Two-Parameter Rényi Information Quantities with Applications to Privacy Amplification and Soft Covering
Shi-Bing Li, Ke Li, Lei Yu
TL;DR
This work introduces a two-parameter Rényi conditional entropy \\widetilde{H}_{\alpha,\beta}(X|Y) and a novel two-parameter Rényi mutual information \\widetilde{I}_{\alpha,\beta}(X:Y), unifying existing Rényi conditional-entropy and mutual-information definitions under a single framework. It proves a suite of structural properties, including monotonicity in the parameters, additivity, data-processing inequalities, variational expressions, and concavity/convexity characteristics, and links these quantities to operational tasks. The authors then apply these quantities to characterize the strong converse exponents for privacy amplification and soft covering under Rényi divergence, providing exact formulas and proving achievability and optimality via variational analyses and type-based methods. The results offer a versatile analytical toolkit for security and channel-resolvability problems and suggest avenues for further exploration of zero-parameter limiting cases and alternative operational interpretations.
Abstract
There are no universally accepted definitions of Rényi conditional entropy and Rényi mutual information, although motivated by different applications, several definitions have been proposed in the literature. In this paper, we consider a family of two-parameter Rényi conditional entropy and a family of two-parameter Rényi mutual information. By performing a change of variables for the parameters, the two-parameter Rényi conditional entropy we study coincides precisely with the definition introduced by Hayashi and Tan [IEEE Trans. Inf. Theory, 2016], and it also emerges naturally as the classical specialization of the three-parameter quantum Rényi conditional entropy recently put forward by Rubboli, Goodarzi, and Tomamichel [arXiv:2410.21976 (2024)]. We establish several fundamental properties of the two-parameter Rényi conditional entropy, including monotonicity with respect to the parameters and variational expression. The associated two-parameter Rényi mutual information considered in this paper is new and it unifies three commonly used variants of Rényi mutual information. For this quantity, we prove several important properties, including the non-negativity, additivity, data processing inequality, monotonicity with respect to the parameters, variational expression, as well as convexity and concavity. Finally, we demonstrate that these two-parameter Rényi information quantities can be used to characterize the strong converse exponents in privacy amplification and soft covering problems under Rényi divergence of order $α\in (0, \infty)$.