Quantum dynamical semigroups, group representations and convolution semigroups

Abstract

Semigroups of operators are known to play an important role in theoretical physics. In particular, quantum dynamical semigroups are fundamental in the theory of open quantum systems. We will describe a class of semigroups of operators which has interesting applications, for instance, in quantum information science. Each of these semigroups of operators is generated, in a suitable way, by a representation (or an antirepresentation) of a group in a Banach space and by a convolution semigroup of probability measures on that group. Some significant examples—including a remarkable type of quantum dynamical semigroups introduced by Kossakowski in the pioneering times of the theory of open quantum systems—and their mutual relations will be discussed.