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

Table of contents
- Solution iii

Solution

Although the physical symmetry generators commute,

V_{a} V_{b} = σ_{x} σ_{z} = - σ_{z} σ_{x} = - V_{b} V_{a} .

Their virtual commutator is

V_{a} V_{b} V_{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.

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