Asymptotic velocity of a position-dependent quantum walk

Abstract

We consider a position-dependent coined quantum walk on $\mathbb {Z}$ and assume that the coin operator C(x) satisfies

$$\begin{aligned} \Vert C(x) - C_0 \Vert \le c_1|x|^{-1-\epsilon }, \quad x \in \mathbb {Z}\setminus \{0\} \end{aligned}$$

with positive $c_1$ and $\epsilon$ and $C_0 \in U(2)$. We show that the Heisenberg operator $\hat{x}(t)$ of the position operator converges to the asymptotic velocity operator $\hat{v}_+$ so that

$$\begin{aligned} \text{ s- }\lim _{t \rightarrow \infty } \mathrm{exp}\left( i \xi \frac{\hat{x}(t)}{t} \right) = \Pi _\mathrm{p}(U) + \mathrm{exp}(i \xi \hat{v}_+) \Pi _\mathrm{ac}(U) \end{aligned}$$

provided that U has no singular continuous spectrum. Here $\Pi _\mathrm{p}(U)$ (resp., $\Pi _\mathrm{ac}(U)$) is the orthogonal projection onto the direct sum of all eigenspaces (resp., the subspace of absolute continuity) of U. We also prove that for the random variable $X_t$ denoting the position of a quantum walker at time $t \in \mathbb {N}$, $X_t/t$ converges in law to a random variable V with the probability distribution

$$\begin{aligned} \mu _V = \Vert \Pi _\mathrm{p}(U)\Psi _0\Vert ^2\delta _0 + \Vert E_{\hat{v}_+}(\cdot ) \Pi _\mathrm{ac}(U)\Psi _0\Vert ^2, \end{aligned}$$

where $\Psi _0$ is the initial state, $\delta _0$ the Dirac measure at zero, and $E_{\hat{v}_+}$ the spectral measure of $\hat{v}_+$.