Proposition: Destructive measure-and-prepare obstruction

Any protocol that first reduces all message-bearing receiver systems to classical records and then prepares a qubit from those records is entanglement breaking. Its recovery half-diamond error on an arbitrary qubit message is at least $1/2$.

At $p=\eta =1$, this lower bound is attained by logical $Z$-measurement followed by preparation of its reported eigenstate, but that protocol is not exact quantum recovery.

Proof. The referenced output is a sum

of positive product operators, hence is separable. For a normalized pure product state, Bell overlap is at most $1/2$ by Cauchy–Schwarz. Convexity gives the same bound for all separable two-qubit states. Testing the Bell projector gives half-diamond error at least $1/2$.

Logical $Z$-measurement and preparation implement ${\Delta }_Z=(\mathrm{id}+{\mathrm{Ad}}_Z)/2$. Lemma lem:Pauli gives error $1/2$.