New Weighted Spectral Geometric Mean and Quantum Divergence

TL;DR

This paper introduces a new class of weighted spectral geometric means, explores their inequalities, relationships with quantum divergences, and studies their barycenter for minimizing quantum divergence sums.

Contribution

It presents new inequalities, relationships with Rényi entropy, and a barycenter concept for the spectral geometric mean, advancing quantum information theory.

Findings

Established inequalities in L"{o}wner order, operator norm, and trace.

Derived log-majorization relationship with Rényi relative entropy.

Analyzed the quantum divergence and barycenter for the new mean.

Abstract

A new class of weighted spectral geometric means has recently been introduced. In this paper, we present its inequalities in terms of the L\"{o}wner order, operator norm, and trace. Moreover, we establish a log-majorization relationship between the new spectral geometric mean, and the R\'{e}nyi relative operator entropy. We also give the quantum divergence of the quantity, given by the difference of trace values between the arithmetic mean and new spectral geometric mean. Finally, we study the barycenter that minimizes the weighted sum of quantum divergences for given variables.