A complete classification of 2d symmetry protected states with symmetric entanglers

Abstract: We consider symmetry protected topological states of 2d quantum spin systems, with a finite symmetry group $G$. It has been conjectured that such states are classified by the cohomology group $H^3(G,U(1))$, but the completeness of this classfication is an open problem. We restrict ourselves to symmetry protected topological states that can be prepared from a product state by a symmetric entangler. For this class of states, we prove that the classification by $H^3(G,U(1))$ is complete.