 Unitary orbit of a density operator

The unitary orbit consists exactly of density operators with the same spectrum. If distinct eigenvalues have multiplicities $n_j$, its stabilizer is ${\prod }_{j}U(n_j)$ and its real dimension is $N^2-{\sum }_j{n}_j^2$. 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.