Irreducible quantum dynamical semigroups

Robert Alicki, K. Lendi

Franco Fagnola, Federico Girotti

We present different characterizations of the notion of irreducibility for Quantum Markov Semigroups (QMSs) and investigate its relationship with other relevant features of the dynamics, such as primitivity, positivity improvement and relaxation; in particular, we show that irreducibility, primitivity and relaxation towards a faithful invariant density are equivalent when the semigroup admits an invariant density. Moreover, in the case of uniformly continuous QMSs, we present several useful ways of checking irreducibility in terms of the operators appearing in the generator in GKLS form. Our exposition is as much self-contained as possible, we present some well known results with elementary proofs (collecting all the relevant literature) and we derive new ones. We study both finite and infinite dimensional evolutions and we remark that many results only require the QMS to be made of Schwarz maps.

Wawrzyniec Guz

Franco Fagnola, Federico Girotti

Abstract We present different characterizations of the notion of irreducibility for Quantum Markov Semigroups (QMSs) and investigate its relationship with other relevant features of the dynamics, such as primitivity, positivity improvement and relaxation; in particular, we show that irreducibility, primitivity and relaxation towards a faithful invariant density are equivalent when the semigroup admits an invariant density. Moreover, in the case of uniformly continuous QMSs, we present several useful ways of checking irreducibility in terms of the operators appearing in the generator in GKLS form. Our exposition is as much self-contained as possible, we present some well known results with elementary proofs (collecting all the relevant literature) and we derive new ones. We study both finite and infinite dimensional evolutions and we remark that many results only require the QMS to be made of Schwarz maps.

Alaa Awad Alzulaibani

In this thesis, we construct a class of quantum dynamical semigroups, called quasifree quantum dynamical semigroups, of the von Neumann algebra generated by an arbitrary Weyl system, with respect to some compatible families of quasi-measures and C0-semigroups. We then describe in concrete terms their associated subproduct systems, in the case of irreducible Weyl systems.

Robert Alicki, K. Lendi