Proposition: Two distinct destructive controls

After internal-label dephasing,

A label-blind occupation measurement that retains only the record “vacuum” or “occupied cell $j$” and discards all quantum carriers gives a constant channel and recovery error $c_d$.

Proof. The coherent-mode decoder gives $(1-q){\Delta }_C+q{\mathcal{T}}_{I/d}$. Since ${\Delta }_C$ is the uniform mixture of the $d$ phase unitaries, this is a Weyl mixture with identity weight $(1-q)/d+q/d^2$. Separating that term gives the upper bound.

On a Bell input, the successful label-dephased branch is separable between radiation and reference. Decoding preserves separability. A separable $d$-by-$d$ state has Bell overlap at most $1/d$: Cauchy–Schwarz proves this for pure product vectors, and mixtures preserve it. The vacuum contribution is $q/d^2$. The Bell test yields the matching lower bound.

The label-blind record probabilities $q,p_1,\dots ,p_N$ depend on the input only through its trace. Discarding every carrier therefore leaves a constant channel. Apply Lemma lem:replacement.