 Single-excitation subspace

For a network of $N$ spin-one-half sites with $Z|0⟩=|0⟩$ and $Z|1⟩=-|1⟩$, the single-excitation subspace has exactly one spin in state $|1⟩$ and dimension $N$. Excitation-conserving exchange interactions preserve it. Its restricted Hamiltonian is a hopping matrix whose entries connect the corresponding sites. Controllability within this subspace need not imply controllability on the full $2^N$-dimensional space.