Workshop in Analysis and Probability Seminar

Date: July 28, 2014

Time: 11:00AM - 12:00PM

Location: Blocker 163

Speaker: Matthew Ziemke, University of South Carolina

Title: Generators of Quantum Markov Semigroups

Abstract: Quantum Markov Semigroups (QMSs) originally arose in the study of the evolutions of irreversible open quantum systems. Mathematically, they are a generalization of classical Markov semigroups where the underlying function space is replaced by a non-commutative operator algebra. In the case when the QMS is uniformly continuous, theorems due to Lindblad, Stinespring, and Kraus imply that the generator of the semigroup has the form $$L(A)= \sum_{n=1}^{\infty} V_n^*AV_n+GA+AG^* $$ where $V_n$ and G are elements of the underlying operator algebra. The form of the generator of a general QMS acting on the bounded operators of a Hilbert space remained open since 1976. In a recent work with George Androulakis we proved the generators of general QMSs (not necessarily uniformly continuous) must also satisfy the form given by Lindblad and Stinespring. In this talk I will explain these results and present some examples in order to clarify these findings.