Theorem: Signalling stability with consistent constants

For channels on the same interfaces,

In particular, if $\Psi$ is nonsignalling and $d_{\diamond }({\Phi }_B,{\Psi }_B)\le \epsilon$, then ${\mathrm{Sig}}_{\Phi }\le \min \{1,2\epsilon \}$.

Proof. For fixed reference, state, and interventions, the two comparison errors each have trace norm at most ${\Vert {\Phi }_B-{\Psi }_B\Vert }_{\diamond }$. The triangle inequality and the $1/2$ trace-distance factor give one such full-diamond norm for the difference of the two distinguishabilities. Take suprema and exchange the channels.