Investigating properties of a family of quantum Rényi divergences
Audenaert and Datta recently introduced a two-parameter family of relative Rényi entropies, known as the \(\alpha \)–\(z\)-relative Rényi entropies. The definition of the \(\alpha \)–\(z\)-relative Rényi entropy unifies all previously proposed definitions of the quantum Rényi divergence of order \(\alpha \) under a common framework. Here, we will prove that the \(\alpha \)–\(z\)-relative Rényi entropies are a proper generalization of the quantum relative entropy by computing the limit of the \(\alpha \)–\(z\) divergence as \(\alpha \) approaches one and \(z\) is an arbitrary function of \(\alpha \). We also show that certain operationally relevant families of Rényi 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.
KeywordsQuantum information Rényi entropy Rényi divergence
The results in this paper were achieved as part of a final year project at the School of Computing, Department of Computer Science at the National University of Singapore under the supervision of Professor Stephanie Wehner, and we want to thank her for discussions and for providing a conducive research environment at the Centre for Quantum Technologies. We would like an anonymous referee for providing us with Example 1. SL would also like to thank Jedrzej Kaniewski, Patrick Coles, and Mischa Woods for discussions leading to the results in this project. Finally, he would like to thank Nelly Ng, his tutor in the Introduction to Information Theory (CS3236) class, for getting him interested in information theory. SL acknowledges support from the Agency for Science, Technology and Research (A*STAR). MT is funded by the Ministry of Education (MOE) and National Research Foundation Singapore, as well as MOE Tier 3 Grant Random numbers from quantum processes (MOE2012-T3-1-009).
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