= Isotropic twirling channel
The <isotropic twirling channel> averages a bipartite <density matrix> over matched local unitaries:
$$
\mathcal T(\rho)=\int(U\otimes\overline U)\rho(U\otimes\overline U)^\dagger\,dU,
$$
where $dU$ is normalized <Haar measure> on the $d$-dimensional <unitary group>. Complex conjugation is taken in the basis defining $|\Phi_d\rangle=d^{-1/2}\sum_j|jj\rangle$.
Writing $P=|\Phi_d\rangle\langle\Phi_d|$ and $F=\operatorname{Tr}(P\rho)$, the result is $FP+(1-F)(I-P)/(d^2-1)$ for $d\geq2$. The averaging preserves $F$, because $U\otimes\overline U$ fixes $|\Phi_d\rangle$, and erases distinctions within its orthogonal complement. Shared classical randomness and local unitaries implement the operation; the resulting <isotropic quantum state> need not be separable.
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