Distribution of chirality in the quantum walk: Markov process and entanglement

TL;DR

This paper analyzes the long-term behavior of quantum walks on a line, focusing on chirality distribution and entanglement, revealing their dependence on initial conditions and their relation to Markovian randomness.

Contribution

It demonstrates the existence of a long-time limit for chirality distribution in quantum walks and links entanglement to the distribution's Markovian properties.

Findings

Long-time chirality distribution depends on initial conditions.

Entanglement correlates with the degree of Markovian randomness.

Initial conditions can be reconstructed from asymptotic distributions.

Abstract

The asymptotic behavior of the quantum walk on the line is investigated focusing on the probability distribution of chirality independently of position. The long-time limit of this distribution is shown to exist and to depend on the initial conditions, and it also determines the asymptotic value of the entanglement between the coin and the position. It is shown that for given asymptotic values of both the entanglement and the chirality distribution it is possible to find the corresponding initial conditions within a particular class of spatially extended Gaussian distributions. Moreover it is shown that the entanglement also measures the degree of Markovian randomness of the distribution of chirality.