Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 343 / 3 / d / Solution

For the Affleck--Kennedy--Lieb--Tasaki state, take the dihedral subgroup of spin rotations generated by $\pi$ rotations about the $x$ and $z$ axes. On each physical spin-one site these generators commute and give an ordinary representation of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$. On the virtual spin-one-half space they may be represented, up to phases, byThey anticommute,so the virtual action is the nontrivial projective representation of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$.

The reduced density operator across a cut commutes with this irreducible projective action. By Schur lemma it is proportional to the identity within each virtual doublet. Consequently every level in the entanglement spectrum of a matrix product state is at least twofold degenerate, the characteristic protected edge-spin degeneracy of the AKLT phase.