Limit measures of inhomogeneous discrete-time quantum walks in one dimension

TL;DR

This paper analyzes inhomogeneous discrete-time quantum walks in one dimension, focusing on various measures such as limit, weak, and stationary measures, revealing coexistence of ballistic and localized behaviors and proposing a universality class.

Contribution

It introduces a universality class of quantum walks based on weak limit measures, including walks with defects, and characterizes stationary measures for these walks.

Findings

Coexistence of ballistic and localized behaviors in quantum walks.

Typical homogeneous walks belong to the proposed universality class.

Stationary measures include the uniform measure and time-averaged limit measures.

Abstract

We treat three types of measures of the quantum walk (QW) with the spatial perturbation at the origin, which was introduced by [1]: time averaged limit measure, weak limit measure, and stationary measure. From the first two measures, we see a coexistence of the ballistic and localized behaviors in the walk as a sequential result following [1,2]. We propose a universality class of QWs with respect to weak limit measure. It is shown that typical spatial homogeneous QWs with ballistic spreading belong to the universality class. We find that the walk treated here with one defect also belongs to the class. We mainly consider the walk starting from the origin. However when we remove this restriction, we obtain a stationary measure of the walk. As a consequence, by choosing parameters in the stationary measure, we get the uniform measure as a stationary measure of the Hadamard walk and a time averaged limit measure of the walk with one defect respectively.