Theorem: Readout-and-drop is exactly local partial trace

For every joint state, including any external reference,

Consequently,

Its radiation-only recovery error is exactly the pre-existing ${\epsilon }_{\mathrm{rec}}({\mathcal{N}}_t)$.

More generally, every trace-preserving operation on $B$, with all its outputs ignored, leaves the state on the untouched reference and $R$ unchanged.

Proof. Partial trace in the computational basis is the sum of the diagonal $B$-blocks, which proves the first equality and (eq:forget-equality).

For the general statement, let ${\Lambda }_B$ be the local channel. Its Heisenberg adjoint is unital. For every bounded observable $A$ on the untouched factors,

All expectations on those factors are unchanged. This argument applies to normal channels on separable spaces; equivalently, in a Kraus representation, ${\sum }_j{L}_j^{†}{L}_j=I_B$.