Proposition: Exact deterministic no-signalling criterion

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

1. every deterministic local intervention on $A$ leaves the $BE$ output unchanged for every input and untouched reference;

2. ${\Phi }_B=\Xi {\mathrm{Tr}}_A$ for some 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 intervention invariance. Conversely, compare identity on $A$ with replacement by its maximally mixed state ${\tau }_A$. Invariance gives

first on states and then on all operators. Taking adjoints proves the equivalence with (iii).