Investigating Properties of a Family of Quantum Renyi Divergences

TL;DR

This paper studies a two-parameter family of quantum Rènyi divergences, proving their relation to quantum relative entropy, analyzing their differentiability at one, and connecting derivatives to the relative entropy variance relevant in quantum hypothesis testing.

Contribution

It establishes the limit behavior of the $ ext{α-}z$-relative Rènyi entropies as α approaches one, unifies previous definitions, and links derivatives at one to operationally significant quantum information quantities.

Findings

The $ ext{α-}z$-relative Rènyi entropies generalize quantum relative entropy.

Certain Rènyi divergences are differentiable at α=1.

Derivative at α=1 equals half the relative entropy variance.

Abstract

Audenaert and Datta recently introduced a two-parameter family of relative R\'{e}nyi entropies, known as the $\alpha$-$z$-relative R\'{e}nyi entropies. The definition of the $\alpha$-$z$-relative R\'{e}nyi entropy unifies all previously proposed definitions of the quantum R\'{e}nyi divergence of order $\alpha$ under a common framework. Here we will prove that the $\alpha$-$z$-relative R\'{e}nyi entropies are a proper generalization of the quantum relative entropy by computing the limit of the $\alpha$-$z$ divergence as $\alpha$ approaches one and $z$ is an arbitrary function of $\alpha$. We also show that certain operationally relevant families of R\'enyi divergences are differentiable at $\alpha = 1$. Finally, our analysis reveals that the derivative at $\alpha = 1$ evaluates to half the relative entropy variance, a quantity that has attained operational significance in second-order quantum hypothesis testing.