A Crossover Between Open Quantum Random Walks to Quantum Walks

Abstract

We propose an intermediate walk continuously connecting an open quantum random walk and a quantum walk with parameters $M\in {{\mathbb {N}}}$ controlling a decoherence effect; if $M=1$, the walk coincides with an open quantum random walk, while $M=\infty$, the walk coincides with a quantum walk. We define a measure which recovers usual probability measures on ${{\mathbb {Z}}}$ for $M=\infty$ and $M=1$ and we observe intermediate behavior through numerical simulations for varied positive values M. In the case for $M=2$, we analytically show that a typical behavior of quantum walks appears even in a small gap of the parameter from the open quantum random walk. More precisely, we observe both the ballistically moving towards left and right sides and localization of this walker simultaneously. The analysis is based on Kato’s perturbation theory for linear operator. We further analyze this limit theorem in more detail and show that the above three modes are described by Gaussian distributions.

Propositions A.1–A.3

The matrix representation for ${\hat{W}}_{s,t}(0)=:{\hat{W}}_{s,t}$ is explicitly written as

$$\begin{aligned} {\hat{W}}_{s,t}= \frac{1}{2} \left[ \begin{array}{cccc|cccc} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} -1 &{} 0 &{} -1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} -1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} -1&{} 0 &{} 0 \\ \hline 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} -1 \\ 0 &{} 0 &{} -1 &{} 0 &{} 1 &{} 0 &{} 0 &{} -1 \\ 0 &{} 0 &{} -1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \end{array} \right] . \end{aligned}$$

for $s=-1$ and $t=0$. Using the formula for $\lambda _1(k)$ and $\lambda _2(k)$, we know that eigenvalues of ${\hat{W}}_{s,t}$ are

$$\begin{aligned} \textrm{spec}({\hat{W}}_{s,t})=\left\{ 0,0,1,1,1,-\frac{1}{2}, \frac{-1\pm i\sqrt{7}}{4}\right\} , \end{aligned}$$

(51)

which shows that the matrix ${\hat{W}}_{s,t}$ possesses eigenvalues with multiplicity greater than 1. By the Cayley-Hamilton theorem, we know that $F_{{\hat{W}}_{s,t}}({\hat{W}}_{s,t})= O$, where

$$\begin{aligned} F_{{\hat{W}}_{s,t}}(\lambda ) = \lambda ^2 (\lambda - 1)^3 (2\lambda ^2 + \lambda + 1)(2\lambda +1). \end{aligned}$$

(52)

Interestingly, we observe the following result.

Lemma A.1

The minimal polynomial $f_{{\hat{W}}_{s,t}}(\lambda )$ of the matrix ${\hat{W}}_{s,t}$ is

$$\begin{aligned} f_{{\hat{W}}_{s,t}}(\lambda ) = \lambda (\lambda - 1) (2\lambda ^2 + \lambda + 1)(2\lambda +1). \end{aligned}$$

Proof

Direct calculations yield that

$$\begin{aligned} {\hat{W}}_{s,t} ({\hat{W}}_{s,t} - I) (2{\hat{W}}_{s,t}^2 + {\hat{W}}_{s,t} + I)(2{\hat{W}}_{s,t}+I) = O. \end{aligned}$$

$\square$

A direct consequence of this lemma is the following.

Proposition A.1

All eigenvalues of ${\hat{W}}_{s,t}$ are semisimple. In particular, the matrix ${\hat{W}}_{s,t}$ is diagonalizable.

Thanks to Proposition A.1, the eigenvalue $\lambda = 1$ of ${\hat{W}}_{s,t}$ is semisimple and hence the associating eigenspace, which is a three-dimensional linear space, is generated by three linearly independent eigenvectors of $\lambda = 1$. We have the following observation for eigenstructures at $\lambda =1$ by a direct computation of the non-perturbed matrix ${\hat{W}}_{s,t}$.

