 Solution

Assume the usual real Hamiltonian controls with freely chosen finite duration, rather than a fixed-time or amplitude-restricted task. Define the Dynamical Lie algebrathe real linear span closed under all nested commutators. Replacing each generator by $-i{H}_m$ gives the same algebra. The associated connected reachable group acts on propagators, on each unitary orbit of a density operator, and on pure states. For $N\ge 2$, the necessary and sufficient conditions are as follows.

For exact unitary operator controllability, $\boxed{\mathfrak{g}=\mathfrak{u}(N)}$. For operator control up to global phase, the possibilities are $\mathfrak{su}(N)$ or $\mathfrak{u}(N)$. If every Hamiltonian is traceless, $\det U(T)=\exp [-i{\int }_0^T\mathrm{Tr}H(t)dt]=1$, so the central phase direction is absent and exact access is to $SU(N)$. The full special unitary Lie algebra has dimension $N^2-1$ and the Lie algebra of the unitary group has dimension $N^2$.

For density operator controllability on every spectral orbit,Scalar phases do not affect conjugation. If only a particular density spectrum is being considered, a smaller group can suffice: for distinct eigenvalue multiplicities $n_j$ and centralizer ${\mathfrak{c}}_{\rho }=\{X\in \mathfrak{u}(N):[X,\rho ]=0\}$, the orbit criterion is $\dim \mathfrak{g}-\dim (\mathfrak{g}\cap {\mathfrak{c}}_{\rho })=N^2-{\sum }_j{n}_j^2$. A maximally mixed state has a one-point orbit and needs no controls. This restricted-orbit condition is weaker than controllability for every density spectrum.

For pure-state controllability, up to a fixed unitary change of basis,where the last two alternatives occur only for even $N$. Here $\mathfrak{sp}(n)$ is the standard compact symplectic Lie algebra acting on ${\mathbb{C}}^{2n}$, not the real noncompact symplectic algebra. These are precisely the unitary matrix algebras whose connected groups act transitively on normalized complex vectors, and also give the ray version of controllability. For $N=2$, $\mathfrak{sp}(1)=\mathfrak{su}(2)$. For even $N\ge 4$, the symplectic cases show why pure-state control can be weaker than density control: $\dim \mathfrak{sp}(n)=n(2n+1)$ is smaller than the nondegenerate density-orbit dimension $N^2-N$, and its scalar center acts trivially on that orbit. For $N=1$, the physical density and pure-state spaces are single points, while exact operator controllability still requires the phase algebra $\mathfrak{u}(1)$.