Density operator controllability (${\rho }_1=U{\rho }_0{U}^{†}$)

A closed system has density operator controllability when every pair of density operators with the same spectrum can be connected by its controls. Under the standard finite-dimensional Hamiltonian assumptions, this holds exactly for Dynamical Lie algebra $\mathfrak{u}(N)$ or $\mathfrak{su}(N)$. This definition ranges over every spectral orbit; a particular degenerate spectrum can be controllable under a smaller group.