Lemma: Depolarizing composition and exact error

For $a,b\in [0,1]$, write $D_a=a\mathrm{id}+(1-a){\mathcal{T}}_d$. Then

In particular, $d_{\diamond }(D_a,\mathrm{id})={\kappa }_d(1-a)$.

Proof. The relation ${{\mathcal{T}}_d}^2={\mathcal{T}}_d$ and commutation with identity give the composition formula.

On a Bell input, ${\mathcal{T}}_d$ gives $I/d^2$. Testing the Bell projector gives $d_{\diamond }({\mathcal{T}}_d,\mathrm{id})\ge 1-d^{-2}$. The uniform average of the $d^2$ Weyl conjugations is ${\mathcal{T}}_d$. Separating the identity conjugation and using the triangle inequality gives the matching upper bound. Finally, $D_a-D_b=(a-b)(\mathrm{id}-{\mathcal{T}}_d)$.