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The Affleck--Kennedy--Lieb--Tasaki state places two virtual spin-one-half degrees of freedom at every site, puts neighboring virtual spins into singlets, and projects the two virtual spins at each site onto their symmetric spin-one triplet. Its local tensor is equivalently proportional to ${\sigma }_x,{\sigma }_y,{\sigma }_z$ in the spin-one Cartesian basis, and its Affleck--Kennedy--Lieb--Tasaki parent Hamiltonian is

The one-dimensional cluster state is the simultaneous $+1$ eigenstate of the commuting stabilizersOn a periodic chain it is obtained by applying a controlled-$Z$ gate on every neighboring pair of the product state $|+{⟩}^{\otimes N}$.

After suitable blocking, the two states have the same nontrivial projective virtual symmetry of a matrix product state for the protecting group ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$. The two virtual symmetry generators can be represented by anticommuting Pauli matrices, so they realize the nontrivial projective class in group cohomology. Consequently the AKLT and cluster states can be connected by a symmetry-preserving gapped path, or equivalently by a symmetry-preserving finite-depth local circuit: this is the Symmetry-protected equivalence of the Affleck--Kennedy--Lieb--Tasaki state and cluster state.

Both states therefore exhibit one-dimensional symmetry-protected topological order. On an open chain their nontrivial virtual representation produces protected edge degrees of freedom and a characteristic degeneracy in the entanglement spectrum of a matrix product state. They do not have intrinsic topological order: if the protecting symmetry is discarded, either state can be connected to a product state by a finite-depth local circuit.

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.