Quantum controllability

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.

A finite-dimensional closed quantum control system with fixed drift and control Hermitian matrices, real control inputs, and dynamics linear in the state at fixed controls and affine in the controls at fixed state. Its Dynamical Lie algebra is generated over the reals by $i{H}_0,i{H}_1,\dots$. Reachability depends on the admissible controls, allowed duration and phase convention.

The dynamical Lie algebra of a finite-dimensional Hamiltonian control system is the real span of its skew-Hermitian drift and control generators and all their nested commutators. The sign choice $iH$ or $-iH$ gives the same algebra. Its connected group determines reachable propagators under the usual freely timed, real-control assumptions. An operator commuting with every generator also commutes with this algebra, imposing invariant sectors.

Let $H_0$ be diagonal and $H_1={\sum }_{j=1}^{N-1}{d}_j(E_{j,j+1}+E_{j+1,j})$, with every $d_j\ne 0$ and the adjacent absolute energy gaps distinct and nonzero. The squared drift adjoint action has distinct eigenvalues on these coupling components, so Lagrange interpolation isolates every adjacent symmetric coupling. One more commutator gives its antisymmetric partner. Their mutual brackets produce adjacent traceless diagonals, while brackets along the connected chain give all longer-range matrix units. Thus the special unitary Lie algebra is generated. Any nonzero-trace generator also supplies the central direction, giving all of $\mathfrak{u}(N)$. This is a sufficient condition in the finite-dimensional freely timed real-control model; connectivity alone does not remove frequency degeneracies.

A drift Quantum Hamiltonian is strongly regular when its energy levels are nondegenerate and distinct unordered transitions have distinct absolute nonzero Bohr frequencies. Sometimes regularity is required only on the graph of allowed control transitions. This lets polynomials in the squared drift adjoint action separate different coupling quadratures and permits frequency-selective resonant quantum control. This notion is unrelated to a strongly regular graph.

For $N\ge 2$, let $\mathfrak{g}$ be a Lie subalgebra of $\mathfrak{u}(N)$ with codimension one. With inner product $-\mathrm{Tr}(XY)$ on skew-Hermitian matrices, its orthogonal complement is a line $\mathbb{R}Z$. Invariance of the inner product shows that $[X,Z]$ lies in that line for every $X\in \mathfrak{g}$; skew-adjointness of the adjoint action makes $[X,Z]$ orthogonal to $Z$, hence zero. A nonscalar skew-Hermitian $Z$ has a centralizer of dimension at most $(N-1{)}^2+1<N^2-1$, by its orthogonal eigenspace decomposition, and cannot centralize all of $\mathfrak{g}$. Thus $Z$ is scalar and $\mathfrak{g}=\mathfrak{su}(N)$. This gives density operator controllability and all determinant-one gates, but not the central phase direction needed for unitary operator controllability.

For a nonsingular skew-symmetric unitary $J$, if each skew-Hermitian control generator obeys $X^{T}J+JX=0$, then so do their real linear combinations and commutators. Differentiating $U^{T}JU$ shows that it stays equal to $J$. The reachable group is therefore contained in a unitary conjugate of the compact symplectic group. This is preservation of an alternating bilinear form, not commutation with $J$.

For $U^{T}JU=J$ with real orthogonal skew-symmetric $J$, the dual $\tilde{\rho }=J{\rho }^T{J}^{†}$ transforms as $\widetilde{U\rho U^{†}}=U\tilde{\rho }{U}^{†}$. Hence $I_J(\rho )$ is constant on the reachable unitary orbit of a density operator. Different values can obstruct density operator controllability even when the ordinary spectra agree. For a four-dimensional $J$ pairing indices $1,4$ and $2,3$, the two diagonal arrangements $(a,a,b,b)$ and $(a,b,b,a)$ have invariants $4ab$ and $2(a^2+b^2)$.

Let $H_0=\mathrm{diag}(-{\omega }_1,0,-{\omega }_2)$ and $H_1=d_1(E_{12}+E_{21})+d_2(E_{23}+E_{32})$, with nonzero $d_j$, distinct positive ${\omega }_j$. The squared adjoint action of $-i{H}_0$ has different eigenvalues $-{\omega }_1^2,-{\omega }_2^2$ on the two adjacent skew-Hermitian couplings. Linear combinations of $-i{H}_1$ and its double commutator with $-i{H}_0$ isolate both couplings. Further commutators give their real antisymmetric partners, two traceless diagonal generators, and both $1$–$3$ generators, hence the special unitary Lie algebra $\mathfrak{su}(3)$. The drift has nonzero trace; subtracting its traceless part supplies $iI$, giving the Lie algebra of the unitary group $\mathfrak{u}(3)$ and unitary operator controllability under the usual freely timed real-control assumptions.

Let $J$ be a symmetric unitary matrix and every skew-Hermitian control generator obey $x^{T}J+Jx=0$. The condition is closed under real linear combinations and commutators, and evolution preserves $U^{T}JU=J$. In a unitary basis where $J$ becomes the identity, the generators are real antisymmetric, so the Dynamical Lie algebra is contained in $\mathfrak{so}(N)$. The state quantity $|{\psi }^{T}J\psi |$ is invariant under the reachable evolution and insensitive to global phase, obstructing unrestricted pure-state controllability. This is a bilinear-form symmetry; it does not assert that $J$ commutes with the Hamiltonians.

For $N\ge 2$, suppose a skew-Hermitian Lie algebra becomes all of $\mathfrak{u}(N)$ after adding the scalar identity direction. Scalar matrices commute, so $[\mathfrak{g},\mathfrak{g}]=[\mathfrak{u}(N),\mathfrak{u}(N)]=\mathfrak{su}(N)$ is contained in $\mathfrak{g}$. If $\mathfrak{g}$ contains any nonzero-trace generator, subtracting its traceless part supplies $iI$, and $\mathfrak{g}$ was already all of $\mathfrak{u}(N)$. This permits using identity-augmented commutator calculations without assuming an additional physical identity control.

For freely chosen control duration in a finite-dimensional closed system, pure-state controllability means the reachable group acts transitively on normalized states or their rays. Up to unitary conjugation, the possible Dynamical Lie algebras are $\mathfrak{u}(N)$, $\mathfrak{su}(N)$, and for even $N=2n$ the standard compact symplectic Lie algebra $\mathfrak{sp}(n)$, with or without $\mathbb{R}iI$. For $N\ge 4$, the symplectic possibilities are pure-state controllable but generally lack density operator controllability.

A closed system has density operator controllability when every pair of density operators with the same spectrum can be connected by its controls. Under the standard finite-dimensional Hamiltonian assumptions, this holds exactly for Dynamical Lie algebra $\mathfrak{u}(N)$ or $\mathfrak{su}(N)$. This definition ranges over every spectral orbit; a particular degenerate spectrum can be controllable under a smaller group.

A finite-dimensional closed-system control model is unitarily operator controllable when its reachable propagators include every unitary operator. For the usual real Hamiltonian controls and freely chosen duration, this requires the Dynamical Lie algebra to be $\mathfrak{u}(N)$. If global phase is ignored, $\mathfrak{su}(N)$ also suffices; exact access to all of $U(N)$ still requires the central phase direction.