Theorem: Classical accumulation cannot produce an entropy fall

For every fixed input state and append-only classical record,

If a quantum factor $K_n$ is also retained, its hybrid output has

In particular, once the record has positive entropy, the raw classical–quantum output cannot later become pure while that record remains unchanged.

Proof. Classical conditional Shannon entropy is nonnegative, giving (eq:recordmonotonicity). For a block-diagonal state ${\sum }_y{p}_y|y⟩⟨y|\otimes {\rho }_y$, its eigenvalues are $p_y$ times those of ${\rho }_y$. Substitution in the entropy definition gives (eq:hybridentropy).