Lemma: Stability of optimal recovery

For channels with the same input and output spaces,

For an implemented decoder $\tilde{\mathcal{D}}$,

Proof. Postprocessing is contractive for diamond distance. Apply the triangle inequality for each decoder and then take infima. The implemented estimate follows by inserting $\mathcal{D}\tilde{\mathcal{N}}$ and $\mathcal{DN}$.