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Infinite Dimensional Analysis Quantum Probability and Related Topics

INTERACTING FOCK SPACES AND THE MOMENTS OF THE LIMIT DISTRIBUTIONS FOR QUANTUM RANDOM WALKS

Chul Ki Ko, Hyun Jae Yoo

Published Mar 1, 2013• 3 citations
Publisher

Abstract

We investigate the limit distributions of the discrete time quantum random walks on lattice spaces via a spectral analysis of concretely given self-adjoint operators. We discuss the interacting Fock spaces associated with the limit distributions. Thereby, we represent the moments of the limit distribution by vacuum expectation of the monomials of the Fock operator. We get formulas not only for one-dimensional walks but also for high-dimensional walks.

DOI
10.1142/s0219025713500033

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We investigate the limit distributions of the discrete time quantum random walks on lattice spaces via a spectral analysis of concretely given self-adjoint operators. We discuss the interacting Fock spaces associated with the limit distributions. Thereby, we represent the moments of the limit distribution by vacuum expectation of the monomials of the Fock operator. We get formulas not only for one-dimensional walks but also for high-dimensional walks.
We investigate one-dimensional three-state quantum walks. We find a formula for the moments of the weak limit distribution via a vacuum expectation of powers of a self-adjoint operator. We use this formula to fully characterize the localization of three-state quantum walks in one dimension. The localization is also characterized by investing the eigenvectors of the evolution operator for the quantum walk. As a byproduct we clarify the concepts of localization differently used in the literature. We also study the continuous part of the limit distribution. For typical examples we show that the continuous part is the same kind as that of two-state quantum walks. We provide with explicit expressions for the density of the weak limits of some three-state quantum walks.
The behavior of one-dimensional quantum random walks is strikingly different from that 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 an 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.
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We introduce a kind of quantum decomposition of generators of a discrete group with a suitable length function and construct a sequence of one-mode interacting Fock spaces associated with a filtration of the group. We show stochastic convergence of such a sequence of interacting Fock spaces and obtain a general stochastic limit theorems on discrete groups. As an application to free groups, we see that the Haagerup states give rise to a Gaussian–Poisson transform through its coherent expression.