= Pure-state controllability
{title2=$U\psi_0=e^{i\chi}\psi_1$}
= Pure-state controllable
{synonym}
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 $N\ge4$, the symplectic possibilities are pure-state controllable but generally lack <density operator controllability>.
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