Unitary operator controllability ($\mathfrak{g}=\mathfrak{u}(N)$)

A finite-dimensional closed-system control model is unitarily operator controllable when its reachable propagators include every unitary operator. For the usual real Hamiltonian controls and freely chosen duration, this requires the Dynamical Lie algebra to be $\mathfrak{u}(N)$. If global phase is ignored, $\mathfrak{su}(N)$ also suffices; exact access to all of $U(N)$ still requires the central phase direction.