Limit theorems of a 3-state quantum walk and its application for discrete uniform measures

TL;DR

This paper establishes long-time limit theorems for a 3-state quantum walk on the line, providing insights into its probability distribution and convergence behavior, and applies these results to derive discrete uniform measures from specific initial states.

Contribution

It introduces two new limit theorems for a 3-state quantum walk and applies them to obtain uniform measures from delocalized initial states.

Findings

Derived limit measure for the walk's probability distribution

Proved convergence in distribution for the walker's position

Obtained discrete uniform limit measures from specific initial states

Abstract

We present two long-time limit theorems of a 3-state quantum walk on the line when the walker starts from the origin. One is a limit measure which is obtained from the probability distribution of the walk at a long-time limit, and the other is a convergence in distribution for the walker's position in a rescaled space by time. In addition, as an application of the walk, we obtain discrete uniform limit measures from the 3-state walk with a delocalized initial state.