QUANTUM RANDOM-WALKS AND TIME-REVERSAL

Classical random walks and Markov processes are easily described by Ho pf algebras. It is also known that groups and Hopf algebras (quantum g roups) lead to classical and quantum diffusions. We study here the mor e primitive notion of a quantum random walk associated with a general Hopf algebra and show that it has a simple physical interpretation in quantum mechanics. This is by means of a representation theorem motiva ted from the theory of Kac algebras: If H is any Hopf algebra, it may be realized in Lin(H) in such a way that DELTAh = W(h x 1)W-1 for an o perator W. This W is interpreted as the time evolution operator for th e system at time t coupled quantum-mechanically to the system at time t + delta. Finally, for every Hopf algebra there is a dual one, leadin g us to a duality operation for quantum random walks and quantum diffu sions and a notion of the coentropy of an observable. The dual system has its time reversed with respect to the original system, leading us to a novel kind of CTP theorem.