Lemma: Replacement and approximate correction

A replacement accessible channel has optimal recovery error $a_d$; every channel has recovery error at most $a_d$. If $d_{\diamond }(\mathcal{N},{\mathcal{T}}_{\omega })\le \eta$, then

For a complete interface, the information–disturbance theorem gives

Proof. On a Bell input, a decoded replacement produces $I_Q/d\otimes {\sigma }_C$, with Bell overlap $d^{-2}$. Preparing $I_C/d$ attains the lower bound: depolarization is the uniform average of the $d^2$ Weyl conjugations, one of which is identity. This also gives the universal constant-decoder cap. The accessible-replacement lower bound follows by contraction and the reverse triangle inequality.

For the last assertion use the finite information–disturbance inequality

and substitute the half-diamond normalization KSW.