Investigating properties of a family of quantum Rényi divergences

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Journal Article
Quantum Information Processing, 2015, 14 (4), pp. 1501 - 1512
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© 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.
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