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Intrinsic topological order survives without a protecting symmetry and is characterized by features such as topology-dependent ground-state degeneracy and anyonic excitations. One-dimensional bosonic gapped systems have no intrinsic topological order, although they can have symmetry-protected topological order.

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.

Apply the theorem to a symmetric injective MPS. The physically transformed tensor $A_g^i={\sum }_{j}U(g{)}_{ij}{A}^j$ generates the same state, so the theorem forcesApplying $g$ and then $h$ shows that $V(g)V(h)$ and $V(gh)$ implement the same gauge transformation. Injectivity makes that gauge unique up to a scalar, hence $V(g)V(h)=\omega (g,h)V(gh)$. Associativity gives the two-cocycle equation for $\omega$.

A continuous symmetry-preserving gapped path changes the tensor and $V(g)$ continuously but cannot change the discrete cohomology class $[\omega ]$ without losing injectivity, breaking the symmetry, or closing the gap. Thus the fundamental theorem of matrix product states turns the virtual projective representation into the invariant classifying one-dimensional symmetry-protected topological phases. With broken symmetry, one first records the permuted ground-state sectors and then applies the same argument to their unbroken subgroup.