Corollary: Single-cell replacement channels

For every $p,\eta$, each individual emitted cell is exactly independent of the message, including references:

Nevertheless, when $p=\eta =1$, the collective channel is exactly quantum-correctable.

Proof. The first marginal is unchanged by a later source interaction. The second follows from (eq:twirl), which applies to ${\Lambda }_{\eta }(X)$ because its trace is $\mathrm{tr}X$. The collective recovery assertion is proved in Theorem thm:recovery.