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Erschienen in:
01.01.2016
Asymptotic velocity of a position-dependent quantum walk
Erschienen in: Quantum Information Processing | Ausgabe 1/2016
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Abstract
We consider a position-dependent coined quantum walk on \(\mathbb {Z}\) and assume that the coin operator C(x) satisfies 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 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 where \(\Psi _0\) is the initial state, \(\delta _0\) the Dirac measure at zero, and \(E_{\hat{v}_+}\) the spectral measure of \(\hat{v}_+\).
$$\begin{aligned} \Vert C(x) - C_0 \Vert \le c_1|x|^{-1-\epsilon }, \quad x \in \mathbb {Z}\setminus \{0\} \end{aligned}$$
$$\begin{aligned} \text{ s- }\lim _{t \rightarrow \infty } \mathrm{exp}\left( i \xi \frac{\hat{x}(t)}{t} \right) = \Pi _\mathrm{p}(U) + \mathrm{exp}(i \xi \hat{v}_+) \Pi _\mathrm{ac}(U) \end{aligned}$$
$$\begin{aligned} \mu _V = \Vert \Pi _\mathrm{p}(U)\Psi _0\Vert ^2\delta _0 + \Vert E_{\hat{v}_+}(\cdot ) \Pi _\mathrm{ac}(U)\Psi _0\Vert ^2, \end{aligned}$$