Dynamics and Quantum Optimal Transport: Three Lectures on Quantum Entropy and Quantum Markov Semigroups

This paper presents the contents of three lectures delivered by the author at the Erdős Center School “Optimal Transport on Quantum Structures”, September 19–23, 2022 in Budapest, Hungary (Lecture One: Quantum Entropy Inequalities; Lecture Two: Reversible QMS as Gradient flow for Relative Entropy; Lecture Three: Applications of the Gradient Flow Structure). It presents a fairly self contained account of an active topic of current research, and this account should be accessible to most graduate students, as befits lectures for a school. The main results are known, but there are a number of new proofs and some new results.