Lemma: Approximate correction and replacement benchmarks

Every accessible channel has ${\epsilon }_{\mathrm{rec}}\le a_d$. A replacement accessible channel has ${\epsilon }_{\mathrm{rec}}=a_d$. Moreover,

Proof. A decoded replacement channel sends a Bell input to $I_Q/d\otimes {\sigma }_L$, with Bell overlap $d^{-2}$. Testing the Bell projector gives error at least $a_d$. Preparing $I_L/d$ attains it, since depolarization is the uniform average of $d^2$ Weyl conjugations, one being identity. This constant decoder also gives the universal upper bound. Equation (eq:lower) follows by contraction and the reverse triangle inequality.

The finite information–disturbance theorem, stated in full diamond normalization, gives

Substituting the half-diamond convention proves (eq:KSW).