Records versus residual instruments

Two one-outcome instruments can report the same symbol while one retains a qubit and the other dephases it. On $|+⟩$, the residuals $|+⟩⟨+|$ and $I/2$ have trace distance $1/2$, although their record laws agree.

This is the separation recorded in En03-A CheungEn03A ({thm:obstruction}). Conversely, equality of complete accessible process data and instruments implies equality of adaptive record laws CheungPsy02B ({thm:process}). A histogram does not reconstruct an unobserved residual mechanism.

Accordingly, the diagnostic priority is the complete isometry, followed by both partial traces. Selected occupation or Bell tests are not substitutes.