Proposition: Exact no-signalling criterion

For input region $A$, output $B$, and channel ${\Phi }_B$, the following are equivalent:

1. all deterministic local interventions on $A$ give the same $BE$ output for every initial state and untouched reference;

2. ${\Phi }_B=\Xi {\mathrm{Tr}}_A$ for a CPTP map $\Xi$;

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

Proof. Factorization proves (i), because a local trace-preserving intervention leaves the $A^{c}E$ reduction unchanged. For the converse, compare identity on $A$ with replacement by $I_A/\dim {\mathcal{H}}_A$. Equality on all states and linearity give

Taking adjoints gives (iii). Conversely (iii) makes all output expectations depend only on the $A^c$ reduction and gives the same factorization.