Quantum Channel Stein's Lemma with an Exponential Strong Converse

Abstract: We establish the quantum channel Stein lemma at every fixed type-I error tolerance in $(0,1)$ and prove an exponential strong converse under arbitrary adaptive strategies. For any finite-dimensional channel pair, the optimal type-II exponent equals the regularized channel relative entropy and is attained by parallel strategies. When this divergence is finite, every strictly larger type-II rate forces the probability of accepting the null hypothesis to decay exponentially, uniformly over strategies and quantum memories. The proof rests on a uniform Stinespring approximation: tensor powers of the null dilation are approximated by tensor powers of the alternative dilation after applying an auxiliary linear map to the environment. The operator-norm error is exponentially small under a squared-norm bound at any rate above the regularized relative entropy. We construct the approximation from a weak testing bound by iterative rate reduction and a fixed-block tensor expansion with an exact residual. It implies right continuity at order one of the regularized sandwiched Rényi channel divergence, which yields the adaptive converse through the channel chain rule. We also obtain a sharp hockey-stick threshold and a subchannel smoothing asymptotic equipartition property for fixed and subexponentially vanishing diamond-norm errors.