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

arXiv:1603.08948Interdisciplinary Information SciencesJan 1, 20176 citations

Edge states in a two-dimensional quantum walk with disorder

Alberto Verga

arXiv:1606.00613The European Physical Journal BMar 1, 201712 citations

Eigenvalues of two-phase three-state quantum walks with one defect

Chusei Kiumi

Mathematical analysis on the existence of eigenvalues is essential because it is equivalent to the occurrence of localization, which is an exceptionally crucial property of quantum walks. We construct the method for eigenvalue problem via the transfer matrix for space-inhomogeneous $n$ -state quantum walks in one dimension with $n-2$ self-loops, which is an extension of the technique in a previous study (Quantum Inf. Process 20(5), 2021). By this method, we reveal the necessary and sufficient condition for the eigenvalue problem of a two-phase three-state quantum walk with one defect whose time evolution varies in the negative part, positive part, and at the origin.Mathematical analysis on the existence of eigenvalues is essential because it is equivalent to the occurrence of localization, which is an exceptionally crucial property of quantum walks. We construct the method for eigenvalue problem via the transfer matrix for space-inhomogeneous $n$-state quantum walks in one dimension with $n-2$ self-loops, which is an extension of the technique in a previous study (Quantum Inf. Process 20(5), 2021). By this method, we reveal the necessary and sufficient condition for the eigenvalue problem of a two-phase three-state quantum walk with one defect whose time evolution varies in the negative part, positive part, and at the origin.

Nov 29, 20211 citation

n

-state quantum walks in one dimension with

n-2

self-loops, which is an extension of the technique in a previous study (Quantum Inf. Process 20(5), 2021). By this method, we reveal the necessary and sufficient condition for the eigenvalue problem of a two-phase three-state quantum walk with one defect whose time evolution varies in the negative part, positive part, and at the origin.