= Solution
A <reachable set> consists of states obtainable from one specified initial state using admissible inputs and allowed durations. <Controllability> means that every target in the declared state space is reachable from every initial state there; the declared space and time convention are essential.
For a closed quantum system, let $U_f$ solve the controlled <Schrödinger equation> with $U_f(0)=I$. The propagator and state reachable sets are
$$
\mathcal R_U=\{U_f(T):f\text{ admissible},\ T\text{ allowed}\},\qquad
\mathcal R_\rho(\rho_0)=\{U\rho_0U^\dagger:U\in\mathcal R_U\}.
$$
A pure-state version acts on normalized vectors up to global phase. <Pure-state controllability> makes every pure target accessible; <unitary operator controllability> permits every desired coherent process, with global phase treated according to the definition.
For <mixed states>, <density operator controllability> means transitivity on each <unitary orbit of a density operator>, not access to all <density operators> regardless of <spectrum>. The <Dynamical Lie algebra> generated by $-iH_0,-iH_1,\ldots$ describes the accessible connected unitary transformations under the usual unconstrained-time control assumptions. Conserved quantities or restricted controls can shrink the <reachable set>. Thus successful preparation of one state is weaker than implementation of an arbitrary gate on all inputs, and coherent <controllability> does not imply purification.
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