Theorem: Optimal flagged-loss candidate

For the channel (eq:erasure), the printed complete decoder has error $3p/4$ and is optimal among all CPTP decoders.

Proof. Its returned channel is $(1-p){\mathrm{id}}_2+p{\mathrm{Dep}}_2$, with error $3p/4$. For any decoder, on a Bell input the non-erased contribution has Bell overlap at most one, while the erasure contribution has overlap exactly $1/4$ after any replacement output. The total overlap is at most $1-3p/4$. The Bell-projector measurement therefore gives error at least $3p/4$.