# Investigating properties of a family of quantum Rényi divergences

Publication Type:
Journal Article
Citation:
Quantum Information Processing, 2015, 14 (4), pp. 1501 - 1512
Issue Date:
2015-04-01
Filename Description Size
1408.6897v2.pdfAccepted Manuscript Version164.97 kB
© 2015, Springer Science+Business Media New York. Audenaert and Datta recently introduced a two-parameter family of relative Rényi entropies, known as the $$\alpha$$α–$$z$$z-relative Rényi entropies. The definition of the $$\alpha$$α–$$z$$z-relative Rényi entropy unifies all previously proposed definitions of the quantum Rényi divergence of order $$\alpha$$α under a common framework. Here, we will prove that the $$\alpha$$α–$$z$$z-relative Rényi entropies are a proper generalization of the quantum relative entropy by computing the limit of the $$\alpha$$α–$$z$$z divergence as $$\alpha$$α approaches one and $$z$$z is an arbitrary function of $$\alpha$$α. We also show that certain operationally relevant families of Rényi divergences are differentiable at $$\alpha = 1$$α=1. Finally, our analysis reveals that the derivative at $$\alpha = 1$$α=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.