Theorem: Typed finite-confidence recovery certificate

Suppose an event of probability at least $1-\alpha$ jointly gives

Define

On that event,

For every target-code effect $F$,

For every target-code Hermitian $O$,

If ${\mathcal{E}}_{\star }=D_{\star }{\mathcal{N}}_{\star }$ is the specified physical factorization, then also

This last bound does not establish optimality of $D_{\star }$.

Proof. Lemma lem:choipure bounds $e({\mathcal{E}}_0)$. The reverse triangle inequality gives

Theorem thm:effect gives the observable conclusions. Optimal recovery is no worse than the specified decoder or the decoder that ignores its input and prepares $I_C/d$; Proposition prop:classical evaluates the latter.