Theorem: Reference-assisted signalling stability

Let $\Phi ,\Psi$ have the same input and output spaces. For a fixed output $B$, suppose

Then

In particular, a half-diamond error bound $d_{\diamond }({\Phi }_B,{\Psi }_B)\le ϵ$ contributes $2ϵ$, not $ϵ$, on the right.

Proof. For a fixed reference, input, and two interventions, let ${\rho }_{\mathcal{I}}$ and ${\rho }_{\mathcal{J}}$ be the resulting states. The two simulation-error terms obey

and the same holds for ${\rho }_{\mathcal{J}}$. The triangle inequality for the output difference therefore adds $2{ϵ}_B$ in trace norm and ${ϵ}_B$ after the prefactor $1/2$. Take the supremum and use ${\mathrm{Sig}}_{\Phi }\le 1$.