Integral formula for quantum relative entropy implies data processing inequality

Abstract

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. The latter result was first proved by Müller-Hermes and Reeb, based on work of Beigi. For a simple application of such monotonicities, we consider any `divergence' that is non-increasing under quantum measurements, such as the concavity of von Neumann entropy, or various known quantum divergences. An elegant argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a `divergence' on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states. Applications of the new integral formulae to the general probabilistic model of information theory, and a related integral formula for the classical Rényi divergence, are also discussed.

arXiv comment16 pages. Accepted to Quantum

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• Quantum Rényi and $f$-divergences from integral representations
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• Continuity of entropies via integral representations
• Entanglement cost for infinite-dimensional physical systems
• Optimising the relative entropy under semidefinite constraints
• Quantum Key Distribution with Imperfections: Recent Advances in Security Proofs
• Reverse-type Data Processing Inequality
• Uhlmann's theorem for measured divergences
• Metric response of relative entropy: A universal indicator of quantum criticality