(Seminar) Quantum modified logarithmic Sobolev inequalities

Abstract: Functional inequalities constitute by now a well-established theory with many connections to other fields of mathematics such as concentration of measure, mixing times of Markov processes or optimal transport to name only a few. Among these inequalities, the modified logarithmic Sobolev inequality (MLSI) controls the exponential entropic convergence of a Markov semigroup towards its stationary measure. Although introduced almost simultaneously, their quantum analogues have long suffered from the loss of certain key properties inherent to the passage to the non-commutative realm. Perhaps the most important of these is the tensorization property, which often allows one to prove a functional inequality for a Markov process on an uncountable state space by reduction to the two-points space. Due to the absence of generic tensorization results for the MLSI in the quantum setting, one is often forced to prove it case by case. However, in the recent years, a new approach to the problem emerged from the communities of operator algebras and quantum information theory. Here, instead of proving the tensorization of MLSI for a product of semigroups, one considers a stronger inequality which naturally tensorizes, namely the complete modified logarithmic Sobolev inequality (CMLSI). The latter consists in proving the inequality for the semigroup tensorized with the identity semigroup over an arbitrarily large matrix algebra. The existence of CMLSI for all quantum Markov semigroups on matrix algebras was however left as an open conjecture. In this talk, I will provide a proof of the conjecture for the class of reversible quantum Markov semigroups. This talk is intended to be self-contained and does not require previous knowledge about quantum mechanics or quantum information theory. It is based on a joint work with Li Gao, a preprint of which is available here: arxiv.org/abs/2102.04146.

analysis of PDEs

Audience: researchers in the topic

Open PDE and analysis seminar and lectures