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arXiv:2111.14300 · Journal of Physics A Mathematical and Theoretical

Localization of space-inhomogeneous three-state quantum walks

Chusei Kiumi

Published Apr 22, 2022• 4 citations
arXivHTMLPDF

Abstract

Abstract Mathematical analysis on the existence of eigenvalues is essential because it is deeply related to localization, which is an exceptionally crucial property of quantum walks (QWs). We construct the method for the eigenvalue problem via the transfer matrix for space-inhomogeneous three-state QWs in one dimension with a self-loop, which is an extension of the technique in a previous study (Kiumi and Saito 2021 Quantum Inf. Process. 20 171). This method reveals the necessary and sufficient condition for the eigenvalue problem of a two-phase three-state QW with one defect whose time evolution varies in the negative part, positive part, and at the origin.

DOI
10.1088/1751-8121/ac6998

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Abstract Mathematical analysis on the existence of eigenvalues is essential because it is deeply related to localization, which is an exceptionally crucial property of quantum walks (QWs). We construct the method for the eigenvalue problem via the transfer matrix for space-inhomogeneous three-state QWs in one dimension with a self-loop, which is an extension of the technique in a previous study (Kiumi and Saito 2021 Quantum Inf. Process. 20 171). This method reveals the necessary and sufficient condition for the eigenvalue problem of a two-phase three-state QW 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 nnn-state quantum walks in one dimension with n−2n-2n−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 of the spectral properties of the time-evolution operator in quantum walks is essential for understanding key dynamical behaviors such as localization and long-term evolution. The inhomogeneous three-state case, in particular, poses substantial analytical challenges due to its higher internal degrees of freedom and the absence of translational invariance. We develop a general framework for the spectral analysis of three-state quantum walks on the one-dimensional lattice with arbitrary time-evolution operators. Our approach is based on a transfer-matrix formulation that reduces the infinite-dimensional eigenvalue problem to a tractable system of two-dimensional recursions, enabling exact characterization of eigenstates. This framework applies broadly to space-inhomogeneous models, including those with finite defects and two-phase structures. We rigorously derive necessary and sufficient conditions for the existence of point spectrum, along with a complete description of the corresponding eigenvalues and eigenstates, which are known to underlie quantum localization phenomena. Furthermore, we give a complete spectral decomposition—discrete spectrum, flat-band eigenvalues (of infinite multiplicity), and absolutely continuous spectrum—with explicit characterization of each component. Using this method, we perform exact numerical analyses of the Fourier walk with spatial inhomogeneity, revealing the emergence of localization despite its delocalized nature in the homogeneous case. Our results provide mathematical tools and physical insights into the structure of quantum walks, offering a systematic path for identifying and characterizing localized quantum states in complex quantum systems.
There is a property called localization, which is essential for applications of quantum walks. From a mathematical point of view, the occurrence of localization is known to be equivalent to the existence of eigenvalues of the time evolution operators, which are defined by coin matrices. A previous study proposed an approach to the eigenvalue problem for space-inhomogeneous models using transfer matrices. However, the approach was restricted to models whose coin matrices are the same in positions sufficiently far to the left and right, respectively. This study shows that the method can be applied to extended models with periodically arranged coin matrices. Moreover, we investigate localization by performing the eigenvalue analysis and deriving their time-averaged limit distribution.
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.
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.