The stationary measure of a space-inhomogeneous quantum walk on the line

TL;DR

This paper analyzes a specific quantum walk with a phase defect at the origin, deriving an explicit stationary measure that exhibits exponential decay and localization, using a novel method called the splitted generating function method.

Contribution

It introduces and applies the SGF method to explicitly determine the stationary measure for the Wojcik quantum walk model, confirming localization properties.

Findings

Stationary measure exhibits exponential decay.

Stationary measure is symmetric and depends on initial coin state.

Results agree with previous studies on the Wojcik model.

Abstract

We study a discrete-time quantum walk (QW) on the line with a single phase at the origin which was introduced and studied by Wojcik et al.[1]. We call the model "Wojcik model" here. Konno et al.[2] investigated other types of QWs with one defect at the origin. They presented a method which gives the stationary measure corresponding to localization for the QWs by use of the generating functions splitted in positive and negative parts respectively. In this paper, we call the method "the splitted generating function method (the SGF method)". To clarify in detail which QW is appropriate for the method in the case of study is one of the important challenges to investigate localization properties for various QWs. As for the Wojcik model, we solve the eigenvalue problem by the SGF method and our results agree with Ref.[1]. From the solution of the problem, we derive a stationary measure with an exponential decay for the position. The explicit expression for the stationary measure is symmetric for the origin and ensures localization depending on the initial coin state.