On multivariate trace inequalities of Sutter, Berta and Tomamichel

TL;DR

This paper analyzes a family of multivariate trace inequalities that extend classical inequalities to multiple matrices using complex powers, with potential applications in quantum information theory.

Contribution

It reformulates multivariate trace inequalities using resolvents and entangled states, facilitating their application in quantum information and perturbation analysis.

Findings

Reformulation of inequalities with resolvents and entangled states

Extension of Lieb's three-matrix inequality to n matrices

Potential applications in quantum information theory, especially in the analysis of the Petz recovery map

Abstract

We consider a family of multivariate trace inequalities recently derived by Sutter, Berta and Tomamichel. These inequalities generalize the Golden-Thompson inequality and Lieb's three-matrix inequality to an arbitrary number of matrices in a way that features complex matrix powers. We show that their inequalities can be rewritten as an $n$-matrix generalization of Lieb's original three-matrix inequality. The complex matrix powers are replaced by resolvents and appropriate maximally entangled states. We expect that the technically advantageous properties of resolvents, in particular for perturbation theory, can be of use in applications of the $n$-matrix inequalities, e.g., for analyzing the rotated Petz recovery map in quantum information theory.