Quantum Channel Stein Theorem beyond Definite Causal Order
arXiv:2609.30268
The paper proves that for any two finite‑dimensional memoryless quantum channels, parallel, adaptive, and general testing strategies all achieve the same Stein exponent (the regularized channel relative entropy) at any fixed type‑I error level, showing that adaptivity gives no asymptotic advantage and providing the exact strong‑converse exponent.
Abstract
Distinguishability is a central concept in information theory and extends naturally to quantum systems. This work shows that, for any two finite-dimensional memoryless quantum channels, parallel, adaptive and general testing strategies achieve the same Stein exponent at every fixed type-I error tolerance , namely the regularized channel relative entropy. When this rate is finite, any larger type-II exponent forces the probability of correctly accepting the null hypothesis to decay exponentially. Thus, adaptivity provides no asymptotic advantage. The key step is proving the continuity of the regularized sandwiched Rényi channel divergence at order one. We also derive the exact strong-converse exponent for general testers.
20 pages