Theorem: Conditional selective relational readout

On the encoded family (eq:phaseencoding), $P_{\mathrm{rel}}$ has complete selective record

with the unit residual on every positive branch. It is invariant under Hypothesis hyp:reference. Under (eq:matching), it is invariant on the whole convolution gauge algebra in the stated coefficient model.

Proof. The amplitudes after CNOT and Hadamard, in raw output order $(b,e)$, are

The raw $e=0$ probabilities depend on $\alpha +\beta$. Thus reporting both raw bits would not be invariant.

For arbitrary two-qubit density operators, put

The path formula gives the actual selective maps

In the first line the two fine maps with bras $⟨{\varphi }_{+}|,⟨{\varphi }_{-}|$ are added, not postselected. Their effects sum to $|00⟩⟨00|+|11⟩⟨11|$. All three maps sum to a trace-preserving map.

For the encoded input, the first line has weight $1/2$. The other weights are ${\left|e^{i\beta }\pm e^{i\alpha }\right|}^2/8$, giving the stated law.

More generally, let

The failure effect commutes with $G_{\gamma }$, and both ${\chi }_{\pm }$ acquire the same phase $e^{i\gamma }$. Consequently

This proves equality of the displayed selective maps, not just an unexamined equality of effects. The reference hypothesis induces exactly this collective phase shift. Condition (eq:matching) supplies the same conclusion for arbitrary convolution gauge directions.