A classification of pure states on quantum spin chains satisfying the split property with on-site finite group symmetries

We consider a set $SPG(\mathcal{A})$ of pure split states on a quantum spin chain $\mathcal{A}$ which are invariant under the on-site action $\tau$ of a finite group $G$. For each element $\omega$ in $SPG(\mathcal{A})$ we can associate a second cohomology class $c_{\omega,R}$of $G$. We consider a classification of $SPG(\mathcal{A})$ whose criterion is given as follows: $\omega_{0}$ and $\omega_{1}$ in $SPG(\mathcal{A})$ are equivalent if there are automorphisms $\Xi_{R}$, $\Xi_L$ on $\mathcal{A}_{R}$, $\mathcal{A}_{L}$ (right and left half infinite chains) preserving the symmetry $\tau$, such that $\omega_{1}$ and $\omega_{0}\circ( \Xi_{L}\otimes \Xi_{R})$ are quasi-equivalent. It means that we can move $\omega_{0}$ close to $\omega_{1}$ without changing the entanglement nor breaking the symmetry. We show that the second cohomology class $c_{\omega,R}$ is the complete invariant of this classification.