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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 740816, 12 pages
http://dx.doi.org/10.1155/2011/740816
Research Article

On Limiting Distributions of Quantum Markov Chains

Department of Mathematics, Bowie State University, 14000 Jericho Park Road, Bowie, MD 20715, USA

Received 15 April 2011; Accepted 15 June 2011

Academic Editor: Pei Yuan Wu

Copyright © 2011 Chaobin Liu and Nelson Petulante. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In a quantum Markov chain, the temporal succession of states is modeled by the repeated action of a “bistochastic quantum operation” on the density matrix of a quantum system. Based on this conceptual framework, we derive some new results concerning the evolution of a quantum system, including its long-term behavior. Among our findings is the fact that the Cesàro limit of any quantum Markov chain always exists and equals the orthogonal projection of the initial state upon the eigenspace of the unit eigenvalue of the bistochastic quantum operation. Moreover, if the unit eigenvalue is the only eigenvalue on the unit circle, then the quantum Markov chain converges in the conventional sense to the said orthogonal projection. As a corollary, we offer a new derivation of the classic result describing limiting distributions of unitary quantum walks on finite graphs (Aharonov et al., 2001).