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.

Characterizations of matrix and operator-valued Φ-entropies, and operator Efron–Stein inequalities

“…Eq. (6.2) is exactly the integral representation for the matrix Brégman divergence proved in [21]. Similarly, Eq.…”

Maps on positive definite matrices preserving Bregman and Jensen divergences

“…Therefore, the function t → Tr e (log t)T 2 +T (log B)T +X , t ≥ 1 can be minorized by a function αt λ 2 +β with some positive α and real number β. Since λ 2 > 1, considering the equality (14) and letting t tend to infinity, we easily obtain a contradiction. Therefore, the eigenvalues of T are all less than or equal to 1.…”

Minimax quantum state estimation under Bregman divergence

“…Third, the convexity of f implies that D f (., .) is convex in its first argument; it is jointly convex if f is operator convex and numerically nonincreasing [18]. Moreover, by generalizing the proof of lower semi-continuity of relative entropy as in reference [19], we obtain the lower semi-continuity of Bregman divergence (see Appendix C for the proof).…”