Lemma: Approximate correction and the two directions

The finite information–disturbance theorem implies

A nearly constant accessible channel instead gives

Proof. The imported information–disturbance inequality, in full diamond normalization, is

Substitution of the half-diamond convention yields (eq:KSW). A constant decoder supplies the $a_d$ cap.

For a replacement accessible channel, any decoded Bell input has Bell overlap $d^{-2}$, hence error at least $a_d$. Preparing $I_L/d$ attains it: depolarization is the uniform average of $d^2$ Weyl conjugations, one of which is identity. The nearly constant lower bound follows by CPTP contraction and the reverse triangle inequality.