Quantum Random Walk Approximation on Locally Compact Quantum Groups

Abstract

Abstract.A natural scheme is established for the approximation of quantum Lévy 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. Mathematics Subject Classification (2000). Primary 46L53, 81S25; Secondary 22A30, 47L25, 16W30.Keywords. quantum random walk, quantum Lévy process, noncommutative probability, locally compact quantum group, C * -bialgebra, stochastic cocycle. IntroductionIn [19] we developed a theory of quantum stochastic convolution cocycles on counital multiplier C * -bialgebras, extending the algebraic theory of quantum Lévy processes created by Schürmann and co-workers (see [25] and references therein, and, for a simplified treatment [17]), and the topological theory of quantum stochastic convolution cocycles on compact quantum groups and operator space coalgebras developed by the authors [18]. Here we apply the results of [19] to introduce and analyse a straightforward scheme for the approximation of such cocycles by quantum random walks. In particular we obtain results on Markovregular quantum Lévy processes on locally compact quantum semigroups, extending and strengthening results in [11] for the compact case. Our analysis exploits a recent approximation theorem of Belton [6], which extends that of [24] (used in [11]). The approximation scheme closely mirrors the way in which Picard iteration operates in the construction of solutions of quantum stochastic differential equations [15].The study of quantum random walks on quantum groups was initiated by Biane in the early 1990s (starting with [7]). Some combinatorial, probabilistic and physical