= Single-excitation subspace
{title2=$\mathcal H_1=\operatorname{span}\{|10\cdots0\rangle,\ldots,|0\cdots01\rangle\}$}
For a network of $N$ spin-one-half sites with $Z|0\rangle=|0\rangle$ and $Z|1\rangle=-|1\rangle$, the single-excitation subspace has exactly one spin in state $|1\rangle$ 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.
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