Limit theorems for open quantum random walks

TL;DR

This paper studies the long-term behavior of open quantum random walks on a line, introducing a dual process to compute their limit distributions explicitly, independent of initial states.

Contribution

It introduces a dual process analogous to Schrödinger-Heisenberg representation, enabling explicit computation of limit distributions for open quantum walks.

Findings

Explicit formulas for limit distributions of quantum walks

Method to eliminate initial state dependence

Application to various example walks

Abstract

We consider the limit distributions of open quantum random walks on one-dimensional lattice space. We introduce a dual process to the original quantum walk process, which is quite similar to the relation of Schr\"odinger-Heisenberg representation in quantum mechanics. By this, we can compute the distribution of the open quantum random walks concretely for many examples and thereby we can also obtain the limit distributions of them. In particular, it is possible to get rid of the initial state when we consider the evolution of the walk, it appears only in the last step of the computation.