Noncommutative Geometry Seminar

We begin with a brief review of important results related to the strong subadditivity of entropy. We show several routes to the proof of the convexity of relative entropy and the monotonicity of Renyi's entropy. One route uses the triple matrix inequality, which will play a role in the sequel, namely the quantum version of the uncertainty principle, which extends the classical Maassen-Uffink inequality. The next act in the play concerns the sandwiched Renyi entropies and their monotonicity under CPTP maps. We then go on to the \alpha-z entropies, their monotonicity and their open problems. Finally, there are some new concavity/convexity theorems extending work of Hiai.

The talk is based on joint works with Eric Carlen and with Elliott Lieb.