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arXiv:1110.3990

Quantum random walk approximation on locally compact quantum groups

J. Martin Lindsay, Adam Skalski

Published Oct 18, 2011• 0 citations
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Abstract

A natural scheme is established for the approximation of quantum Levy processes on locally compact quantum groups by quantum random walks. We work in the somewhat broader context of discrete approximations of completely positive quantum stochastic convolution cocycles on C*-bialgebras.
DOI
10.48550/arxiv.1110.3990

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We give a simple and direct treatment of the convergence of quantum random walks to quantum stochastic operator cocycles, using the semigroup method. The pointwise product of two such quantum random walks is shown to converge to the quantum stochastic Trotter product of the respective limit cocycles. Since such Trotter products themselves reduce to pointwise products when the cocycles inhabit commuting subspaces of the system algebra, this yields an elementary approach to the quantum random walk approximation of the 'tensorisation' of cocycles with common noise dimension space. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.
We summarise recent results concerning quantum stochastic convolution cocycles in two contexts—purely algebraic and C∗C^*C∗-algebraic. In each case the class of cocycles arising as the solution of a quantum stochastic differential equation is charact