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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.

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.

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.

The stated relation places every generator, and therefore the whole Dynamical Lie algebra, inside the standard compact symplectic Lie algebra $\mathfrak{sp}(\ell )\subset \mathfrak{su}(2\ell )$. Indeed the relation is closed under linear combinations and commutators, and differentiating evolution givesThus it supplies a symplectic dynamical symmetry in quantum control, rather than a claim that $J$ commutes with every Quantum Hamiltonian.

For $\ell \ge 2$, b1 is false: even the full compact symplectic group does not act transitively on every mixed-state spectral orbit. Its algebra has dimension $\ell (2\ell +1)<(2\ell {)}^2-1$. An explicit obstruction is the symplectic invariant of a density operatorwhich is unchanged by a symplectic conjugation. For $N=4$ and the printed $J$, let $a,b>0$, $a\ne b$, $2a+2b=1$. The isospectral operators $\mathrm{diag}(a,a,b,b)$ and $\mathrm{diag}(a,b,a,b)$ have invariants $4ab$ and $2(a^2+b^2)$, respectively, and cannot be interconverted. For the exceptional case $\ell =1$, $\mathfrak{sp}(1)=\mathfrak{su}(2)$, so b1 cannot be decided from inclusion alone; equality gives density-state controllability.

b2 is not implied by the displayed relation: more information is needed. It permits the full symplectic algebra but also a single commuting generator, or even zero generators. The full $Sp(\ell )$ is pure-state controllable: identify ${\mathbb{C}}^{2\ell }$ with ${\mathbb{H}}^{\ell }$, extend any unit vector to a quaternionic orthonormal basis, and map one such basis to another. A one-axis diagonal subgroup, in contrast, cannot change level populations.

The required extra information is the generated algebra, obtained by computing its real span of iterated commutators. Within this specified symplectic inclusion, equality $\mathfrak{g}=\mathfrak{sp}(\ell )$ gives pure-state controllability; its dimension is $\ell (2\ell +1)$. The mere inclusion does not establish equality. For $N=2$, equality also answers b1 affirmatively. None of these strictly symplectic generators supplies arbitrary global phase.

A real Lie algebra is a real vector space with a bilinear antisymmetric bracket satisfying the Jacobi identity. For matrices, the bracket is the commutator $[X,Y]=XY-YX$. The Dynamical Lie algebra isIt is found by repeatedly adjoining commutators and taking real linear spans until no new independent generators appear. Let $G$ be its connected dynamical Lie group. Under the standard assumptions of freely timed controls with unrestricted real amplitudes, the following are the relevant Lie criteria.

For pure-state controllability, $G$ must act transitively on normalized state vectors, or on their rays when global phase is disregarded. The standard finite-dimensional classification is, up to a unitary change of basis,Here $\mathfrak{sp}(n)$ is the compact symplectic Lie algebra in its defining complex $2n$-dimensional representation. Equivalently, the infinitesimal action spans the tangent directions to the whole pure-state orbit. The symplectic cases are important because pure-state reachability alone need not give complete mixed-state reachability.

For density operator controllability, the action must be transitive on every unitary orbit of a density operator, meaning on every fixed-spectrum class. The criterion is $\mathfrak{g}=\mathfrak{su}(N)$ or $\mathfrak{u}(N)$. This concerns all density operators, not merely one unusually degenerate spectrum.

For exact unitary operator controllability, the criterion is $\mathfrak{g}=\mathfrak{u}(N)$. If only the physical gate up to global phase matters, it is enough that the image after removing the scalar direction is $\mathfrak{su}(N)$, equivalently that $\mathfrak{g}+\mathbb{R}iI=\mathfrak{u}(N)$. A traceless system can implement all special-unitary gates while lacking independent control of the overall phase.

Set $X_j=i{H}_j$. The symplectic Lie condition is $X_j^{T}J+J{X}_j=0$, equivalently $H_j^{T}J+J{H}_j=0$. For the supplied diagonal $H_0$, the nonzero entries of $J$ pair levels $1,4$ and $2,3$; the corresponding energy sums are $-\alpha +\alpha =0$ and $-\beta +\beta =0$. Thus $H_{0}J+J{H}_0=0$ for all real $\alpha ,\beta$.

For the supplied real symmetric control matrix, direct multiplication givesBoth generators therefore satisfy the same alternating-form condition. That condition is closed under real linear combinations, and for two such generatorsas follows by substituting $X^{T}J=-JX$ and $Y^{T}J=-JY$ into $(XY{)}^{T}J-(YX{)}^{T}J$. Hence the whole Dynamical Lie algebra obeys it.

