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Investigating Properties of a Family of Quantum Renyi Divergences

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