Corollary: Identical marginal diagnostics do not certify recovery

For the Bell input, the noiseless full-transfer experiment $(\eta ,\lambda )=(1,1)$ and the completely dephased experiment $(1,0)$ have identical accessible density matrices. They therefore have identical accessible occupation probabilities, purity, and every other observable depending only on that marginal.

Nevertheless their optimal recovery errors are respectively $0$ and $1/2$, their decoded reference concurrences are $1$ and $0$, and their reference–accessible mutual informations are $2$ and $1$.

Proof. Apply Theorem thm:fullNoise at its two endpoints.