Theorem: Finite complete-dilation certificate and continuum lift

Let $\hat{V}:C\to \hat{R}\otimes \hat{B}$ be a finite-dimensional isometry embedded into a common complete output space. Suppose

For a finite residual density operator $\hat{\sigma }$, put

Then

There exists a finite CPTP decoder, completed on the full radiation space, for which

Accordingly,

Independently, if a specified finite CPTP decoder $\hat{\mathcal{D}}$ has

its complete extension satisfies

Proof. Partial trace of the complete-channel comparison gives

The triangle inequality and Lemma lem:choi prove (eq:delta-certificate).

For the finite channel, $\widehat{{\delta }_{\mathrm{full}}}\le de$. The finite decoder set is compact, so an optimal decoder exists. Equation (eq:KSW) gives its recovery error at most $\sqrt{2de}$.

If $P_R$ is the projection onto the embedded $\hat{R}$, complete this decoder by

The map is CP, and its two traces sum to $\mathrm{Tr}Y$. Postprocessing contraction adds at most $b$, proving (eq:epsilon-certificate). The constant decoder supplies the additional bound $a_d$.

For the specified decoder, Lemma lem:choi gives the finite error $d{r}_{\mathrm{dec}}$. The same extension and comparison prove (eq:direct-certificate).