Theorem: Exact no-signalling criterion

The following are equivalent:

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

2. there is a channel $\Xi :L({\mathcal{H}}_{A^c})\to L({\mathcal{H}}_B)$ with

   $${\Phi }_B=\Xi \circ {\mathrm{Tr}}_A;$$
3. $${\Phi }_B^{\ast }(L({\mathcal{H}}_B))\subseteq I_A\otimes L({\mathcal{H}}_{A^c}).$$

The factorization implies equality with every untouched reference.

Proof. Factorization implies (i), since a local trace-preserving intervention leaves the $A^{c}R$ reduction unchanged.

Suppose (i) holds. Compare the identity intervention with the replacement channel

With no reference, equality for every input state gives equality of the corresponding linear maps:

Define

This is CPTP and gives (ii).

Taking adjoints proves (ii) implies (iii). Conversely, (iii) makes the output expectation depend only on the $A^c$ reduction. The same replacement construction gives (ii). Tensoring the factorization with ${\mathrm{id}}_R$ proves the final assertion.