Proposition: Complete table instrument

The map (eq:evaluation) is CPTP and has conditional sign mean $v_{ki}$. It preserves the payload in each subnormalized branch, including its correlations with an untouched reference. A finite controlled probe preparation, followed by destructive probe readout, realizes this instrument under specified finite unitary permissions.

Proof. Its Kraus operators are

Their squared moduli sum to identity. The reference statement is the same operator identity tensored with the reference identity.

Prepare the probe vector $\sqrt{p_{+}(k,i)}|+⟩+\sqrt{p_{-}(k,i)}|-⟩$, controlled by the selected classical indices, and read the probe. The difference of its outcome probabilities is $v_{ki}$. Every finite normalized probe preparation extends to a finite unitary with a declared blank input.