Lemma: Identity versus complete depolarization

For ${\mathcal{T}}_d(Y)=\mathrm{Tr}(Y)I_d/d$,

Proof. The Weyl twirl represents ${\mathcal{T}}_d=d^{-2}{\sum }_r{\mathrm{Ad}}_{U_r}$. The identity term has weight $d^{-2}$. The triangle inequality and $D_{\diamond }({\mathrm{Ad}}_U,\mathrm{id})\le 1$ give the upper bound $c_d$. On the Bell input, the $d^2$ Weyl outputs are orthogonal Bell states, so the matching trace distance is $c_d$.

For any constant channel ${\mathcal{T}}_{\sigma }$, the Bell output is $I_S/d\otimes \sigma$. Its Bell-projector expectation is $d^{-2}$. The effect bound gives distance at least $c_d$.