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Audenaert and Datta recently introduced a two-parameter family of relativeR\'{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 allpreviously 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 thequantum relative entropy by computing the limit of the $\alpha$-$z$ divergenceas $\alpha$ approaches one and $z$ is an arbitrary function of $\alpha$. Wealso show that certain operationally relevant families of R\'enyi divergencesare differentiable at $\alpha = 1$. Finally, our analysis reveals that thederivative at $\alpha = 1$ evaluates to half the relative entropy variance, aquantity that has attained operational significance in second-order quantumhypothesis testing.

Investigating Properties of a Family of Quantum Renyi Divergences