Quantum Conditional Entropies from Convex Trace Functionals

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 theory. 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-Thirring inequalities, variational characterizations of trace functionals, and spectral pinching techniques.