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arXiv:1510.03454 · Journal of Statistical Physics

Open Quantum Random Walks: Ergodicity, Hitting Times, Gambler’s Ruin and Potential Theory

Carlos F. Lardizabal, Rafael R. Souza

Published Jul 11, 2016• 21 citations
arXivHTMLPDF
DOI
10.1007/s10955-016-1578-9

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We make use of the Open Quantum Random Walk setting due to S. Attal, F. Petruccione, C. Sabot and I. Sinayskiy in order to discuss hitting times and a quantum version of the Mean Hitting Time Formula from classical probability theory. We study an open quantum notion of hitting probability on a finite collection of sites and with this we are able to describe the problem in terms of linear maps and its matrix representations. After setting an open quantum version of the fundamental matrix for ergodic Markov chains we are able to prove our main result and as consequence a version of the Random Target Lemma. We also study a mean hitting time formula in terms of the minimal polynomial associated to the matrix representation of the quantum walk. We discuss applications of the results to open quantum dynamics on graphs together with open questions.