Proposition: Permanent classical recovery obstruction

Every purely classical channel with code input dimension $d\ge 2$ satisfies

The bound is sharp for computational-basis measurement. Its complete complement is never constant.

Proof. Following a classical channel by any quantum decoder gives a measure-and-prepare channel. Its Bell output is separable HorodeckiShorRuskai. For product states,

Convexity extends the bound to separable states. The Bell-projector test gives error at least $1-1/d$.

Basis measurement followed by basis-state preparation is dephasing. Dephasing is the average of the $d$ phase conjugations, so separating its identity term gives matching error $1-1/d$. A constant complete complement would imply exact recovery by Theorem thm:exact, a contradiction.