  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.
