Lemma: Contraction and isometric comparison

The following statements hold.

1. CPTP pre- and postprocessing do not increase $d_{\diamond }$.

2. For replacement channels with the same input,

   $$d_{\diamond }({\mathcal{T}}_{\sigma },{\mathcal{T}}_{\omega })=d_{\mathrm{tr}}(\sigma ,\omega ).$$

3. If $V,W:C\to R\otimes B$ are isometries, then

   $$d_{\diamond }({\mathrm{Ad}}_V,{\mathrm{Ad}}_W)\le \min \{1,{\Vert V-W\Vert }_{\infty }\}.$$

   The same bound holds for either of their two partial-trace channels.

4. If ${\mathcal{N}}_1,{\mathcal{N}}_2$ have common interfaces, then

   $$\left|{\epsilon }_{\mathrm{rec}}({\mathcal{N}}_1)-{\epsilon }_{\mathrm{rec}}({\mathcal{N}}_2)\right|\le d_{\diamond }({\mathcal{N}}_1,{\mathcal{N}}_2).$$

   The analogous inequality holds for ${\delta }_{\mathrm{leak}}$ of two channels with a common residual output.

Proof. A channel has diamond norm one, so submultiplicativity proves (i). For (ii), applying the replacement-channel difference to a referenced state ${\rho }_{QC}$ gives ${\rho }_Q\otimes (\sigma -\omega )$. Its trace norm is ${\Vert \sigma -\omega \Vert }_1$.

For (iii), after adjoining any reference, expand

The trace norm is at most $2{\Vert V-W\Vert }_{\infty }{\Vert X\Vert }_1$. Divide by two; channel distances are at most one. Part (i) gives the marginal statements.

For (iv), use any decoder for ${\mathcal{N}}_1$ also on ${\mathcal{N}}_2$, apply the triangle inequality and (i), minimize, and interchange the two channels. The leakage statement follows by using the same replacement state in both comparisons.