The non-uniform stationary measure for discrete-time quantum walks in one dimension

TL;DR

This paper explores stationary measures of one-dimensional discrete-time quantum walks, revealing that non-uniform measures are stationary except when the coin matrix is diagonal, where only uniform measures are stationary.

Contribution

It demonstrates that non-uniform stationary measures exist for non-diagonal coin matrices and provides a simple proof that uniform measures are stationary in more general quantum walks.

Findings

Non-uniform measures are stationary for non-diagonal coin matrices.

Diagonal matrices only admit uniform stationary measures.

Uniform measures are stationary in broader quantum walk models.

Abstract

We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 times 2 unitary matrix U. In our previous paper [15], we proved that any uniform measure becomes the stationary measure of the QW by solving the corresponding eigenvalue problem. This paper reports that non-uniform measures are also stationary measures of the QW except U is diagonal. For diagonal matrices, we show that any stationary measure is uniform. Moreover, we prove that any uniform measure becomes a stationary measure for more general QWs not by solving the eigenvalue problem but by a simple argument.