Lemma: Constancy and quantitative witnesses

For a finite input, ${\delta }_{\mathrm{full}}=0$ if and only if $\mathcal{C}$ is constant. If two code states satisfy

then

If $O=O^{†}$, $\Vert O\Vert \le 1$, and

then

Proof. For every replacement state, the triangle inequality gives

Also $w\le {\Vert \mathcal{C}({\rho }_0-{\rho }_1)\Vert }_1=2x$.

If leakage is zero, choose ${\sigma }_n$ with $d_{\diamond }(\mathcal{C},{\mathcal{T}}_{{\sigma }_n})\to 0$. Evaluation at ${\tau }_C$ gives ${\sigma }_n\to \mathcal{C}({\tau }_C)$ in trace norm. Replacement-channel distance equals the corresponding state distance, so $\mathcal{C}={\mathcal{T}}_{\mathcal{C}({\tau }_C)}$.