Abstract
We prove a sharp remainder term for Hölder’s inequality for traces as a consequence of the uniform convexity properties of the Schatten trace norms. We then show how this implies a novel family of Pinsker type bounds for the quantum Rényi entropy. Finally, we show how the sharp form of the usual quantum Pinsker inequality for relative entropy may be obtained as a fairly direct consequence of uniform convexity.
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Acknowledgements
These results in this paper were obtained while the author was visiting at the I.M.A. in Minnesota during Spring 2015. The author thanks the anonymous referee for a careful reading of the paper.
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This work is partially supported by U.S. National Science Foundation Grant DMS 1501007.
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Carlen, E.A. A remainder term for Hölder’s inequality for matrices and quantum entropy inequalities. Arch. Math. 109, 365–371 (2017). https://doi.org/10.1007/s00013-017-1066-8
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DOI: https://doi.org/10.1007/s00013-017-1066-8