Limit distributions of three-state quantum walks: the role of coin eigenstates

TL;DR

This paper investigates how the initial coin state, especially its coherence, influences the long-term behavior of three-state quantum walks with localization, using eigenbasis decomposition for clearer analysis.

Contribution

It introduces a simplified eigenbasis approach to analyze the limit distributions and the role of coherence in three-state quantum walks.

Findings

Limit distributions are more easily expressed in the eigenbasis.

Even moments and localization at the origin depend only on incoherent probabilities.

Odd moments and localization outside the origin depend on initial coherence.

Abstract

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.