Quantum channel divergences approach relative entropy in quantum tests

Continuity of Regularized Channel Rényi Divergences

Information Theory

Summary

This paper shows that a specific way to measure differences between quantum processes, called the regularized sandwiched Rényi divergence, smoothly changes into another well-known measure called the relative entropy as a certain parameter gets close to one. The authors use a new technical tool, the channel hockey-stick divergence, combined with known math bounds to prove this convergence. This result helps improve how we distinguish between different quantum channels in tasks involving many uses, guaranteeing very clear success or failure outcomes and strong limits on error rates. Their findings support better understanding of quantum channel discrimination and related information properties in quantum computing.

What this means in practice

  • For quantum communication engineers: Enhance designs for quantum communication systems by using precise error limits for distinguishing quantum channels in multiple-use scenarios.
  • For quantum algorithm developers: Improve algorithms that test quantum operations by applying sharp success-failure boundaries derived from channel divergence analysis.

A theory result. No direct application yet.

Authors

Jinzhao Wang, Yuxiang Yang

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.
View PDFOpen arXiv