= Solution
For <reachability in control theory>, a state $x_1$ is reachable from $x_0$ if an admissible input and a permitted duration drive the dynamical system from $x_0$ to $x_1$. <Controllability> requires this for every initial-target pair in the specified state space. Restrictions on duration or field amplitudes change the reachable set, so the admissible controls are part of the definition.
For a closed quantum system, $\dot U=-iH[f(t)]U$, $U(0)=I$, and states evolve as $\psi(t)=U(t)\psi(0)$ or $\rho(t)=U(t)\rho(0)U(t)^\dagger$. This gives three notions of <quantum controllability>. <Unitary operator controllability> requires every desired $U_d\in U(N)$ to be reachable from the identity; when a <global phase> is immaterial, one instead asks for every special unitary operation. <Density operator controllability> requires every pair of <density operators> with the same spectrum to be connected. The spectral qualification is necessary because <unitary time evolution> cannot change eigenvalues; the natural state space is a <unitary orbit of a density operator>. <Pure-state controllability> requires any normalized initial state to be transferred to any target ray, $U(T)\psi_0=e^{i\chi}\psi_d$.
Thus $\boxed{\text{operator controllability}\Rightarrow\text{density operator controllability}\Rightarrow\text{pure-state controllability}}$, with the operator implication also valid if <global phase> is discarded. The converse implications need not hold. A pure-state task concerns one vector, whereas an operator task must act correctly on every input vector at once.
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