Functional inequalities are potent tools in analysing the convergence time of quantum Markov semigroups. Specifically, the modified log-Sobolev inequality (MLSI) concerns the (exponential) convergence of the time evolution in terms of relative entropy as a quantitative measure. In this talk, I will present an estimate of modified log-Sobolev constant using completely positive mixing time. Our proof uses only entropic inequalities, which gives a unified information-theoretic approach that applies in both classical and quantum setting, even the Type III von Neumann algebras. For a quantum analogue of birth-and-death process, our estimate is tight up to a factor of absolute constant. As an application, I will talk about the use of (complete) modified log-Sobolev constant in estimating the decay of relative entropy of entanglement.
This is based on joint work with Marius Junge, Nicholas LaRacuente, and Haojian Li.
This video was produced by the International Centre for Mathematical Sciences, as part of the workshop Mathematical Physics in Quantum Technology: From Finite to Infinite Dimensions.
