Proposition: Carrier contraction does not contract an external persistent label

Let ${\omega }_0=\rho \otimes |0⟩⟨0|$ and ${\omega }_1=\rho \otimes |1⟩⟨1|$. For every carrier channel $\Lambda$,

The label may influence a future carrier observation through a later controlled operation, even when $\Lambda$ is complete depolarization.

Proof. The environmental supports are orthogonal, so the two joint states remain perfectly distinguishable. For the future observation, replace or allocate the carrier in $|0⟩$, apply the label-controlled ${\sigma }_x$, and measure $Z$. The resulting answer reveals the label.