Quantum conditional entropies from convex trace functionals

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.

💡 Research Summary

This paper introduces a three‑parameter family of quantum conditional entropies, denoted (H^{\alpha,z}_{\lambda}(A|B)), which simultaneously generalizes the two canonical families derived from the (\alpha!-!z) relative entropy (the Petz‑type and the sandwiched‑type). The new parameter (\lambda) interpolates (or extrapolates) between the two canonical cases: (\lambda=0) recovers the Petz‑type conditional entropy, while (\lambda=1) yields the sandwiched‑type. By allowing (\lambda) to take any value in (