Investigating Properties of a Family of Quantum Renyi Divergences

Abstract

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.

15 pages, v2: journal version

Cited by in corpus (21)

• Quantum Information Processing with Finite Resources -- Mathematical Foundations
• Correlation Detection and an Operational Interpretation of the Renyi Mutual Information
• Moderate deviation analysis for classical communication over quantum channels
• Inequalities for quantum divergences and the Audenaert-Datta conjecture
• Non-Asymptotic Classical Data Compression with Quantum Side Information
• Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions
• Quantum work statistics and resource theories: bridging the gap through Renyi divergences
• Quantum Rényi and $f$-divergences from integral representations
• Duality of channels and codes
• Sharp Continuity Bounds for Entropy and Conditional Entropy
• Some continuity properties of quantum Rényi divergences
• Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond
• Geometric relative entropies and barycentric Rényi divergences
• Quantum hypothesis testing in many-body systems
• An invitation to the sample complexity of quantum hypothesis testing
• Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing
• Multivariate Fidelities
• Efficient fidelity estimation: Alternative derivation and related applications
• Rényi-Holevo inequality from $\alpha$-$z$-Rényi relative entropies
• A class of permutation-invariant measurements and their relation to quantum relative entropies
• Riemannian-geometric generalizations of quantum fidelities and Bures-Wasserstein distance