On the joint convexity of the Bregman divergence of matrices
arXiv:1405.7885 · doi:10.1007/s11005-015-0757-y
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.
v2: 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