Theorem: Unobserved readout-and-drop is partial trace

For every joint state, including an external reference,

Consequently,

with exactly the pre-existing radiation recovery error.

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

Proof. Partial trace is the sum of the diagonal blocks in any orthonormal basis. For a general local channel, its Heisenberg adjoint is unital. It therefore leaves every observable $A\otimes I_B$ on the untouched factors unchanged.