Records and residuals remain different certificates

Two record-retaining instruments may have the same outcome law and different residuals. For example, with one coarse outcome, the identity qubit channel and computational dephasing have identical records but, on $|+⟩⟨+|$, output $|+⟩⟨+|$ and $I/2$. Their trace distance is $1/2$. Implementation errors ${\epsilon }_1,{\epsilon }_2$ leave a gap at least $1/2-{\epsilon }_1-{\epsilon }_2$, by the triangle inequality. This is the observation-boundary discipline compared with VII-E, VII-G, and VII-E-ST CheungObservation.

More generally, a complete recorded instrument that recovers every input in the full quantum algebra can have only input-independent record effects. Indeed, for Kraus operators $A_i$ of the instrument and $B_j$ of a proposed recovery, all $B_j{A}_i$ are scalar multiples of $I$, because the identity channel has rank-one Choi matrix. Completeness of the recovery then gives ${A_i}^{†}{A}_k={\lambda }_{ik}I$. Summing the diagonal terms for one record makes its effect scalar. This full-input criterion is consistent with standard quantum error correction KnillLaflamme,Watrous. It does not prohibit transport of the surviving payload.