Twirling channels have minimal mixed-unitary rank

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