Proposition: Hybrid output and terminal discard

For any returned block-diagonal output ${\rho }_{YK}={\sum }_y{p}_y|y⟩⟨y|\otimes {\rho }_{K|y}$,

Thus an unchanged positive-entropy record prevents purity of the raw hybrid output.

Furthermore, for all referenced operators,

Terminal residual Enter followed by drop leaves the already-returned accessible channel unchanged.

Proof. The spectrum of a diagonal block is $p_y$ times the spectrum of ${\rho }_{K|y}$, proving the entropy formula. Writing an operator as ${\sum }_{y,z}{X}_{yz}\otimes |y⟩⟨z{|}_A$, readout followed by discard returns ${\sum }_y{X}_{yy}$, precisely the partial trace. The blocks may contain an arbitrary reference.