Lemma: Residual-response witness

If

then ${\delta }_{\mathrm{full}}(\mathcal{E})\ge x/2$. Also ${\delta }_{\mathrm{full}}(\mathcal{E})=0$ if and only if $\mathcal{E}$ is a replacement channel.

Proof. For any $\sigma$, the triangle inequality gives

Take the infimum. In finite dimensions the density operators $\sigma$ form a compact set, so the infimum defining zero leakage is attained. Zero distance is channel equality.