Theorem: Conditional complete relational instrument

The protocol has selective maps, on arbitrary two-qubit input operators,

They form a complete instrument. On (eq:phaseinput),

For

the selective-map identities

hold on all operators and with every untouched reference.

Consequently the declared complete output is witness-orbit invariant under Assumption ass:reference; under (eq:matching) the same conclusion holds for the whole stated first-order coefficient algebra.

Proof. With $U=(H\otimes I)\mathrm{CNOT}$, the four destructive path bras are $⟨00|U=⟨{\varphi }_{+}|$, $⟨10|U=⟨{\varphi }_{-}|$, $⟨01|U=⟨{\chi }_{+}|$, and $⟨11|U=⟨{\chi }_{-}|$. Summing the two fine failure maps gives the first line of (eq:relmaps). The four Bell projectors sum to the identity, proving completeness.

The two odd-parity amplitudes have squared moduli ${\left|e^{i\beta }\pm e^{i\alpha }\right|}^2/8$. The even-parity total weight is $1/2$. This gives (eq:relprob).

For $F$, the equivalent computational-basis Kraus bras $⟨00|,⟨11|$ acquire only scalar phases under $G_{\gamma }$; each phase cancels in its selective map. Both ${\chi }_{\pm }$ acquire the common phase $e^{i\gamma }$. Thus every map in (eq:relmaps) is invariant. Tensoring the linear-map identity with a reference identity proves its complete version.

Under the witness translations, both $\alpha ,\beta$ increase by $\lambda c_{\epsilon }/\sigma$. Equation (eq:matching) gives the same collective phase for every permitted gauge direction.