Theorem: Leakage-certificate stability

Let $\mathcal{W}$ and $\hat{\mathcal{W}}$ be complete channels $C\to A\otimes B$, each induced by an isometry, and suppose

Their accessible and complementary channels satisfy

If ${\delta }_{\mathrm{leak}}(\hat{\mathcal{C}})\le \lambda$, then

Proof. Partial trace gives $d_{\diamond }(\mathcal{N},\hat{\mathcal{N}})\le e$ and $d_{\diamond }(\mathcal{C},\hat{\mathcal{C}})\le e$. For every ${\tau }_B$,

Take the infimum and exchange the two channels to obtain (eq:leakstable). For every decoder, postcomposition contracts distance; the same argument proves (eq:recstable). Apply Proposition prop:approximate first to the comparison channel and then to the actual channel.