Continuity of Regularized Channel Rényi Divergences

2026-09-23

AI Summary

This paper investigates the asymptotic properties of finite-dimensional quantum channel discrimination. It first establishes that the regularized sandwiched Rényi divergence converges to the relative entropy as the order approaches one. Subsequently, by integrating the hockey-stick divergence, Stinespring approximation, and Gour’s method, it derives exponential decay bounds under higher-order thresholds. The core contribution lies in elevating asymptotic bounds to an exponential strong converse theorem, thereby establishing a zero-one testing law and the asymptotic equipartition property (AEP) for subchannels. Furthermore, this work provides a unified characterization of exponential strong converse results and AEP across both parallel and adaptive channel discrimination scenarios.

Abstract

We prove that the regularized, stabilized sandwiched Rényi divergence of finite-dimensional quantum channels converges to their regularized relative entropy as the Rényi order tends to one. The key tool is the channel hockey-stick divergence: Gour's Stinespring approximation bound and a Schatten norm estimate amplify an asymptotic bound below one into exponential decay at higher threshold rates. For channel pairs with finite max-relative entropy, known operational connections then give exponential strong converses for parallel and adaptive discrimination, a sharp zero--one testing law, and the subchannel asymptotic equipartition property.

Problem

• Quantum channels
• Sandwiched Rényi divergence
• Regularized relative entropy
• Continuity

Innovation

• Regularized sandwiched Rényi divergence
• Channel hockey-stick divergence
• Quantum channel discrimination
• Exponential strong converse
• Asymptotic equipartition property