Proposition: Reference-stable approximate no-signalling

Suppose $d_{\diamond }(\tilde{\mathcal{C}},\mathcal{C})\le e$, where $\mathcal{C}$ obeys the scheduled support bound. For two states with equal ancestor reductions, the actual output reductions differ in trace distance by at most

The same bound holds with an untouched reference when the joint ancestor–reference reductions agree.

Proof. The ideal outputs agree. Insert this common output between the two actual outputs. Each actual–ideal discrepancy is at most $e$, so the triangle inequality gives $2e$. Trace distance between states is at most one. The diamond norm supplies the referenced version.