Investigating properties of a family of quantum Renyi divergences

Audenaert and Datta recently introduced a two-parameter family of relative Renyi entropies, known as the alpha-z-relative Renyi entropies. The definition of the alpha-z- relative Renyi entropy unifies all previously proposed definitions of the quantum Renyi divergence of order a under a common framework. Here, we will prove that the alpha-z-relative Renyi entropies are a proper generalization of the quantum relative entropy by computing the limit of the alpha-z divergence as a approaches one and z is an arbitrary function of a. We also show that certain operationally relevant families of Renyi 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 and channel coding for finite block lengths.