Open quantum random walks with decoherence on coins with $n$ degrees of freedom

TL;DR

This paper introduces partially open quantum random walks (POQRW) that interpolate between unitary and open quantum walks, analyzing their limiting distributions under decoherence for coins with multiple degrees of freedom.

Contribution

It defines POQRW models with decoherence parameter, establishes conditions for their limiting distributions, and computes explicit distributions for specific coin dimensions.

Findings

Limiting distributions are convex combinations of normal distributions.

Eigenvalue conditions determine the convergence behavior.

Explicit distributions are derived for coins with 2 and 3 degrees of freedom.

Abstract

In this paper, we define a new type of decoherent quantum random walks with parameter $0\le p\le 1$, which becomes a unitary quantum random walk (UQRW) when $p=0$ and an open quantum random walk (OPRW) when $p=1$ respectively. We call this process a partially open quantum random walk (POQRW). We study the limiting distribution of a POQRW on $Z^1$ subject to decoherence on coins with $n$ degrees of freedom, which converges to a convex combination of normal distributions if the superoperator $\mathcal{L}_{kk}$ satisfies the eigenvalue condition, that is, 1 is an eigenvalue of $\mathcal{L}_{kk}$ with multiplicity one and all other eigenvalues have absolute values less than 1. A Perron-Frobenius type of theorem is provided in determining whether or not the superoperator satisfies the eigenvalue condition. Moreover, we compute the limiting distributions of characteristic equations of the position probability functions for $n=2$ and 3.