Corollary: An explicit complementary witness

Suppose states $\rho ,\sigma$ and a complementary effect $H_B$ give

Then every deterministic decoder obeys

A proved lower confidence bound $q_L$ can replace $q$ on its confidence event.

Proof. The complementary output trace distance is at least $q$. For every ${\tau }_B$, its two distances from the corresponding outputs are each at most $d_{\diamond }(\mathcal{C},{\mathcal{T}}_{{\tau }_B})$. Thus $q\le 2\lambda (\mathcal{C})$. Apply Theorem thm:leakage.