Proposition: Exact deterministic no-signalling

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 all inputs and references;

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 gives intervention invariance. Conversely compare identity on $A$ with replacement by ${\tau }_A=I_A/\dim {\mathcal{H}}_A$. Invariance gives

on states and hence, by linearity, on all operators. Adjoints give equivalence with the algebra containment.