The declared external approximate-correction input

For finite-dimensional complementary channels, the KSW information–disturbance theorem gives

Indeed its full-norm inequality

becomes the first inequality after substituting $2{\epsilon }_{\mathrm{rec}}$ and $2{\delta }_{\mathrm{full}}$. The corresponding converse gives the second KSW. We apply these bounds to finite proxies below; no unstated infinite-dimensional attainment assertion is needed.