Example: An input-independent noise label is not free decoder assistance

For the preceding implementation,

is input independent. If the decoder nevertheless receives $a$, it can apply ${U_a}^{†}$ and undo the corresponding Pauli error.

Thus at $\eta =0$ the unlabelled carrier channel is complete depolarization, while the labelled channel permits exact correction before RAC decoding. The delay threshold assumes no such label access.

An input-independent label marginal does not establish a product joint state or authorize an omitted recovery wire.