Some inequalities for spectral geometric mean with applications

TL;DR

This paper investigates inequalities related to the spectral geometric mean of positive invertible operators, establishing new bounds, relations, and applications in quantum information theory.

Contribution

It introduces new Hölder type inequalities, operator norm relations, and log-majorization results for the spectral geometric mean, with applications to quantum Tsallis relative entropy.

Findings

Established Hölder type inequality for spectral geometric mean.

Derived order relations among quantum Tsallis relative entropies.

Provided a new lower bound for Tsallis relative entropy.

Abstract

Recently, the spectral geometric mean has been studied by some papers. In this paper, we firstly estimate the H\"{o}lder type inequality of the spectral geometric mean of positive invertible operators on the Hilbert space for all real order in terms of the generalized Kantorovich constant and show the relation between the weighted geometric mean and the spectral geometric one under the usual operator order. Moreover, we show their operator norm version. Next, in the matrix case, we show the log-majorization for the spectral geometric mean and their applications. Among others, we show the order relation among three quantum Tsallis relative entropies. Finally we give a new lower bound of the Tsallis relative entropy.