Curvature-dimension conditions for symmetric quantum Markov semigroups

The curvature-dimension condition consists of the lower Ricci curvature bound and upper dimension bound of the Riemannian manifold, which has a number of geometric consequences and is very helpful in proving many functional inequalities. The Bakry-Émery theory and Lott-Sturm-Villani theory allow to extend this notion beyond the Riemannian manifold setting and have seen great progress in the past decades. In this talk, I will first review several notions around lower Ricci curvature bounds in the noncommutative setting and present our work on gradient estimates. Then I will speak about two noncommutative versions of curvature-dimension conditions. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet-Myers theorem, and concavity of entropy power in the noncommutative setting. I will also give some examples coming from noncommutative analysis, quantum information and noncommutative probability. This is based on joint work (arXiv:2007.13506, arXiv:2105.08303) with Melchior Wirth (IST Austria).