Investigating Properties of a Family of Quantum Renyi Divergences

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.