Central limit theorems for open quantum random walks on the crystal lattices

TL;DR

This paper extends the central limit theorem for open quantum random walks from integer lattices to crystal lattices, including hexagonal examples, and develops Fourier analysis tools for these structures.

Contribution

It proves the CLT for open quantum random walks on crystal lattices and introduces Fourier analysis and dual processes for these walks, advancing the mathematical framework.

Findings

CLT established for walks on crystal lattices

Fourier analysis developed for these structures

Dual processes constructed for open quantum random walks

Abstract

We consider the open quantum random walks on the crystal lattices and investigate the central limit theorems for the walks. On the integer lattices the open quantum random walks satisfy the central limit theorems as was shown by Attal, {\it et al}. In this paper we prove the central limit theorems for the open quantum random walks on the crystal lattices. We then provide with some examples for the Hexagonal lattices. We also develop the Fourier analysis on the crystal lattices. This leads to construct the so called dual processes for the open quantum random walks. It amounts to get Fourier transform of the probability densities, and it is very useful when we compute the characteristic functions of the walks. In this paper we construct the dual processes for the open quantum random walks on the crystal lattices providing with some examples.