Proposition: One accessible marginal does not determine recovery

The identity qubit channel and complete $Z$-dephasing produce the same accessible output $I_2/2$ for the system half of a Bell input. Their optimal deterministic referenced recovery errors are, respectively,

in half-diamond distance to the identity.

Proof. Both channels fix $I_2/2$. Identity is recovered exactly. Complete dephasing is entanglement breaking, so every decoder leaves its Bell-input output separable across reference and receiver. A separable two-qubit state has Bell-projector expectation at most $1/2$, by Cauchy–Schwarz for pure product states and convexity. The Bell test therefore gives error at least $1/2$.

For attainment,

so $d_{\diamond }({\Delta }_Z,\mathrm{id})\le 1/2$.