×

Weak limit theorem for a one-dimensional split-step quantum walk. (English) Zbl 1438.47023

The authors study a split-step quantum walk on \(\mathbb{Z}\), whose evolution operator is \(U=SC\). Here, the shift operator \(S\) on \(\ell^2(\mathbb{Z})\oplus\ell^2(\mathbb{Z})\) is given by \[ S= \begin{pmatrix} p & qL \\ qL^* & -p \end{pmatrix}, \quad p,q>0, \quad p^2+q^2=1. \] The coin operator \(C\) is defined as the multiplication operator by a family of unitary \(2\times 2\)-matrices \(\{C(x)\}_{x\in\mathbb{Z}}\). The main assumption on this family is \[ \Vert C(x)-C_0\Vert\le \frac{k}{\vert x\vert^{1+\varepsilon}}, \ \ x\in\mathbb{Z}\backslash\{0\}, \ \ C_0= \begin{pmatrix} a & b \\ b & -a \end{pmatrix},\] \(a,b,k,\varepsilon>0\). The position \(X_t\) of a walker at time \(t\) with an initial state \(\Psi_0\) is a random variable with the distribution \[ P(X_t=x)=\Vert U^t\Psi_0\Vert^2, \quad x\in\mathbb{Z}. \] The authors show that, under the above assumptions, \(X_t/t\) converges in law to a random variable \(V\) with precisely computed distribution.

MSC:

47A40 Scattering theory of linear operators
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
60F05 Central limit and other weak theorems
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
PDFBibTeX XMLCite
Full Text: arXiv Link