Logarithmic Sobolev Inequalities (LSI) were introduced on Gaussian measures by L.Gross in 70s as a reformulation of Nelson's Hypercontractivity in quantum field theory. Later they have been studied on manifolds, graphs, and matrices (non-commutative spaces). A natural framework of classical LSIs given by the theory of Markov Semigroups. The quantum, or matrix-valued, version LSIs are based on the non-commutative generalizations of Markov semigroups, where the underlying probability spaces are operator algebra. The advantage and also the difficulty of matrix-valued LSIs are the tensorization property. Matrix-valued LSIs have been found rich connections to quantum information theory, quantum optimal transportation, quantum many-body systems, and quantum machine learning theory. In this talk, I will present some recent progress on matrix-valued LSIs.
