Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 57 / 3 / a / Solution

The Pauli matrices obey $X|0⟩=|1⟩$, $X|1⟩=|0⟩$, $Y|0⟩=i|1⟩$, and $Y|1⟩=-i|0⟩$. Thus $X\otimes X$ and $Y\otimes Y$ contribute opposite amplitudes on $|00⟩,|11⟩$ and equal amplitudes on $|01⟩,|10⟩$. For the XY exchange interaction $E=(X\otimes X+Y\otimes Y)/2$,Equivalently $E=|01⟩⟨10|+|10⟩⟨01|$. It transfers an excitation across a link without creating or destroying excitations, so a sum of such links preserves the single-excitation subspace.