= Solution
A <reachable set> contains all states attainable from a specified initial state by admissible <control inputs> in the allowed times, and <controllability> means that every target in the declared state space is reachable from every initial state. For a closed <quantum control> system, distinguish <pure-state controllability> on state rays, <density operator controllability> on fixed-spectrum unitary orbits, and <unitary operator controllability> for entire propagators. State engineering asks for a particular state transfer; process engineering asks for an entire <quantum gate>, which must act correctly on every input. A closed <Hamiltonian> evolution preserves density-operator <eigenvalues>, so arbitrary mixed states of different spectra cannot be connected even by a fully operator-controllable system.
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