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.
Quantum Information Processing – Springer Journals
Published: Feb 3, 2015
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