Open quantum random walks with decoherence on coins with $n$ 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.

20 pages, no figures

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• Decoherence in quantum walks - a review
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• Microscopic derivation of Open Quantum Walks
• Site recurrence of open and unitary quantum walks on the line
• Unusual scaling in a discrete quantum walk with random long range steps
• The Open Quantum Brownian Motion and continual measurements
• The localization of quantum random walks on sierpinski gaskets