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Controllability asks whether the available control inputs can move a dynamical system between every pair of states in its declared state space. Fixing a duration $T>0$ gives the precise condition ${\mathcal{R}}_T(x_0)=X$ for every $x_0\in X$, where $X$ is the state space and ${\mathcal{R}}_T(x_0)$ its reachable set. With freely chosen duration, every pair must instead be connectable in some permitted finite time, which may depend on the pair. Always specify admissible control inputs, deadlines and any constraints along the path. ^[ref]

The surprising part is that one control input can control several state variables by acting at different times. For the double integrator, acceleration changes velocity directly, while the accumulated velocity changes position. Starting from rest, accelerate during the first half of the journey and brake during the second: the final velocity can vanish although the final position has changed. The figure is an analytic mathematical example in dimensionless quantities, with no experimental measurements.

Controllability is useful before trying to design a controller: a requested endpoint outside the reachable set cannot be attained by clever choice of feedback control. The Kalman rank criterion makes this an exact linear algebra calculation for the stated linear time-invariant system model. It concerns whole states, so attaining one plant output is not enough to establish controllability. In quantum control, the declared state space may be an orbit constrained by conservation laws; see quantum controllability and unitary orbit of a density operator.

Assume piecewise-constant real controls with freely chosen duration, as in the usual finite-dimensional quantum controllability theorem. We can establish the strongest notion here: the Dynamical Lie algebra is $\mathfrak{u}(3)$, so the system has unitary operator controllability, and consequently both pure-state and fixed-spectrum density operator controllability.

To verify the hypotheses explicitly, use matrix units $E_{jk}=|j⟩⟨k|$ and put $K_0=-i{H}_0$, $X_{jk}=-i(E_{jk}+E_{kj})$, $Y_{jk}=E_{jk}-E_{kj}$. Then $K_1=-i{H}_1=d_1{X}_{12}+d_2{X}_{23}$ andSince $d_1{d}_2\ne 0$ and ${\omega }_1^2\ne {\omega }_2^2$, linear combinations of these two elements isolate $X_{12}$ and $X_{23}$. Their commutators with $K_0$ give ${\omega }_1{Y}_{12}$ and $-{\omega }_2{Y}_{23}$, respectively. Further commutators give two independent traceless diagonal generators, since $[X_{jk},Y_{jk}]=2i(E_{jj}-E_{kk})$, as well as the two $1$–$3$ generators. These eight generators span the special unitary Lie algebra $\mathfrak{su}(3)$.

Finally $\mathrm{Tr}{K}_0=i({\omega }_1+{\omega }_2)\ne 0$. Subtract its traceless component, already in the generated algebra, to obtain a nonzero multiple of $iI$. This proves the Lie algebra of the unitary group is all of $\mathfrak{u}(3)$, not merely $\mathfrak{su}(3)$. The compact-group controllability theorem then gives access to every unitary with a suitable control and some duration. This does not assert controllability at every predetermined short time or under additional bandwidth constraints. This proves that two adjacent transitions generate the qutrit unitary Lie algebra.

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 $\rho (t)=U(t)\rho (0)U(t{)}^{†}$. 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.

Quantum controllability concerns the transformations reachable under specified quantum control inputs. Unitary operator controllability concerns entire propagators; density operator controllability concerns conjugation on every spectral orbit; pure-state controllability concerns normalized state vectors or their physical rays. These notions need not coincide.