Lemma A.2

The eigenspace of ${\hat{W}}_{s,t}$ associated with $\lambda = 1$ is generated by the following vectors:

$$\begin{aligned}{}[0, 0, 0, 0, 1, 0, 0, 1]^\top ,\quad [1, 0, 0, 1, 0, 0, 0, 0]^\top , \quad [1, 0, 1, 0, 1, 1, 0, 0]^\top . \end{aligned}$$

One of their orthonormal choice is

$$\begin{aligned} \nonumber \phi _1&= \left[ 0, 0, 0, 0, \frac{1}{\sqrt{2}}, 0, 0, \frac{1}{\sqrt{2}}\right] ^\top ,\;\;\phi _2 = \left[ \frac{1}{\sqrt{2}}, 0, 0, \frac{1}{\sqrt{2}}, 0, 0, 0, 0\right] ^\top \\ \phi _3&= \left[ \frac{1}{2\sqrt{3}}, 0, \frac{1}{\sqrt{3}}, -\frac{1}{2\sqrt{3}}, \frac{1}{2\sqrt{3}}, \frac{1}{\sqrt{3}}, 0, -\frac{1}{2\sqrt{3}}\right] ^\top . \end{aligned}$$

(53)

Next consider the perturbation of these eigenstructures. Namely, consider eigenpairs of ${\hat{W}}_{s,t}(\delta )$ with sufficiently small $\delta$. Thus to apply the reduction process [7], the next task is to obtain an explicit expression for the matrix ${\tilde{T}}^{(1)}$ because its eigenvalue $\lambda _j^{(1)}$ gives the coefficient of the second order of the eigenvalue of ${\hat{W}}_{s,t}(\delta )$. To this end, we need to prepare an explicit expression for the eigenprojection $\Pi$ of the eigenvalue 1 of ${\hat{W}}_{s,t}(0)$. Our strategy here is to compute eigenvalue $\lambda _j^{(1)}$ of ${\tilde{T}}^{(1)}$. Since $\Pi$ is an eigenprojection, it holds that $\Pi ^2=\Pi$ and $\Pi {\hat{W}}_{s,t}={\hat{W}}_{s,t}\Pi$. In addition, it can be shown that the property $\Pi =\Pi ^*$ holds by [6] which treats more general setting of the truncated unitary matrix.

Proposition A.2

Let the centered generalized eigenspace of ${\hat{W}}_{s,t}$ be defined by

$$\begin{aligned} {{\mathcal {H}}}_c:=\{\psi \in \ell ^2(\{s,\dots ,t\};{{\mathbb {C}}}^4) \;|\; ({\hat{W}}_{s,t}-\lambda )^m\psi =0 \mathrm {\;for\; some\;} |\lambda |=1,\;m\ge 1\} \end{aligned}$$

Then we have the following properties of ${{\mathcal {H}}}_c$.

• 1. The centered generalized eigenspace of ${\hat{W}}_{s,t}:={\hat{W}}_{s,t}(0)$ is expressed by

  $$\begin{aligned} {{\mathcal {H}}}_c&= \bigoplus _{|\lambda |=1}\ker (\lambda -{\hat{W}}_{s,t}) \\&= \textrm{span}\{ \chi _{s,t}^*\psi \;|\; \textrm{supp}(\psi )\subset \{s,\dots ,t\},\;\psi \mathrm {\;is\;an\;eigenvector\;of\;} {\hat{W}}_{-\infty ,\infty }(0)={\hat{U}}(0) \} \end{aligned}$$

• 2. The complementary invariant subspace of ${{\mathcal {H}}}_c$ can be expressed by ${{\mathcal {H}}}_c^\perp$.

Proof

This is a direct consequence of Lemma 3.3, (3.12) and Lemma 3.4 in [6]. $\square$

By the first part of Proposition A.2 implies that ${{\mathcal {H}}}_c=\ker (1-{\hat{W}}_{s,t})$. Moreover the second part of Proposition A.2 implies that the eigenprojection $\Pi$ of the eigenvalue 1 can be computed by using Lemma A.2 as follows.

$$\begin{aligned} \Pi&= \phi _1\phi _1^*+\phi _2\phi _2^*+\phi _3\phi _3^* \\&= \frac{1}{12}\begin{bmatrix} 7 &{}\quad 0 &{}\quad 2 &{}\quad 5 &{}\quad 1 &{}\quad 2 &{}\quad 0 &{}\quad -1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 2 &{}\quad 0 &{}\quad 4 &{}\quad -2 &{}\quad 2 &{}\quad 4 &{}\quad 0 &{}\quad -2\\ 5 &{}\quad 0 &{}\quad -2 &{}\quad 7 &{}\quad -1 &{}\quad -2 &{}\quad 0 &{}\quad 1\\ 1 &{}\quad 0 &{}\quad 2 &{}\quad -1 &{}\quad 7 &{}\quad 2 &{}\quad 0 &{}\quad 5\\ 2 &{}\quad 0 &{}\quad 4 &{}\quad -2 &{}\quad 2 &{}\quad 4 &{}\quad 0 &{}\quad -2\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ -1 &{}\quad 0 &{}\quad -2 &{}\quad 1 &{}\quad 5 &{}\quad -2 &{}\quad 0 &{}\quad 7\\ \end{bmatrix}, \end{aligned}$$

