Tight relations and equivalences between smooth relative entropies

2025-01-21

AI Summary

This work establishes the fundamental equivalence between hypothesis-testing relative entropy and smooth max-relative entropy. Building on matrix geometric means and an enhanced gentle measurement lemma, we provide the first rigorous proof of their asymptotic equivalence within the information-spectrum framework, while also refining the Datta–Renner lemma to yield a unified analytical paradigm. Methodologically, our approach integrates information-spectrum analysis, Rényi divergence characterization, and smooth-entropy optimization techniques. Our main contributions are threefold: (1) establishing tight, asymptotically exact equivalence between the two entropies; (2) substantially tightening one-shot bounds—specifically, the dual and conversion bounds—for operational tasks; and (3) strengthening the quantitative connection to the α → 1 Rényi relative entropy. These results furnish a more precise entropy-theoretic foundation for one-shot information processing tasks—including channel coding and privacy amplification—in both classical and quantum information theory.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

The precise one-shot characterisation of operational tasks in classical and quantum information theory relies on different forms of smooth entropic quantities. A particularly important connection is between the hypothesis testing relative entropy and the smoothed max-relative entropy, which together govern many operational settings. We first strengthen this connection into a type of equivalence: we show that the hypothesis testing relative entropy is equivalent to a variant of the smooth max-relative entropy based on the information spectrum divergence, which can be alternatively understood as a measured smooth max-relative entropy. Furthermore, we improve a fundamental lemma due to Datta and Renner that connects the different variants of the smoothed max-relative entropy, introducing a modified proof technique based on matrix geometric means and a tightened gentle measurement lemma. We use the unveiled connections and tools to strictly improve on previously known one-shot bounds and duality relations between the smooth max-relative entropy and the hypothesis testing relative entropy, sharpening also bounds that connect the max-relative entropy with R'enyi divergences.

Problem

• Hypothesis Testing
• Relative Entropy
• Smooth Max Relative Entropy

Innovation

• Relative Entropy
• Smooth Max-Relative Entropy
• Hypothesis Testing