{r_{x}^{-}}^{-}\right)^{-m_{x}^{-}}\ \left(\prod_{i=r_{x}^{-}}^{n_{-}-1}T_{i}^{-}\right)^{-1}T_{-}\varphi,&x<x_{-},\end{cases}$

(2)

with

$$\tilde{T}_{k}^{\pm}=\prod_{i=0}^{k-1}T_{i}^{\pm}\prod_{i=k}^{n_{\pm}-1}T_{i}^{\pm},$$

for $k\in\{0,\dotsc,n_{\pm}-1\}.$ Here, $\tilde{\Psi}:\mathbb{Z}\rightarrow\mathbb{C}^{2}$ is a map constructed by transfer matrices, where $\Psi=J^{-1}\tilde{\Psi}$ satisfies equation (1) (but not necessarily satisfies $\sum_{x\in\mathbb{Z}}\|\Psi(x)\|_{\mathbb{C}^{2}}^{2}<\infty$). For $\lambda\in[0,2\pi)$, $\tilde{\Psi}$ has $\varphi\in\mathbb{C}^{2}$ as a parameter. We let $W_{\lambda}$ be a set of all possible $\tilde{\Psi}$ obtained by varying $\varphi$:

$$W_{\lambda}=\left\{\tilde{\Psi}:\mathbb{Z}\rightarrow\mathbb{C}^{2}\ \middle|\ \tilde{\Psi}(x)\text{ is given by }(\ref{eq:tilde_psi}),\ \varphi\in\mathbb{C}^{2}\right\}.$$

Note that $\Psi=J^{-1}\tilde{\Psi}$ where $\tilde{\Psi}\in W_{\lambda}\setminus\{\mathbf{0}\}$ but not necessarily $\tilde{\Psi}\in\mathcal{H}$ is the stationary measure of the quantum walk studied in [27, 28, 21, 20, 29]. We define sign function for real numbers $u$ as follows:

$$\operatorname{sgn}(u)=\begin{cases}1,&u>0,\\ 0,&u=0,\\ -1,&u<0.\end{cases}$$

Then, two eigenvalues of $\tilde{T}_{k}^{\pm}\ (k\in\{0,\dotsc,n_{\pm}-1\})$ can be written as $\zeta_{\pm}^{>},\zeta_{\pm}^{<}$ expressed as below:

	$\displaystyle\zeta_{\pm}^{>}=\frac{A^{\pm}+\operatorname{sgn}\left(A^{\pm}\right)\sqrt{\left(A^{\pm}\right)^{2}-\left|\alpha_{0}^{\pm}\right|^{2}\left|\alpha_{1}^{\pm}\right|^{2}\cdots\left|\alpha_{n_{\pm}-1}^{\pm}\right|^{2}}}{\alpha_{0}^{\pm}\alpha_{1}^{\pm}\cdots\alpha_{n_{\pm}-1}^{\pm}},$
	$\displaystyle\zeta_{\pm}^{<}=\frac{A^{\pm}-\operatorname{sgn}\left(A^{\pm}\right)\sqrt{\left(A^{\pm}\right)^{2}-\left|\alpha_{0}^{\pm}\right|^{2}\left|\alpha_{1}^{\pm}\right|^{2}\cdots\left|\alpha_{n_{\pm}-1}^{\pm}\right|^{2}}}{\alpha_{0}^{\pm}\alpha_{1}^{\pm}\cdots\alpha_{n_{\pm}-1}^{\pm}},$

where $\displaystyle A^{\pm}=\frac{1}{2}\alpha_{0}^{\pm}\alpha_{1}^{\pm}\cdots\alpha_{n_{\pm}-1}^{\pm}\mathrm{tr}\left(\prod_{i=0}^{n_{\pm}-1}T_{i}^{\pm}\right)$. Note that the eigenvalues are independent of $k$.