which is an orthogonal projection, in which case $\Pi ^2 = \Pi$ and $\Pi ^*= \Pi$ hold.

On the other hand,

$$\begin{aligned} T^{(1)} = \begin{bmatrix} t^{(1)} &{} O \\ O &{} t^{(1)} \end{bmatrix},\quad t^{(1)}= -\frac{i}{2} \begin{bmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad -1 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad -1 \\ \end{bmatrix} \end{aligned}$$

and hence

$$\begin{aligned} \Pi T^{(1)}\Pi&= \frac{-i}{6} \begin{bmatrix} 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad -1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad -1 \\ 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad -1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad -1 \\ \end{bmatrix} \end{aligned}$$

Direct calculations yield the following result.

Proposition A.3

The matrix $\Pi T^{(1)}\Pi$ is a skew-Hermitian matrix. Eigenvalues of the matrix $\Pi T^{(1)}\Pi$ on the eigenspace $\textrm{R}(\Pi )$ are 0, $i / \sqrt{3}$ and $-i/\sqrt{3}$. In particular, all these eigenvalues are simple. The corresponding eigenprojections are $\Pi ^{(1)}_1:=v_1v_1^{^*}$, $\Pi ^{(1)}_2:=v_2v_2^*$ and $\Pi ^{(1)}_3:=v_3v_3^*$, respectively. Here

$$\begin{aligned} v_1&=[1/2, 0,0,1/2,-1/2,0,0,-1/2]^\top ,\\ v_2&=\frac{1}{\sqrt{2}\;(3-\sqrt{3})}[2-\sqrt{3},0,1-\sqrt{3},1,2-\sqrt{3},1-\sqrt{3},0,1]^\top ,\\ v_3&=\frac{1}{\sqrt{2}\;(3+\sqrt{3})}[2+\sqrt{3},0,1+\sqrt{3},1,2+\sqrt{3},1+\sqrt{3},0,1]^\top . \end{aligned}$$

Remark A.1

• 1. $\Pi =\Pi _1^{(1)}+\Pi _2^{(1)}+\Pi _3^{(1)}$ and $\Pi _i^{(1)}\Pi _j^{(1)}=\delta _{ij}\Pi _i^{(1)}$ holds.

• 2. The eigenvalue on $\textrm{R}(1-\Pi )$ is also 0. The eigenvector of eigenvalue $\epsilon$ for the matrix $\Pi (T^{(1)}-\epsilon I)\Pi$ corresponds to that of $\textrm{R}(\Pi )$. Taking $\epsilon \rightarrow 0$ to this eigenvector, we obtain the eigenvector $v_1$. See [7] for more detail.

• 3. The coefficients of the second orders of the eigenvalues of $\lambda _1(\delta )$ for $j=0$ in (27) and $\lambda _2(\delta )$ for $j=1,2$ in (30) identify with the eingenvalues of $\Pi T^{(1)}\Pi$ in $R(\Pi )$. This derives from nothing but the contribution of the reduction process of the perturbation theory [7]. Moreover, it is possible to obtain the coefficients of more than the third orders by repeating the reduction process. For example, the coefficient of $\lambda _2(\delta )$ for $j=1$ can be computed by the same procedure by replacing ${\tilde{T}}^{(1)}$ with ${\tilde{T}}^{(2)}$, where

  $$\begin{aligned} {\tilde{T}}^{(2)}:=\Pi T^{(2)}\Pi -\Pi T^{(1)}\Pi T^{(1)} S- \Pi T^{(1)}S T^{(1)} \Pi -ST^{(1)}\Pi T^{(1)}\Pi . \end{aligned}$$

  Here S is the reduced resolvent of ${\hat{W}}_{s,t}$ at the eigenvalue 1.