Proposition: Operational content of the recovery metric

If a decoder satisfies $D_{\diamond }({\mathcal{D}}_t{\mathcal{N}}_t,{\mathrm{id}}_A)\le ϵ$, then, for every reference $S^{\prime }$ and every joint state ${\rho }^{S^{\prime }A}$,

Conversely, it is sufficient to verify this uniform state bound for a reference of dimension $d$.

Proof. Apply the definition of the stabilized trace norm to the channel difference. For a Hermiticity-preserving difference of channels, the diamond norm is attained on a state with reference dimension equal to the input dimension. Larger references do not increase the supremum Watrous2018.