Curvature-dimension in noncommutative analysis

The curvature-dimension condition of a Riemannian manifold consists of the lower Ricci curvature bound and upper dimension bound. Apart from some geometric consequences, this geometric condition also has many applications in analysis. These include a number of functional inequalities such as Poincaré inequality and logarithmic Sobolev inequality, which play an important role in probability, optimal transport, mathematical physics, and many other areas. Two classical theories that allow us to extend this curvature-dimension condition beyond the Riemannian setting are the Bakry-Émery theory that uses diffusion Markov semigroups, and Lott-Sturm-Villani theory that relies on optimal transport. The past decades have witnessed great progress of both theories, and their extensions to more general frameworks remain still an active research direction. The aim of this project is to investigate the quantized curvature-dimension condition and its applications. In the quantized analysis, functions are replaced by operators that do not commute in general. This brings new difficulties, to overcome which we need new tools and ideas. In recent years, several notions around curvature-dimension conditions in the noncommutative setting have emerged. In this project, we will study their relations, properties, and apply them to examples coming from noncommutative analysis and quantum information theory.

The most significant results of this project are concerned with the analysis of the Boolean functions in the quantum world. In the classical setting, Boolean functions $f:\{-1,1\}^n\to \{-1,1\}$ are of great importance in theoretical computer science, social choice theory, and many other areas. One fundamental problem in the analysis of Boolean analysis is to understand the structure of Boolean functions with small complexity. Two basic complex measures are the influence and the degree, which can be represented using the Fourier spectrum. The Fourier analysis tools have played an essential role in studying Boolean functions with small complexity. In this project, the PI developed some new ideas and methods to tackle similar problems in the quantum realm, where functions are replaced with matrices that are not commutative in general. The difficulty in the quantum setting is the lack of the concepts of points or the non-commutativity of matrices. In the present project, the PI and his collaborators studied quantum analogs of Boolean functions with small influences or degrees. In particular, the PI and his collaborators obtained quantum analogs of a KKL theorem, Talagrand's inequality, and Friedgut's Junta Theorem, which are three fundamental results on the analysis of Boolean functions with small influences. The main tools come from the study of curvature-dimension conditions in the non-commutative analysis. This condition carries information about two important features of a geometric object, namely, the Ricci curvature lower bound and dimension upper bound of a Riemannian manifold. The study of this condition on more general (non-commutative) structures has seen great progress in past decades. The PI and his collaborators also extended a recent breakthrough on the sample complexity of learning Boolean functions of low degrees to the quantum setting. They discovered a connection between the Fourier analysis in the classical and quantum worlds, which allows them to reduce the problems in the quantum realm into their classical analogs.