Proposition: Complete evaluation instrument

The map (eq:tablechannel) is CPTP and acts as identity on the surviving payload in each subnormalized classical branch. It preserves arbitrary untouched reference correlations in that channel sense. Its conditional sign mean on catalogue entry $x$ and clock code $a$ is $v_{a,x}$.

It is implementable by a controlled finite probe isometry, probe readout, and classical report generation.

Proof. Kraus operators are

Since ${\sum }_y{p}_y(a,x)=1$, their squared moduli sum to $I_{CGP}$. The formula on matrix units proves the reference claim.

For the probe, use the normalized vector

as a controlled preparation, followed by destructive readout. The resulting finite isometry extends to a unitary on a finite larger register. The sign expectation is $p_{+}-p_{-}=v_{a,x}$.