Twirling channels have minimal mixed-unitary rank

For a positive integer $d$ and unitary representation $\rho:G\rightarrow\mathrm{U}(d)$ of compact group $G$, the twirling channel for this is linear mapping $\Phi: M_d\rightarrow M_d$ defined as $\Phi(X)=\int_{G}\mathrm{d}\mu(g)\,\rho(g)X\rho(g^{-1})$ every $X\in M_d$, where $\mu$ Haar measure on $G$. Such channels are examples mixed-unitary channels, they in convex hull set fixed size. By Carath\'eodory's theorem, these can always be expressed finite combination channels. We consider rank channels---which minimum number distinct conjugations required to express channels---and show that equal its Choi rank, both which dimension von Neumann algebra generated by representation. Moreover, we how explicitly construct minimal decompositions types provide some examples.