Quantum conditional entropies from convex trace functionals

TL;DR

This paper introduces a new family of quantum conditional entropies based on convex trace functionals, establishing their fundamental properties and operational relevance in quantum information theory.

Contribution

It develops novel convexity results and proves key properties like data-processing inequalities and additivity for these quantum entropies, advancing theoretical understanding.

Findings

Established convexity and monotonicity properties

Proved data-processing inequalities and additivity

Demonstrated operational significance in quantum information

Abstract

We study geometric properties of trace functionals that generalize those in [Zhang, Adv. Math. 365:107053 (2020)], arising from a novel family of conditional entropies with applications in quantum information. Building on new convexity results for these functionals, we establish data-processing inequalities and additivity properties for our entropies, demonstrating their operational significance. We further prove completeness under duality, chain rules, and various monotonicity properties for this family. Our proofs draw on tools from complex interpolation theory, multivariate Araki--Lieb and Lieb--Thirring inequalities, variational characterizations of trace functionals, and spectral pinching techniques.