Subsequently, since $\displaystyle\left|\det T_{k}^{\pm}\right|=\left|\frac{\overline{\alpha_{k}^{\pm}}}{\alpha_{k}^{\pm}}\right|=1$ for $k\in\{0,\dotsc,n_{\pm}-1\},$

$$\left|\det\tilde{T}_{k}^{\pm}\right|=\left|\frac{\overline{\alpha_{0}^{\pm}}\overline{\alpha_{2}^{\pm}}\cdots\overline{\alpha_{n_{\pm}-1}^{\pm}}}{\alpha_{0}^{\pm}\alpha_{2}^{\pm}\cdots\alpha_{n_{\pm}-1}^{\pm}}\right|=1$$

holds, which also means $|\zeta_{\pm}^{>}||\zeta_{\pm}^{<}|=1$. From Corollary 2.2, we know that $|\zeta_{\pm}^{>}|\geq 1$ and $|\zeta_{\pm}^{<}|\leq 1$ hold, and we have the main theorem.

This theorem implies that the eigenvalue problem is to find the solution $\lambda\in[0,2\pi)$ of a single equation obtained from the second condition of the theorem in the range of $\left(A^{\pm}\right)^{2}-\left|\alpha_{0}^{\pm}\right|^{2}\left|\alpha_{1}^{\pm}\right|^{2}\cdots\left|\alpha_{n_{\pm}-1}^{\pm}\right|^{2}>0$. We can also see that if $e^{i\lambda}\in\sigma(U)$, the associated eigenvector $\Psi\in\ker(U-e^{i\lambda})\setminus\{\mathbf{0}\}$ is given as $\Psi=J^{-1}\tilde{\Psi}$ where

$\displaystyle\tilde{\Psi}(x)=\begin{cases}\displaystyle(\zeta_{+}^{<})^{m_{x}^{+}}\prod_{i=0}^{r_{x}^{+}-1}T_{i}^{+}T_{+}\varphi,&x>x_{+},\\ \displaystyle\prod_{i=0}^{x-1}T_{i}\varphi,&0<x\leq x_{+},\\ \varphi,&x=0,\\ \displaystyle\prod_{i=1}^{|x|}T_{-i}^{-1}\varphi,&x_{-}\leq x<0,\\ \displaystyle(\zeta_{-}^{>})^{-m_{x}^{-}}\ \left(\prod_{i=r_{x}^{-}}^{n_{-}-1}T_{i}^{-}\right)^{-1}T_{-}\varphi,&x<x_{-},\end{cases}$

with $\displaystyle\varphi\in\ker\left(\left(\prod_{i=0}^{n_{+}-1}T_{i}^{+}-\zeta_{+}^{<}\right)T_{+}\right)\cap\ker\left(\left(\prod_{i=0}^{n_{-}-1}T_{i}^{-}-\zeta_{-}^{>}\right)T_{-}\right).$

Furthermore, we can quantitatively evaluate localization by deriving time-averaged limit distribution defined by

$$\overline{\nu}_{\infty}(x)=\lim_{T\rightarrow\infty}\frac{1}{T}\sum_{t=0}^{T-1}\left\|\left(U^{t}\Psi_{0}\right)(x)\right\|_{\mathbb{C}^{2}}^{2}.$$

The time-averaged limit distribution can be calculated by the eigenvectors of $U$ [12]. For multiplicity $m_{\lambda}=\dim\ker(U-e^{i\lambda})$ and complete orthonormal basis $\Psi_{j}^{\lambda}\in\ker(U-e^{i\lambda}),\ j=1,2,\dotsc,m_{\lambda}$, the following holds:

$$\overline{\nu}_{\infty}(x)=\sum_{e^{i\lambda}\in\sigma(U)}\sum_{j,k=1}^{m_{\lambda}}\overline{\left<\Psi_{k}^{\lambda},\Psi_{0}\right>}\left<\Psi_{j}^{\lambda},\Psi_{0}\right>\left<\Psi_{k}^{\lambda}(x),\Psi_{j}^{\lambda}(x)\right>.$$

Here, we show some important facts for calculating $\overline{\nu}_{\infty}$.

Therefore, $\overline{\nu}_{\infty}(x)$ can be written as

$\displaystyle\overline{\nu}_{\infty}(x)=\sum_{e^{i\lambda}\in\sigma(U)}\left|\left<\Psi^{\lambda},\Psi_{0}\right>\right|^{2}\left\|\Psi^{\lambda}(x)\right\|_{\mathbb{C}^{2}}^{2}.$

(3)