Appears in collection : 2017 - T3 - WS1 - Operator algebras and quantum information theory

In this talk we present a relation between generalized entropies and operator means. For example, as pointed by Furuichi \cite{SF11}, two upper bounds on the Tsallis entropies suggest the following inequality: for positive operators X and Y and \nu\in[0,1], Tr(X#_\nu Y)\geq 1/2 Tr(X+Y-|X-Y|), where the symbole stands for the weighted geometric mean , that is, X^{1/2}(X^{-1/2}YX^{-1/2})^\nu X^{1/2}. Unfortunately, this inequality does not hold in general, but this is true when XY + YX \geq 0. We can extend this inequality for a general operator mean and it is called the generalized reverse Cauchy inequality. We also give a formulation of new Rényi relative entropies by Mosonyi and Ogawa using operator means.