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 \[ \|C(x) - C_0 \| \leq c_1|x|^{-1-ε}, \quad x \in \mathbb{Z} \] 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 \[ \mbox{s-}\lim_{t \to \infty} {\rm exp}\left( i ξ\frac{\hat x(t)}{t} \right) = Π_{\rm p}(U) + {\rm exp}(i ξ\hat v_+) Π_{\rm ac}(U) \] 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 \[ μ_V = \|Π_{\rm p}(U)Ψ_0\|^2δ_0 + \|E_{\hat v_+}(\cdot) Π_{\rm ac}(U)Ψ_0\|^2, \] where ${\Psi }_0$ is the initial state, ${\delta }_0$ the Dirac measure at zero, and $E_{{\hat{v}}_{+}}$ the spectral measure of ${\hat{v}}_{+}$.

References in corpus (1)

• Trapping a particle of a quantum walk on the line

Cited by in corpus (1)

• Resonances in finitely perturbed quantum walks, resonance expansion and generic simplicity