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Infinite Dimensional Analysis Quantum Probability and Related Topics

One-dimensional three-state quantum walks: Weak limits and localization

Chul Ki Ko, Etsuo Segawa, Hyun Jae Yoo

Published Nov 23, 2016• 6 citations
Publisher

Abstract

We investigate one-dimensional three-state quantum walks. We find a formula for the moments of the weak limit distribution via a vacuum expectation of powers of a self-adjoint operator. We use this formula to fully characterize the localization of three-state quantum walks in one dimension. The localization is also characterized by investing the eigenvectors of the evolution operator for the quantum walk. As a byproduct we clarify the concepts of localization differently used in the literature. We also study the continuous part of the limit distribution. For typical examples we show that the continuous part is the same kind as that of two-state quantum walks. We provide with explicit expressions for the density of the weak limits of some three-state quantum walks.

DOI
10.1142/s0219025716500259

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We investigate one-dimensional three-state quantum walks. We find a formula for the moments of the weak limit distribution via a vacuum expectation of powers of a self-adjoint operator. We use this formula to fully characterize the localization of three-state quantum walks in one dimension. The localization is also characterized by investing the eigenvectors of the evolution operator for the quantum walk. As a byproduct we clarify the concepts of localization differently used in the literature. We also study the continuous part of the limit distribution. For typical examples we show that the continuous part is the same kind as that of two-state quantum walks. We provide with explicit expressions for the density of the weak limits of some three-state quantum walks.
We analyze two families of three-state quantum walks which show the localization effect. We focus on the role of the initial coin state and its coherence in controlling the properties of the quantum walk. In particular, we show that the description of the walk simplifies considerably when the initial coin state is decomposed in the basis formed by the eigenvectors of the coin operator. This allows us to express the limit distributions in a much more convenient form. Consequently, striking features which are hidden in the standard basis description are easily identified. Moreover, the dependence of moments of the position distribution on the initial coin state can be analyzed in full detail. In particular, we find that in the eigenvector basis the even moments and the localization probability at the origin depend only on incoherent combination of probabilities. In contrast, odd moments and localization outside the origin are affected by the coherence of the initial coin state.
Quantum walks behave differently from what we expect and their probability distributions have unique structures. They have localization, singularities, a gap, and so on. Those features have been discovered from the view point of mathematics and reported as limit theorems. In this paper we focus on a time-dependent three-state quantum walk on the line and demonstrate a limit distribution. Three coin states at each position are iteratively updated by a coin-flip operator and a position-shift operator. As the result of the evolution, we end up to observe both localization and a gap in the limit distribution.
We treat quantum walk (QW) on the line whose quantum coin at each vertex tends to the identity as the distance goes to infinity. We obtain a limit theorem that this QW exhibits localization with not an exponential but a ``power-law" decay around the origin and a ``strongly" ballistic spreading called bottom localization in this paper. This limit theorem implies the weak convergence with linear scaling whose density has two delta measures at x=0x=0x=0 (the origin) and x=1x=1x=1 (the bottom) without continuous parts.
A two-dimensional discrete-time quantum walk (DTQW) can be realized by alternating a two-state DTQW in one spatial dimension followed by an evolution in the other dimension. This was shown to reproduce a probability distribution for a certain configuration of a four-state DTQW on a two-dimensional lattice. In this work we present a three-state alternate DTQW with a parametrized coin-flip operator and show that it can produce localization that is also observed for a certain other configuration of the four-state DTQW and nonreproducible using the two-state alternate DTQW. We will present two limit theorems for the three-state alternate DTQW. One of the limit theorems describes a long-time limit of a return probability, and the other presents a convergence in distribution for the position of the walker on a rescaled space by time. We find that the spatial entanglement generated by the three-state alternate DTQW is higher than that by the four-state DTQW. Using all our results, we outline the relevance of these walks in three-level physical systems.