Localization and limit laws of a three-state alternate quantum walk on a two-dimensional lattice
Phys. Rev. A 92, 062307 – Published 3 December, 2015
DOI: https://doi.org/10.1103/PhysRevA.92.062307
Abstract
A two-dimensional discrete-time quantum walk (DTQW) can be realized by alternating a two-state DTQW in one spatial dimension followed by an evolution in the other dimension. This was shown to reproduce a probability distribution for a certain configuration of a four-state DTQW on a two-dimensional lattice. In this work we present a three-state alternate DTQW with a parametrized coin-flip operator and show that it can produce localization that is also observed for a certain other configuration of the four-state DTQW and nonreproducible using the two-state alternate DTQW. We will present two limit theorems for the three-state alternate DTQW. One of the limit theorems describes a long-time limit of a return probability, and the other presents a convergence in distribution for the position of the walker on a rescaled space by time. We find that the spatial entanglement generated by the three-state alternate DTQW is higher than that by the four-state DTQW. Using all our results, we outline the relevance of these walks in three-level physical systems.