Proposition: Half-diamond conversion

For a complete interface,

In particular, a purely classical accessible channel satisfies

Proof. Substitute ${\inf }_{\mathcal{D}}{\Vert \mathcal{DN}-\mathrm{id}\Vert }_{\diamond }=2{\epsilon }_{\mathrm{rec}}$ and ${\inf }_{\tau }{\Vert \mathcal{C}-{\mathcal{T}}_{\tau }\Vert }_{\diamond }=2{\delta }_{\mathrm{full}}$ into the displayed information–disturbance inequalities. The universal cap $a_d$ is Lemma lem:replacement. The classical lower bound follows from ${{\epsilon }_{\mathrm{rec}}}^2\le 2{\delta }_{\mathrm{full}}$ and Theorem thm:classical.