Asymptotic distributions of quantum walks on the line with two entangled coins

TL;DR

This paper investigates the asymptotic behavior of quantum walks on a line with two entangled coins, deriving limit distributions and analyzing localization effects influenced by initial states and coin parameters.

Contribution

It provides explicit formulas for stationary distributions and explores conditions under which the quantum walk exhibits localization, extending previous studies with new analytical results.

Findings

Weak limit distributions are derived for the quantum walks.

Stationary probability distributions depend on coin parameters and initial states.

Localization leads to exponential decay of stationary probability with position.

Abstract

We advance the previous studies of quantum walks on the line with two coins. Such four-state quantum walks driven by a three-direction shift operator may have nonzero stationary distributions (localization), thus distinguishing themselves from the quantum walks on the line in the basic scenario (i.e., driven by a single coin). In this work, asymptotic position distributions of the quantum walks are examined. We derive a weak limit for the quantum walks and explicit formulas of stationary probability distribution, whose dependencies on the coin parameter and the initial state of quantum walks are presented. In particular, it is shown that the weak limit for the present quantum walks can be of the form in the basic scenario of quantum walks on the line, for certain initial states of the walk and certain values of the coin parameter. In the case where localization occurs, we show that the stationary probability exponentially decays with the absolute value of a walker's position, independent of the parity of time.