Theorem: Finite-proxy certificate and lift

Let $\hat{V}:C\to \hat{R}\otimes \hat{B}$ be a finite isometry, embedded factorwise into the output space of $V$. Suppose

For a specified proxy residual state $\hat{\sigma }$, set

Then

A finite decoder, extended CPTP to all of $R$, has error

Consequently ${\epsilon }_{\mathrm{rec}}(\mathcal{N})$ is also at most $a_d$.

For a specified proxy decoder with

its extension satisfies

Proof. The complete comparison gives allowance $b$ for both reduced channels. Equation (eq:Choi-bound) bounds proxy leakage by $de$. The triangle inequality proves (eq:leak-certificate). Finite decoder Choi matrices form a compact set, so an optimal proxy decoder exists. Apply (eq:KSW) to the proxy.

Let $P_R$ project onto the embedded finite accessible subspace. Extend the decoder by

with the compression identified with $\hat{R}$. This map is CPTP and agrees on proxy outputs. Postprocessing contraction adds $b$, outside the square root. The constant decoder supplies $a_d$. The direct certificate follows by applying the Choi bound to the specified decoded proxy channel, followed by the same lift.