If $\dot{U}=X(t)U$ and $U(0)=I$, differentiating givesThus every reachable propagator satisfies $U^{T}JU=J$. Also $J^T=-J$, $J^{†}J=I$, so this is a symplectic dynamical symmetry in quantum control: the generated group lies in the corresponding compact symplectic group, unitarily equivalent to $Sp(2)$. It is not necessary to claim equality with $Sp(2)$, and special choices of $\alpha ,\beta$ can give a smaller group.

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.

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 areA 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 $-i{H}_0,-i{H}_1,\dots$ 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.

Assume the usual real Hamiltonian controls with freely chosen finite duration, rather than a fixed-time or amplitude-restricted task. Define the Dynamical Lie algebrathe real linear span closed under all nested commutators. Replacing each generator by $-i{H}_m$ gives the same algebra. The associated connected reachable group acts on propagators, on each unitary orbit of a density operator, and on pure states. For $N\ge 2$, the necessary and sufficient conditions are as follows.

For exact unitary operator controllability, $\boxed{\mathfrak{g}=\mathfrak{u}(N)}$. For operator control up to global phase, the possibilities are $\mathfrak{su}(N)$ or $\mathfrak{u}(N)$. If every Hamiltonian is traceless, $\det U(T)=\exp [-i{\int }_0^T\mathrm{Tr}H(t)dt]=1$, so the central phase direction is absent and exact access is to $SU(N)$. The full special unitary Lie algebra has dimension $N^2-1$ and the Lie algebra of the unitary group has dimension $N^2$.

For density operator controllability on every spectral orbit,Scalar phases do not affect conjugation. If only a particular density spectrum is being considered, a smaller group can suffice: for distinct eigenvalue multiplicities $n_j$ and centralizer ${\mathfrak{c}}_{\rho }=\{X\in \mathfrak{u}(N):[X,\rho ]=0\}$, the orbit criterion is $\dim \mathfrak{g}-\dim (\mathfrak{g}\cap {\mathfrak{c}}_{\rho })=N^2-{\sum }_j{n}_j^2$. A maximally mixed state has a one-point orbit and needs no controls. This restricted-orbit condition is weaker than controllability for every density spectrum.

For pure-state controllability, up to a fixed unitary change of basis,where the last two alternatives occur only for even $N$. Here $\mathfrak{sp}(n)$ is the standard compact symplectic Lie algebra acting on ${\mathbb{C}}^{2n}$, not the real noncompact symplectic algebra. These are precisely the unitary matrix algebras whose connected groups act transitively on normalized complex vectors, and also give the ray version of controllability. For $N=2$, $\mathfrak{sp}(1)=\mathfrak{su}(2)$. For even $N\ge 4$, the symplectic cases show why pure-state control can be weaker than density control: $\dim \mathfrak{sp}(n)=n(2n+1)$ is smaller than the nondegenerate density-orbit dimension $N^2-N$, and its scalar center acts trivially on that orbit. For $N=1$, the physical density and pure-state spaces are single points, while exact operator controllability still requires the phase algebra $\mathfrak{u}(1)$.

Both the drift and local $Z_k$ control commute with $S$, so $[H_S+f(t)Z_k,S]=0$ at every time. The propagator consequently obeys $U(t{)}^{†}SU(t)=S$, as follows by differentiating this expression. Thus every reachable unitary operator preserves the magnetization sectors of a spin chain. If $P_q$ is a spectral projector of $S$, then $U$ commutes with $P_q$ and $\mathrm{Tr}(P_q\rho (t))=\mathrm{Tr}(P_q\rho (0))$.

The eigenspaces are nontrivial proper subspaces: $|0\cdots 0⟩$ has eigenvalue $N$, while $|10\cdots 0⟩$ has eigenvalue $N-2$. One cannot transfer the first pure state to the second, although their density operators have the same spectrum. Nor can one implement a unitary that mixes these two sectors. Hence $\boxed{\text{the full system has neither operator, density, nor pure-state controllability}}$. Its Dynamical Lie algebra lies in the block-diagonal commutant of $S$. Controllability within a fixed sector, such as the single-excitation subspace, is still possible and must be tested separately.

Assume $c_1{c}_2\ne 0$, as required for a connected chain and the supplied divisions. The single-excitation subspace has dimension three, so its Dynamical Lie algebra ${\mathfrak{g}}_1={\mathrm{Lie}}_{\mathbb{R}}\{iA,iB\}$ is contained in $\mathfrak{u}(3)$. The intended conclusion from nine independent skew-Hermitian directions is $\boxed{{\mathfrak{g}}_1=\mathfrak{u}(3)}$, giving unitary operator controllability, density operator controllability and pure-state controllability on this sector. Here is a direct generation argument that does not assume an independent identity control or rely on the normalization of the printed intermediate matrices.

Here $c_n$ denotes the hopping entry of the supplied reduced $A$. With the stated Pauli matrices, restricting $c_n(X_n{X}_{n+1}+Y_n{Y}_{n+1})$ instead gives hopping $2c_n$; absorbing that common factor into the reduced coupling does not change the Dynamical Lie algebra.

