Proposition: Record obstruction and append-only entropy

For every purely classical output channel on a $d$-dimensional code,

Computational-basis measurement attains the bound. For an unedited append-only record,

Moreover,

on all referenced operators.

Proof. Any decoder following a classical-output channel is measure-and-prepare, so its Bell output is separable HorodeckiShorRuskai. Every separable state has Bell overlap at most $1/d$, since a product state has overlap $\mathrm{Tr}({{\rho }_Q}^{\mathsf{T}}{\sigma }_C)/d\le 1/d$. The Bell-projector test gives error at least $b_d$. Basis-state preparation gives dephasing, the average of $d$ phase conjugations, and has matching error $b_d$.

The entropy assertion is the classical chain rule. For the last assertion, expand $X={\sum }_{y,z}{X}_{yz}\otimes |y⟩⟨z{|}_A$. Readout followed by discard returns ${\sum }_y{X}_{yy}$, the partial trace. The blocks may include any reference.