Jointly convex quantum Jensen divergences

TL;DR

This paper explores the joint convexity of quantum Jensen divergences and establishes that they are generated by functions within the recently defined Matrix Entropy Class, linking divergence properties to matrix entropy theory.

Contribution

It identifies the set of functions producing jointly convex quantum Jensen divergences as exactly those in the Matrix Entropy Class, connecting divergence convexity with matrix entropy.

Findings

Quantum Jensen divergences are characterized by the Matrix Entropy Class.

The set of generating functions for jointly convex divergences is fully described.

This links divergence convexity properties to matrix entropy concepts.

Abstract

We investigate the quantum Jensen divergences from the viewpoint of joint convexity. It turns out that the set of the functions which generate jointly convex quantum Jensen divergences on positive matrices coincides with the Matrix Entropy Class which has been introduced by Chen and Tropp quite recently in [Electron. J. Probab. 19 (2014), 1-30].