= Unitary orbit of a density operator
{title2=$\mathcal O(\rho)=\{U\rho U^\dagger:U\in U(N)\}$}
The unitary orbit consists exactly of <density operators> with the same spectrum. If distinct eigenvalues have multiplicities $n_j$, its stabilizer is $\prod_jU(n_j)$ and its real dimension is $N^2-\sum_jn_j^2$. A reachable compact group with <Lie algebra> $\mathfrak g$ is transitive on this orbit exactly when $\dim\mathfrak g-\dim(\mathfrak g\cap\{X:[X,\rho]=0\})$ equals the orbit dimension. Global phases act trivially by conjugation.
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