Introduction and scope

An operational recovery statement has more content than an operator identity or an existence theorem. It specifies what information is to be returned, what interface is accessible, which decoder is implemented, what happens on every branch, and how the resulting error is bounded.

Operator-algebra quantum error correction already provides the appropriate mathematical language for quantum, classical, and hybrid targets BenyKempfKribs,BenyKempfKribsPRA. Approximate recovery has corresponding worst-case entanglement-fidelity formulations and constructive recovery results BenyOreshkov. Perfect-tensor networks provide finite bulk-to-boundary code models PastawskiEtAl. The present contribution is an explicit certification interface connecting a supplied finite code to a specified apparatus. It does not replace those theories.

The internal source notes supplied for this manuscript attribute related interfaces to N. Cheung's manuscripts En02-B/C, En05-C, Oai265-B, Oai290-A, Em05-B, XVII-A, and Oai280-A/B CheungEn02B,CheungEn02C,CheungEn05C,CheungOai265B, CheungOai290A,CheungEm05B,CheungXVIIA,CheungOai280AB. Their full texts and publication metadata were not supplied. Accordingly, these citations identify provenance and intended comparisons, not independently checked published theorems. Every mathematical result used below is proved here or attributed to an identified external article. Appendix app:provenance records this distinction.

outcome. For finitely many predeclared tasks we construct intervals

such that, under the printed access, preparation, measurement, and sampling hypotheses,

Here ${\Lambda }_{\star ,j}$ is the channel actually claimed, ${\mathcal{T}}_j$ is the declared target, ${\alpha }_{\mathrm{s}}$ is the validation failure budget, and $\beta$ covers separately justified calibration or transfer assumptions. This is a statement about complete channels, not selected successful branches.