Theorem: Conditional complete CNOT–Hadamard instrument

The selective maps are

They form a complete instrument, with unit signal quantum output, and

Each selective map is invariant under

The map identities hold with arbitrary untouched references and arbitrary passive surviving finite factors.

Therefore the declared coarse output is invariant on the witness orbit under Assumption ass:reference; condition (eq:matching) supplies the corresponding whole-algebra coefficient statement.

Proof. The four destructive path bras after the gates are the four Bell bras. The two even-parity projectors sum to $|00⟩⟨00|+|11⟩⟨11|$; the odd ones are $|{\chi }_{+}⟩⟨{\chi }_{+}|$ and $|{\chi }_{-}⟩⟨{\chi }_{-}|$. Their sum is the identity.

The odd amplitudes are proportional to $e^{i\beta }\pm e^{i\alpha }$, giving the probabilities. For the failure map use Kraus bras $⟨00|,⟨11|$; each acquires only a scalar phase under $G_{\gamma }$. Both ${\chi }_{\pm }$ acquire the same scalar phase. These phases cancel in the selective maps, including their reference extensions.

The two witness phases increase by the same amount. Equation (eq:matching) gives the same conclusion for each stated algebra direction.