Letters in Mathematical Physics

, Volume 105, Issue 5, pp 675–692 | Cite as

On the Joint Convexity of the Bregman Divergence of Matrices

  • József Pitrik
  • Dániel VirosztekEmail author


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.

Mathematics Subject Classification

46N50 46L30 81Q10 


joint convexity Bregman divergence Tsallis entropy monotonicity 


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisBudapest University of Technology and EconomicsBudapestHungary

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