Abstract: We derive new characterizations of the matrix Phi-entropies introduced in [Electron. J. Probab., 19(20): 1-30, 2014 ]. These characterizations help to better understand the properties of matrix Phi-entropies, and are a powerful tool for establishing matrix concentration inequalities for matrix-valued functions of independent random variables. In particular, we use the subadditivity property to prove a Poincare inequality for the matrix Phi-entropies. We also provide a new proof for the matrix Efron-Stein inequality. Furthermore, we derive logarithmic Sobolev inequalities for matrix-valued functions defined on Boolean hypercubes and with Gaussian distributions. Our proof relies on the powerful matrix Bonami-Beckner inequality. Finally, the Holevo quantity in quantum information theory is closely related to the matrix Phi-entropies. This allows us to upper bound the Holevo quantity of a classical-quantum ensemble that undergoes a special Markov evolu

Date: 29 July 2015