Write $R_{mn}=E_{mn}-E_{nm}$ and $Y_{mn}=i(E_{mn}+E_{nm})$ using matrix units. The physical generators giveNext isolate the second transition and generate its other direction:Finally $R_{13}=[R_{12},R_{23}]$ and $Y_{13}=[R_{12},Y_{23}]$. The three $R_{mn}$, three $Y_{mn}$ and two diagonal differences are eight independent traceless skew-Hermitian matrices, hence span $\mathfrak{su}(3)$. Because $\mathrm{Tr}(iB)=i\ne 0$, subtracting the traceless part of $iB$ also supplies $i{I}_3$, completing $\mathfrak{u}(3)$. This is the trace-direction step in identity augmentation in quantum controllability and the three-site instance of Endpoint control of an XX spin chain. If a coupling vanishes, the chain disconnects and the full-sector conclusion fails.

There is a redundant direction in the printed final pair of commutators: with the printed intermediate definitions, both are proportional to $E_{13}-E_{31}$. Thus that pair cannot itself establish the stated independence. The mixed commutator $[R_{12},Y_{23}]=i(E_{13}+E_{31})$ above provides the missing direction and establishes the required conclusion independently.

Let ${\mathfrak{g}}_{\mathrm{full}}$ be the algebra on the full eight-dimensional space. Restriction to the invariant single-excitation sector is a surjective Lie algebra homomorphism $R:{\mathfrak{g}}_{\mathrm{full}}\to \mathfrak{u}(3)$. If $\dim {\mathfrak{g}}_{\mathrm{full}}=9$, the rank-nullity theorem gives $\ker R=0$, soThis is an abstract algebra isomorphism; on the full Hilbert space it is a reducible eight-dimensional representation preserving sectors of dimensions $1,3,3,1$. The sector actions are linked by the same nine generators, rather than independently controllable blocks. In particular, this is not $\mathfrak{u}(8)$ or $\mathfrak{su}(8)$, so it does not imply full-system controllability. The full spin Hamiltonians are traceless, so their algebra is embedded in $\mathfrak{su}(8)$ even though its abstract type is $\mathfrak{u}(3)$; its central direction is not an unrestricted overall phase on the full space.

Both $A$ and $C$ are real symmetric and couple only neighboring sites whose diagonal entries in $J$ have opposite signs. For an entry $x_{mn}$ of $x=iA$ or $iC$, symmetry gives $(x^{T}J+Jx{)}_{mn}=(J_{nn}+J_{mm})x_{mn}$. Every nonzero entry joins signs $+1$ and $-1$, while the diagonal entries vanish. Hence $\boxed{x^{T}J+Jx=0\text{ for }x=iA,iC}$.

The condition is preserved by real linear combinations and commutators, so the entire Dynamical Lie algebra satisfies it. For the physical generator $x(t)=-i(A+f(t)C)$, differentiation gives $\frac{d}{dt}(U^{T}JU)=U^T(x^{T}J+Jx)U=0$, and therefore $\boxed{U(t{)}^{T}JU(t)=J}$. This is an orthogonal dynamical symmetry in quantum control: a preserved symmetric bilinear form. It need not give eigenspaces of a commuting operator; indeed $J$ does not commute with $A$ or $C$.

Set $D=\mathrm{diag}(1,i,1)$. Then $D^{T}JD=I$, so the transformed generators $x^{\prime }=D^{†}xD$ satisfy $x^{\prime T}+x^{\prime }=0$ as well as $x^{\prime †}+x^{\prime }=0$. They are consequently real antisymmetric, lying in $\mathfrak{so}(3)$. The original generators already supply the independent directions $p=iC=i(E_{12}+E_{21})$ and $q=i(A-C)=i(E_{23}+E_{32})$, and their commutator is $[p,q]=E_{31}-E_{13}$. These three independent directions exhaust the three-dimensional algebra. ThusHere $E_{mn}$ denotes a matrix unit. The reachable connected group is a unitary conjugate of $SO(3)$, a proper subgroup of $SU(3)$, so there is no unitary operator controllability.

Pure-state control also fails. For every reachable state ${\psi }^{\prime }=U\psi$, the quantity ${\psi }^{\prime T}J{\psi }^{\prime }={\psi }^{T}J\psi$ is invariant, and its absolute value is unchanged even by an overall state phase. The normalized states $e_1$ and $(e_1+e_2)/\sqrt{2}$ have respectively $|{\psi }^{T}J\psi |=1$ and $0$, so no allowed control connects their rays. Therefore $\boxed{\text{neither pure-state nor density operator controllability holds}}$ for this reduced three-level system, despite the absence of a simple excitation-sector obstruction within the sector.

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 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.