Full-code recovery and the complete complement

This is the full-code Knill–Laflamme criterion KnillLaflamme1997, also locally reported in Oai280-C CheungOai280C. “Complete” means the environment of a Stinespring dilation, including purifying factors, not a selected environmental record.

For example, $\mathcal{N}={\mathrm{Tr}}_S$ on $Q_L\otimes S$ exactly returns the subsystem $Q_L$, but its complement returns the input-dependent $S$ marginal. Subsystem recovery is not a left inverse on the full $Q_L\otimes S$ algebra.