Proposition: Classical transmission has a referenced error floor

If the complete message-to-receiver channel $\Lambda :L(M)\to L(M)$ is measure-and-prepare, then

Local source preprocessing and receiver decoding do not remove this floor when the designated connection remains measure-and-prepare and no additional communication or entanglement resource is admitted.

Proof. For a maximally entangled message-reference input $|{\Omega }_d⟩=d^{-1/2}{\sum }_j|j⟩|j⟩$, a measure-and-prepare channel produces a separable output HSR. Its overlap with $|{\Omega }_d⟩⟨{\Omega }_d|$ is at most $1/d$: Cauchy–Schwarz proves this for pure products, and convexity extends it to separable states. The Bell-projector probability difference is at least $1-1/d$. This is a lower bound on the referenced trace distance and hence on the half-diamond distance.

Preprocessing changes the measured effects, and decoding changes the prepared states. The resulting channel remains measure-and-prepare.