A Survey on Invariance and Ergodicity of Quantum Markov Semigroups

Abstract

This article surveys recent results on invariances and ergodicity of general quantum Markov semigroups of bounded linear maps acting on C * -or von Neumann algebra A.In particular, we consider existence and uniqueness of invariant (stationary) quantum states as well as ergodicity and mean ergodicity of quantum states via heavy usage of the GNS representation. This survey is made self-contained by also reviewing relevant concepts and results necessary for the subsequent developments.

Stability Analysis of Quantum Systems: a Lyapunov Criterion and an Invariance Principle

“…The convergence of probability measures in metric spaces has been studied since the mid-1950's, starting from the seminal work of Prokhorov (Prokhorov, 1956;Parthasarathy, 1967). In fact, in the mathematical physics literature, the stability of quantum systems has been analyzed using semigroups and the quantum analog of probability measure convergence (Fagnola and Rebolledo, 2001, 2003Chang, 2014;Deschamps et al, 2016). In essence, (Fagnola and Rebolledo, 2001, 2002, 2003 establish conditions for the existence of invariant states and the convergence to these states given that the invariant state is faithful; i.e., for any positive operator A, tr (Aρ) = 0 if and only if A = 0.…”