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

TL;DR

This paper explores the properties of the right mean in the context of the newly introduced $oldsymbol{ ext{α-z}}$ Bures-Wasserstein quantum divergence, including operator inequalities and bounds for matrix means.

Contribution

It investigates the right mean's properties, presents new operator inequalities involving matrix power and Cartan means, and establishes bounds related to the Wasserstein mean.

Findings

Operator inequalities involving the right mean and matrix power mean.

Bounds for the Hadamard product of two right means.

Verification of trace inequality with the Wasserstein mean.

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.