Theorem: Independent recovery extrapolation

Suppose actual and proxy complete channels are compared under access-preserving identifications with distance at most $ϵ$. If

then

This is an information-theoretic decoder-existence conclusion.

Alternatively, let a specified proxy decoder $\hat{\mathcal{D}}$, possibly (eq:petz), have discrepancy

If its actual implementation has $d_{\diamond }({\mathcal{D}}_{\mathrm{imp}},\hat{\mathcal{D}})\le e_{\mathrm{imp}}$, then that fixed implementation has error at most

Proof. Partial traces contract the complete-channel comparison. The replacement triangle inequality proves leakage. Apply a proxy-optimal decoder to the actual accessible channel and use Lemma lem:correction. The independent constant decoder gives the $a_d$ cap. For the specified decoder, use Lemma lem:choi and telescope the channel and implementation differences.