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On the Joint Convexity of the Bregman Divergence of Matrices

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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.

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  1. Department of Mathematical Analysis, Budapest University of Technology and Economics, Egry József u. 1., Budapest, 1111, Hungary

    József Pitrik & Dániel Virosztek

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  1. József Pitrik
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  2. Dániel Virosztek
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Correspondence to Dániel Virosztek.

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Pitrik, J., Virosztek, D. On the Joint Convexity of the Bregman Divergence of Matrices. Lett Math Phys 105, 675–692 (2015). https://doi.org/10.1007/s11005-015-0757-y

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  • DOI: https://doi.org/10.1007/s11005-015-0757-y

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