A localized quantum walk with a gap in distribution

TL;DR

This paper investigates a specific three-state quantum walk on a line, revealing a limit distribution characterized by both localization and a gap, highlighting unique quantum behaviors not seen in classical walks.

Contribution

It introduces a time-dependent three-state quantum walk model and demonstrates the simultaneous occurrence of localization and a distribution gap in its limit behavior.

Findings

Observation of localization in the quantum walk

Identification of a gap in the limit distribution

Demonstration of unique quantum features in the model

Abstract

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.