Properties of Open Quantum Walks on $\mathbb{Z}$

Abstract

A connection between the asymptotic behavior of the open quantum walk and the spectrum of a generalized quantum coins is studied. For the case of simultaneously diagonalizable transition operators an exact expression for probability distribution of the position of the walker for arbitrary number of steps is found. For a large number of steps the probability distribution consist of maximally two "soliton"-like solution and a certain number of Gaussian distributions. The number of different contributions to the final probability distribution is equal to the number of distinct absolute values in the spectrum of the transition operators. The presence of the zeros in spectrum is an indicator of the "soliton"-like solutions in the probability distribution.

4 pages, 2 figures; contribution to FQMT'11: Frontiers of Quantum and Mesoscopic Thermodynamics (Prague, Czech Republic, 25-30 July 2011)

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• Open Quantum Walks: a short introduction
• Central limit theorem for reducible and irreducible open quantum walks
• Microscopic derivation of Open Quantum Walks
• Quantum Hidden Markov Models based on Transition Operation Matrices
• Generalized open quantum walks on Apollonian networks
• On a generalized Central Limit Theorem and Large Deviations for Homogeneous Open Quantum Walks
• Lazy Open Quantum Walks
• Phase transition of an open quantum walk
• Attractors and asymptotic dynamics of open discrete-time quantum walks on cycles