Lemma: Contraction and recovery stability

CPTP preprocessing and postprocessing contract $d_{\diamond }$. On fixed accessible output spaces,

Replacement channels satisfy

Moreover, ${\delta }_{\mathrm{full}}(\mathcal{C})=0$ if and only if $\mathcal{C}$ is constant.

Proof. CPTP maps have completely bounded trace norm one. The contraction statements follow by composing the reference-stable norm comparison.

For any fixed decoder, the triangle inequality changes its error by at most $d_{\diamond }({\mathcal{N}}_1,{\mathcal{N}}_2)$. Taking infima in both directions proves recovery stability.

On a referenced input state, a replacement channel returns the reference marginal tensored with its replacement state. This proves the replacement-channel equality; a product input already attains it.

If leakage is zero, choose states ${\sigma }_n$ with $d_{\diamond }(\mathcal{C},{\mathcal{T}}_{{\sigma }_n})\to 0$. Evaluation at ${\tau }_C=I_C/d$ gives ${\sigma }_n\to \mathcal{C}({\tau }_C)$ in trace norm. The replacement-channel equality and the triangle inequality then give

The converse is immediate.