The Limiting Distribution of Decoherent Quantum Random Walks

Abstract

The behaviors of one-dimensional quantum random walks are strikingly different from those of classical ones. However, when decoherence is involved, the limiting distributions take on many classical features over time. In this paper, we study the decoherence on both position and ``coin'' spaces of the particle. We propose a new analytical approach to investigate these phenomena and obtain the generating functions which encode all the features of these walks. Specifically, from these generating functions, we find exact analytic expressions of several moments for the time and noise dependence of position. Moreover, the limiting position distributions of decoherent quantum random walks are shown to be Gaussian in an analytical manner. These results explicitly describe the relationship between the system and the level of decoherence.

38 pages. 3 figures. Based on a Ph.D. thesis at Temple University

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• Classical-like behavior in quantum walks with inhomogeneous, time-dependent coin operators
• Tunneling effects in a one-dimensional quantum walk
• Analytical expression for variance of homogeneous-position quantum walk with decoherent position
• The localization of quantum random walks on sierpinski gaskets
• Directional quantum random walk induced by coherence
• On the von Neumann entropy of certain quantum walks subject to decoherence
• Time-inhomogeneous Quantum Walks with Decoherence on Discrete Infinite Spaces
• Limit theorems for the interference terms of discrete-time quantum walks on the line
• Time-inhomogeneous Quantum Markov Chains with Decoherence on Finite State Spaces