Theorem: Permanent classical-record obstruction

For every purely classical output channel,

The bound is tight: complete computational-basis measurement has optimal recovery error exactly $b_d$. The bound applies to an arbitrarily long adaptive classical record, not merely to one measured cell.

Proof. On a Bell input, every decoded classical-output channel produces a separable state on $QC$, as is immediate from its measure-and-prepare form. For a product state,

Convexity gives the same bound for every separable state. The Bell-projector test therefore gives recovery error at least $1-1/d$. This is the entanglement-breaking mechanism HorodeckiShorRuskai; compare the local bounds in CheungHawk04B,CheungHawk04C.

For tightness, measure the basis and prepare the observed basis state. The composite is dephasing:

For $Z={\sum }_i{e}^{2\pi ii/d}|i⟩⟨i|$,

Separating the identity term gives $d_{\diamond }(\Delta ,{\mathrm{id}}_C)\le 1-1/d$. The Bell lower bound gives equality.

Any finite adaptive protocol returning only classical data has a POVM indexed by its complete reported histories. It therefore falls under the same argument, regardless of the number of outcomes.