Layer Cake Representations for Quantum Divergences

TL;DR

This paper introduces a layer cake representation for quantum divergences, providing new proofs, insights, and applications for quantum information measures like Renyi and f-divergences, advancing the theoretical understanding of quantum information theory.

Contribution

It proposes a novel layer cake approach to quantum divergences, establishing equivalence with existing integral representations and offering new proofs, representations, and applications.

Findings

Provided an alternative proof of the relative entropy integral representation.

Proved a conjecture on a trace expression for Renyi divergence.

Introduced applications to error exponents and new integral and variational representations.

Abstract

Defining suitable quantum extensions of classical divergences often poses a challenge due to the non-commutative nature of quantum information. In this work, we propose a new approach via what we call the layer cake representation. The resulting quantum R\'enyi and $f$-divergences are then proven to be equivalent to those recently defined via integral representations. Nevertheless, the approach can provide several insights. We give an alternative proof of the integral representation of the relative entropy by Frenkel and prove a conjecture regarding a trace expression for the R\'enyi divergence. Additionally, we give applications to error exponents in hypothesis testing, a new Riemann-Stieltjes type integral representation and a variational representation.