Characterisation of matrix entropies

Abstract

The notion of matrix entropy was introduced by Tropp and Chen with the aim of measuring the fluctuations of random matrices. It is a certain entropy functional constructed from a representing function with prescribed properties, and Tropp and Chen gave some examples. We give several abstract characterisations of matrix entropies together with a sufficient condition in terms of the second derivative of their representing function.

Major revision. We found an error in the previous version that we cannot repair. It implies that we no longer can be certain that the sufficient condition of operator convexity of the second derivative of a matrix entropy is also necessary. We added more abstract characterisations of matrix entropies and improved the analysis of the concrete examples

References in corpus (2)

• Moment inequalities for functions of independent random variables
• Trace functions as Laplace transforms

Cited by in corpus (7)

• Subadditivity of Matrix phi-Entropy and Concentration of Random Matrices
• Characterisations of Matrix and Operator-Valued $\Phi$-Entropies, and Operator Efron-Stein Inequalities
• Divergence and Sufficiency for Convex Optimization
• Maximum Entropy and Sufficiency
• Quantum entropy derived from first principles
• A note on quantum entropy
• Jointly convex quantum Jensen divergences