Theorem: Exact no-signalling criterion

For a finite channel $\Phi$, the following are equivalent:

1. ${\mathrm{Sig}}_{\Phi }(A\to B)=0$;

2. a channel $\Xi :\mathcal{B}({\mathcal{H}}_{A^c})\to \mathcal{B}({\mathcal{H}}_B)$ satisfies ${\Phi }_B=\Xi {\mathrm{Tr}}_A$;

3. $${\Phi }_B^{\ast }(\mathcal{B}({\mathcal{H}}_B))\subseteq I_A\otimes \mathcal{B}({\mathcal{H}}_{A^c}).$$

The factorization holds with every untouched reference.

Proof. For (ii), local trace preservation leaves the $A^{c}E$ reduction unchanged, proving (i). Conversely, compare the identity intervention with the local replacement

Condition (i), even without a reference, implies ${\Phi }_B={\Phi }_B({\mathcal{R}}_A\otimes \mathrm{id})$ on all states and therefore on all operators. Set

It is CPTP and proves (ii). Taking adjoints gives (iii). Conversely, (iii) makes every output expectation depend only on the $A^c$ reduction; the same replacement argument gives (ii). Tensor the factorization with ${\mathrm{id}}_E$.