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arXiv:2310.08898

Stationary Measures of Space-Inhomogeneous Three-State Quantum Walks on Line: Revisited

Shengsheng Liu, Caishi Wang, Jijun Zhao

Published Oct 13, 2023• 0 citations
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Abstract

Of a quantum walk, its stationary measures play an important role in understanding its evolution behavior. In this paper we investigate stationary measures of two models of space-inhomogeneous three-state quantum walk on the line. By using the method of reduced matrix, we find out stationary measures of the two models under some mild conditions. Our results generalize the corresponding ones existing in the literature.

DOI
10.48550/arxiv.2310.08898

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Of a quantum walk, its stationary measures play an important role in understanding its evolution behavior. In this paper we investigate stationary measures of two models of space-inhomogeneous three-state quantum walk on the line. By using the method of reduced matrix, we find out stationary measures of the two models under some mild conditions. Our results generalize the corresponding ones existing in the literature.
The discrete-time quantum walk (QW) is a quantum version of the random walk (RW) and has been widely investigated for the last two decades. Some remarkable properties of QW are well known. For example, QW has a ballistic spreading, i.e., QW is quadratically faster than RW. For some cases, localization occurs: a walker stays at the starting position forever. In this paper, we consider stationary measures of two-state QWs on the line. It was shown that for any space-homogeneous model, the uniform measure becomes the stationary measure. However, the corresponding result for space-inhomogeneous model is not known. Here, we present a class of space-inhomogeneous QWs on the line and cycles in which the uniform measure is stationary. Furthermore, we briefly discuss a difference between QWs and RWs.
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.
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.