Right mean for the $\alpha -z$ Bures-Wasserstein quantum divergence

Abstract

A new quantum divergence induced from the $\alpha -z$ Renyi relative entropy, called the $\alpha -z$ Bures-Wasserstein quantum divergence, has been recently introduced. We investigate in this paper properties of the right mean, which is a unique minimizer of the weighted sum of $\alpha -z$ Bures-Wasserstein quantum divergences to each points. Many interesting operator inequalities of the right mean with the matrix power mean including the Cartan mean are presented. Moreover, we verify the trace inequality with the Wasserstein mean and provide bounds for the Hadamard product of two right means.