  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.
