Algebraic theory of quantum diffusions

David Applebaum

RL Hudson

Wojciech De Roeck

Shahn Majid

Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more 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 motivated from the theory of Kac algebras: If H is any Hopf algebra, it may be realized in Lin(H) in such a way that Δh=W(h⊗1)W−1 for an operator W. This W is interpreted as the time evolution operator for the system at time t coupled quantum-mechanically to the system at time t+δ. Finally, for every Hopf algebra there is a dual one, leading us to a duality operation for quantum random walks and quantum diffusions 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.

J. Amran Sussmann

M. Evans, R. L. Hudson

Quantum diffusions are non-commutative generalisations of classical diffusions on manifolds, in which the manifold i s ‘quantised’ by deforming the algebra of smooth functions thereon to a non-commutative algebra, and in which quantum stochastic calculus replaces the Itô calculus of Brownian motion. Quantum diffusions fall into two categories; inner diffusions, which are in a certain sense cohomologically trivial, and which act by conjugating by a unitary process, and outer diffusions, which do not. A natural perturbation procedure is described for linking two cohomologically equivalent quantum diffusions; it involves the construction of a new type of intertwining unitary process.