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Integral representations of quantum relative entropy, and of the directional second and higher order derivatives of von Neumann entropy, are established, and used to give simple proofs of fundamental, known data processing inequalities: the Holevo bound on the quantity of information transmitted by a quantum communication channel, and, much more generally, the monotonicity of quantum relative entropy under trace-preserving positive linear maps -- complete positivity of the map need not be assumed.
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R. Blume-Kohout, H. K. Ng, D. Poulin, L. Viola: Information-preserving structures: A general framework for quantum zero-error information. Physical Review A 82 (6), 062306
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I. H. Kim, M. B. Ruskai: Bounds on the concavity of quantum entropy. J. Math. Phys. 55 (2014), no. 9, 092201, 5 pp
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A. Müller-Hermes, D. Reeb: Monotonicity of the quantum relative entropy under positive maps. Ann. Henri Poincaré 18 (2017), no. 5, 1777–1788
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M. M. Wilde, Optimized quantum f f -divergences and data processing, J. Phys. A: Math. Theor. 51 (2018) 374002
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F. Hiai, M. Mosonyi: Different quantum f f -divergences and the reversibility of quantum operations. Reviews in Mathematical Physics 29 (7), 1750023
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A. Jenčová: Recoverability of quantum channels via hypothesis testing, e-print arXiv:2303.11707
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H. Li, Monotonicity of optimized quantum f f -divergence, arXiv:2104.12890
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D. Virosztek: The metric property of the quantum Jensen-Shannon divergence. Advances in Mathematics 380:107595
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