Symmetry-protected equivalence of the Affleck--Kennedy--Lieb--Tasaki state and cluster state

After suitable blocking, the AKLT and one-dimensional cluster states realize the same nontrivial ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry-protected topological phase. Their virtual symmetries carry the same nontrivial projective class and produce protected edge degrees of freedom.