Proposition: Fixed-interface approximation

The instrument (eq:table-instrument) is CPTP. Suppose normalized approximating coefficient vectors satisfy

On the same finite interfaces, the approximating complete instrument obeys

Exact discrete outputs such as acceptance and incidence guards must be retained exactly, not inferred from approximate scores.

Proof. Completeness of the displayed Kraus family proves the channel assertion. Define

These are isometries and ${\Vert {\tilde{V}}_a-V_a\Vert }_{\mathrm{op}}\le \delta$, because distinct $x$-columns have orthogonal code factors. Use one joint controlled isometry for all clock blocks, then the common record dephasing/readout. For isometries $V,W$, expansion into two terms gives $\frac{1}{2}{\Vert {\mathrm{Ad}}_V-{\mathrm{Ad}}_W\Vert }_{\diamond }\le {\Vert V-W\Vert }_{\mathrm{op}}$, also with references. Channel postprocessing contracts the norm. The last assertion is the guard distinction of Proposition prop:guard.