Proposition: Approximate correction input

The finite information–disturbance theorem gives

Proof. Use the Stinespring continuity theorem and its information–disturbance consequence in raw diamond normalization KSW (Theorems 1 and 3):

Here the replacement state is optimized, and the converse requires existence of an appropriate replacement state, not every state. Substitute ${ϵ}_{\mathrm{raw}}=2{\epsilon }_{\mathrm{rec}}$ and ${\delta }_{\mathrm{raw}}=2{\delta }_{\mathrm{full}}$. The independent constant-decoder cap is Lemma lem:benchmarks.