On the joint convexity of the Bregman divergence of matrices

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.

arXiv commentv2: some new results about the relation of joint convexity and monotonicity under stochastic maps has been added and the proof of the main theorem has been changed essentially v3: some typos corrected v4: essentially the published version

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• Quantum Hellinger distances revisited
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• Maps on positive definite matrices preserving Bregman and Jensen divergences
• Maximum Entropy and Sufficiency
• Divergence and Sufficiency for Convex Optimization
• Minimax quantum state estimation under Bregman divergence
• Maps on quantum states preserving Bregman and Jensen divergences
• Quantum $f$-divergence preserving maps on positive semidefinite operators acting on finite dimensional Hilbert spaces
• A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means
• Complete Sobolev Type Inequalities
• From Poincaré Inequalities to Nonlinear Matrix Concentration
• On The Chain Rule Optimal Transport Distance
• Jointly convex quantum Jensen divergences