On the joint convexity of the Bregman divergence of matrices

TL;DR

This paper characterizes when Bregman divergences for matrices are jointly convex, derives a quantum entropy inequality, and explores the relationship between convexity and monotonicity in quantum information theory.

Contribution

It provides a complete characterization of joint convexity for matrix Bregman divergences and links this property to quantum entropy inequalities and monotonicity.

Findings

Derived a sharp inequality for quantum Tsallis entropy.

Showed joint convexity does not imply monotonicity under stochastic maps.

Established that all monotone Bregman divergences are jointly convex.

Abstract

We characterize the functions for which the corresponding Bregman divergence is jointly convex on matrices. As an application of this characterization, we derive a sharp inequality for the quantum Tsallis entropy of a tripartite state, which can be considered as a generalization of the strong subadditivity of the von Neumann entropy. (In general, the strong subadditivity of the Tsallis entropy fails for quantum states, but it holds for classical states.) Furthermore, we show that the joint convexity of the Bregman divergence does not imply the monotonicity under stochastic maps, but every monotone Bregman divergence is jointly convex.