Proposition: Recovery, implementation, and model triangle

Let ${\mathcal{N}}_{\mathrm{mod}}$ and ${\mathcal{N}}_{\mathrm{act}}$ have the same accessible output space, and let $D$ and $\tilde{D}$ have the same input and output spaces. If

then

Proof. Insert $D{\mathcal{N}}_{\mathrm{mod}}$ and $\tilde{D}{\mathcal{N}}_{\mathrm{mod}}$ and use the triangle inequality. Pre- and post-composition by channels contract diamond distance.