Classification and decomposition of Quantum Markov Semigroups

Abstract

We show that a QMS on a σ-finite von Neumann algebra A can be decomposed as the sum of several “sub”-semigroups corresponding to transient and recurrent projections. We discuss two applications to physical models

Irreducible and periodic positive maps

“…it is an ideal of . The appropriate extension to the non-commutative framework, however, is still the notion of hereditary * -subalgebra as it is clear from [6] and the decomposition of quantum Markov semigroups developed in [13].…”

A Survey on Invariance and Ergodicity of Quantum Markov Semigroups

“…The remaining part of this subsection will be devoted to the study of the structure of the infinitesimal generator of a quantum dynamical semigroup, which is already revealed by the examples in the previous section. See Umanita [68] and [69] We introduce first a notion related to conditional complete positivity. …”

“…Recent contributions in this area include works done by Carbone [8], Luczak [50], Fagnola [31], Rebolledo [56] or both (see [32][33][34][35][36]) and recent results obtained by Choi [10], and Umanita [68,69] and Emel'syanov and Wolff [25]. The main purpose of this article is to explore and survey recent results on invariances, mean ergodicity and mean ergodicity of the semigroup {T * t , t ≥ 0}.…”

“…The main purpose of this article is to explore and survey recent results on invariances, mean ergodicity and mean ergodicity of the semigroup {T * t , t ≥ 0}. The material presented in this survey is largely based on the works done by (see [32][33][34]) and recent results obtained by Umanita [68,69] and Emel'syanov and Wolff [25].…”

Van Hove Limit for Infinite Systems

“…Although phenomenological markovian laws for such systems have been studied (see e.g. [8][9][10]), a general recipe to construct a proper markovian generator in the van Hove limit, given the hamiltonian perturbation, is lacking.…”