= Solution
In units $\hbar=1$, a <bilinear Hamiltonian control system> has
$$
\dot U(t)=-i\left(H_0+\sum_{k=1}^m u_k(t)H_k\right)U(t),\qquad U(0)=I,
$$
where the <Hermitian matrices> $H_0,H_k$ are fixed, and the real functions $u_k$ are admissible <control inputs>. The corresponding vector or <density operator> dynamics are $\dot\psi=-iH(u)\psi$ or $\dot\rho=-i[H(u),\rho]$. The dynamics are linear in the state at fixed input and affine in the inputs at fixed state; this is the bilinear terminology.
The <reachable set> from an initial state $x_0$ at time $T$ is $\mathcal R_T(x_0)=\{x(T;x_0,u):u\text{ admissible}\}$. With freely chosen duration, take $\mathcal R(x_0)=\bigcup_{T\geq0}\mathcal R_T(x_0)$. \b[Controllability means every allowed target is reachable from every allowed initial state]. The allowed space and phase convention must be specified. For closed <quantum control>, <density operators> remain on their initial <unitary orbit of a density operator>, rather than becoming arbitrary density operators with different <eigenvalues>.
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