Proposition: Named-effect transfer

For an encoding channel $\mathcal{E}$, suppose

for effects $F_{\mathrm{b}},F_{\mathrm{m}}$. Then the corresponding expectation error is at most $e(t)+a_F$. The converse is false.

Proof. Insert the expectation in $\mathcal{E}{\Lambda }_{\mathrm{m},t}(\rho )$. An effect changes by at most trace distance, and the operator mismatch contributes at most $a_F$. Identity and complete $Z$-dephasing agree on the effect $|0⟩⟨0|$ for every input but are different channels.