Proposition: Complete-complement certificate from Bell overlap

For a specified complete finite isometry $V:C\to R\otimes B$, with $C$ a qubit, let ${\mathcal{C}}_B$ be its complement. A CPTP decoder with Bell overlap $F_e$ implies existence of ${\tau }_B$ such that

A valid lower bound $F_L$ can replace $F_e$ in the upper expression.

Proof. Dilate the decoder and evolve the Bell input to a pure vector $\Psi$ on reference, decoded output, and enlarged environment. Projection onto the target Bell vector has norm $\sqrt{F_e}$. For $F_e>0$, normalize the environmental projection to $\xi$. The distance between $\Psi$ and ${\Phi }_2\otimes \xi$ is $\sqrt{1-F_e}$. Tracing decoder systems gives

Lemma lem:choi gives the factor two. For $F_e=0$, the cap one suffices.