Isotropic twirling channel

The isotropic twirling channel averages a bipartite density matrix over matched local unitaries:where $dU$ is normalized Haar measure on the $d$-dimensional unitary group. Complex conjugation is taken in the basis defining $|{\Phi }_d⟩=d^{-1/2}{\sum }_j|jj⟩$.

Writing $P=|{\Phi }_d⟩⟨{\Phi }_d|$ and $F=\mathrm{Tr}(P\rho )$, the result is $FP+(1-F)(I-P)/(d^2-1)$ for $d\ge 2$. The averaging preserves $F$, because $U\otimes \overline{U}$ fixes $|{\Phi }_d⟩$, 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.