Integral formula for quantum relative entropy implies data processing inequality

TL;DR

This paper introduces integral formulas for quantum relative entropy and derivatives of von Neumann entropy, providing simplified proofs of key data processing inequalities and exploring their applications in quantum information theory.

Contribution

The authors develop new integral representations for quantum relative entropy and derivatives, enabling straightforward proofs of data processing inequalities without requiring complete positivity.

Findings

Proved monotonicity of quantum relative entropy under trace-preserving positive maps.

Showed the infimum of certain divergences on quantum states equals that on classical binary states.

Extended integral formula applications to general probabilistic models and classical Rényi divergence.

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 M\"uller-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 R\'enyi divergence, are also discussed.