Preprint proves regularized channel Rényi divergences converge to relative entropy at order 1
The preprint demonstrates continuity of regularized channel Rényi divergences at order one. It supplies operational strong converses for quantum channel discrimination. Evidence rests on a single mathematical derivation without numerical validation.
The work derives an asymptotic bound below order one that amplifies into exponential decay via Gour's Stinespring approximation and Schatten-norm estimates. For channel pairs with finite max-relative entropy, this yields exponential strong converses for parallel and adaptive discrimination plus a sharp zero-one testing law. The result supplies the subchannel asymptotic equipartition property previously missing from the literature.
Earlier continuity statements for states did not extend to channels because regularization and stabilization interact with the completely-positive trace-preserving constraint. The new proof closes that gap by embedding the hockey-stick divergence as an intermediary that inherits both regularization and the data-processing inequality.
Quantum communication protocols relying on Rényi-based converses, such as private capacity bounds and hypothesis testing, now inherit uniform continuity at the relative-entropy limit. This tightens finite-blocklength estimates used in repeater design and entanglement distillation.
Next steps include extending the argument to infinite-dimensional channels and testing numerical sharpness on qubit amplitude-damping pairs within the next twelve months.
Wang et al.: Within 18 months a follow-up will numerically verify the exponential decay rate on two-qubit channels to within 5 percent of the analytic bound.
Sources (3)
- [1]Primary Source(https://arxiv.org/abs/2609.28635)
- [2]Supporting Source(https://arxiv.org/abs/2007.07437)
- [3]Supporting Source(https://quantum-journal.org/papers/q-2021-03-10-417/)