Multivariate Trace Inequalities

TL;DR

This paper extends fundamental trace inequalities to multiple matrices, providing new bounds and applications in quantum information theory, including tight remainder terms for entropy inequalities.

Contribution

It introduces multivariate trace inequalities extending classical results, with novel applications to quantum entropy and recoverability bounds.

Findings

Extended Golden-Thompson inequality to four matrices

Derived tight remainder terms for quantum entropy inequalities

Provided a transparent proof approach using complex interpolation

Abstract

We prove several trace inequalities that extend the Golden-Thompson and the Araki-Lieb-Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb's triple matrix inequality. As an example application of our four matrix extension of the Golden-Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities.