The Stationary Measure for Diagonal Quantum Walk with One Defect

Abstract

This study is motivated by previous work [14]. We treat 3 types of one-dimensional quantum walks (QWs), whose time evolutions are described by diagonal unitary matrix, and diagonal unitary matrices with one defect. In this paper, we call QW defined by diagonal unitary matrices, the diagonal and we consider stationary distributions of generally 2-state diagonal QW with one defect, 3-state space-homogeneous diagonal QW, and 3-state diagonal QW with one defect. One of purposes of our study is to characterize QWs by stationary measure, which may lead to answer basic and natural question, What stationary measure is for one-dimensional QWs ?. In order to analyze stationary distribution, we focus on corresponding eigenvalue problems and definition of stationary measure.