Proposition: Exact flagged-erasure recovery error

The unrestricted optimal error for this model is

A two-slot decoder attains it: recover on the $\mathrm{ok}$ branch and prepare $I_2/2$ on the lost branch.

Proof. On a logical Bell input, the good branch has Bell overlap at most one. The erased branch produces a constant output with Bell overlap $1/4$. The total overlap is at most $1-3\ell /4$, giving the lower bound.

The stated decoder implements $(1-\ell )\mathrm{id}+\ell \mathcal{T}$. Lemma lem:constant and homogeneity give error $3\ell /4$.