Suitable bases for quantum walks with Wigner coins

TL;DR

This paper simplifies the analysis of Wigner quantum walks by identifying a suitable basis, revealing hidden dynamical regimes, and providing explicit results up to five dimensions, including effects like trapping and peak reduction.

Contribution

It introduces a basis transformation that simplifies the limit density of Wigner walks and uncovers new dynamical behaviors not visible in the standard basis.

Findings

Limit density depends on only one physical parameter.

A basis change simplifies the limit density expression.

Integer j models exhibit trapping effects with asymmetric probabilities.

Abstract

The analysis of a physical problem simplifies considerably when one uses a suitable coordinate system. We apply this approach to the discrete-time quantum walks with coins given by $2j+1$-dimensional Wigner rotation matrices (Wigner walks), a model which was introduced in T. Miyazaki et al., Phys. Rev. A 76, 012332 (2007). First, we show that from the three parameters of the coin operator only one is physically relevant for the limit density of the Wigner walk. Next, we construct a suitable basis of the coin space in which the limit density of the Wigner walk acquires a much simpler form. This allows us to identify various dynamical regimes which are otherwise hidden in the standard basis description. As an example, we show that it is possible to find an initial state which reduces the number of peaks in the probability distribution from generic $2j+1$ to a single one. Moreover, the models with integer $j$ lead to the trapping effect. The derived formula for the trapping probability reveals that it can be highly asymmetric and it deviates from purely exponential decay. Explicit results are given up to the dimension five.