Pure-state controllability ($U{\psi }_0=e^{i\chi }{\psi }_1$)

For freely chosen control duration in a finite-dimensional closed system, pure-state controllability means the reachable group acts transitively on normalized states or their rays. Up to unitary conjugation, the possible Dynamical Lie algebras are $\mathfrak{u}(N)$, $\mathfrak{su}(N)$, and for even $N=2n$ the standard compact symplectic Lie algebra $\mathfrak{sp}(n)$, with or without $\mathbb{R}iI$. For $N\ge 4$, the symplectic possibilities are pure-state controllable but generally lack density operator controllability.