Theorem: Uniform local-system subsystem recovery

Let ${\mathcal{N}}_{12}$ be the accessible channel obtained from (eq:encoding) by tracing $B$, followed by the character-coordinate transformation. Then

on ${\mathcal{A}}_{\mathrm{in}}$, including every external reference and arbitrary correlations of $C$ with $K$.

The label $k=1$ and the full qubit coefficient subsystem of ${\mathcal{L}}_1$ are recovered with zero ideal error.

Proof. The encoder and partial trace are CPTP. The label compression and output-tag preparation in (eq:decoder) are CP. Unitary conjugation and partial trace are CP. Summing the label branches preserves trace, so ${\mathcal{D}}_{12}$ is CPTP.

For each input block ${\rho }_k$, conjugation by ${U_k}^{†}$ restores

Tracing $R_2$ and the inaccessible gauge gives precisely ${\mathrm{Tr}}_K{\rho }_k$, with label $k$. The calculation is an equality of linear maps, so it holds after tensoring with ${\mathrm{id}}_Q$.