Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 50 / 3

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.

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 $\mathfrak{u}(1)$.

The Pauli matrices obey $[X,Z]=-2iY$ and $[Y,Z]=2iX$, and operators on different tensor factors commute. HenceAdding gives $\boxed{[X_m{X}_n+Y_m{Y}_n,Z_m+Z_n]=0}$. Each $Z_m{Z}_n$ commutes separately with both $Z_m$ and $Z_n$, so $\boxed{[Z_m{Z}_n,Z_m+Z_n]=0}$. Every interaction on sites $m,n$ also commutes with all $Z_{\ell }$ for $\ell \notin \{m,n\}$. Therefore equal $XX$ and $YY$ coefficients make their contributions cancel pair by pair, and $\boxed{[H_S,S]=0\text{ when }{\alpha }_{mn}={\beta }_{mn}\text{ for every pair}}$. This is Excitation-number conservation in XXZ spin chains: $S$ is the total spin magnetization in Pauli normalization.

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.

Use the computational basis ordered as $|000⟩,|001⟩,|010⟩,|011⟩,|100⟩,|101⟩,|110⟩,|111⟩$. Since $Z|0⟩=|0⟩$ and $Z|1⟩=-|1⟩$,For arbitrary $N$, a computational basis vector with $q$ ones has $N-q$ contributions $+1$ and $q$ contributions $-1$, giving eigenvalue $N-2q$. All $q=0,\dots ,N$ occur, soThe binomial coefficient counts which $q$ sites are excited. Thus there are exactly $N+1$ distinct magnetization sectors of a spin chain, and their dimensions sum to ${\sum }_q\left(\genfrac{}{}{0ex}{}{N}{q}\right)=2^N$, the full Hilbert-space dimension.

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.