Although the physical symmetry generators commute,
$$
V_aV_b=\sigma_x\sigma_z=-\sigma_z\sigma_x=-V_bV_a.
$$
Their virtual commutator is
$$
\boxed{
V_aV_bV_a^{-1}V_b^{-1}=-I}.
$$
This phase cannot be removed by rephasing $V_a$ and $V_b$. The virtual matrices therefore form a nontrivial projective representation, which is the nontrivial <projective virtual symmetry of a matrix product state> and proves that the <cluster state> lies in a nontrivial SPT phase.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 343 / 3 / b / iii / Solution

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The one-dimensional cluster state is the common positive eigenstate of commuting stabilizers $Z_{j-1}X_jZ_{j+1}$ and realizes a nontrivial $\mathbb Z_2\times\mathbb Z_2$ SPT phase.


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 Incoming links: Cluster state