Quantitative Signalling-to-Product Rigidity for Two-Qubit Unitary Channels

Abstract

For a bipartite unitary channel, exact absence of one-way signalling is known to force product structure. We give an explicit quantitative version for two-qubit unitaries. Let Sigma_{A->B}(U) be the unhalved diamond-norm distance of Bob's reduced output channel from the set of channels depending only on Bob's input, and let D_prod(U) be the operator-norm distance of U from product unitaries. We prove (1/2) Sigma_{A->B}(U) <= D_prod(U) <= 2 sqrt(2) sin(3 pi/16) Sigma_{A->B}(U), where 2 sqrt(2) sin(3 pi/16) is approximately 1.5713899168. The proof combines local-unitary invariance, the Cartan/Weyl decomposition of two-qubit gates, two explicit signalling witnesses, a scalar product-unitary comparator, and an analytic optimization over the Weyl chamber. On the Ising edge U_theta = exp(-i theta Z tensor Z) we additionally obtain the exact signalling value Sigma_{A->B}(U_theta)=sin(2 theta) for 0 <= theta <= pi/4, which shows that the universal lower coefficient 1/2 is asymptotically sharp. We do not claim that the displayed upper constant is the optimal constant of the theorem. Preprint / quantitative theorem candidate with strong internal audit; external independent verification and novelty closure remain pending; not peer reviewed.