# Investigating properties of a family of quantum Rényi divergences

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 $$\alpha$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# Investigating properties of a family of quantum Rényi divergences

, Volume 14 (4) – Feb 3, 2015
12 pages

/lp/springer_journal/investigating-properties-of-a-family-of-quantum-r-nyi-divergences-0kObPSravm
Publisher
Springer Journals
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-0935-y
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Feb 3, 2015

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