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Twirling operation w.r.t. a group $U(d)$ is equivalent to a depolarizing channel
I will give an extended explanation of Nielsen's proof, i.e. your first ref link. The idea is that, \(\rho=\sum_ip_i|i\rangle\langle i|\), we can prove it's depolarizing channel for each \(|i\rangle\langle i|\) with same \(p\), then