Bulk-Edge correspondence of entanglement spectrum in 2D spin ground states

General local spin $S$ ground states, described by a Valence Bond Solid (VBS) on a two dimensional lattice are studied. The norm of these ground states is mapped to a classical O(3) model on the same lattice. Using this quantum-to-classical mapping we obtain the partial density matrix $\rho_{A}$ associated with a subsystem ${A}$ of the original ground state. We show that the entanglement spectrum of $\rho_{\rm A}$ in a translation invariant lattice is given by the spectrum of a quantum spin chain at the boundary of region $A$, with local Heisenberg type interactions between spin 1/2 particles.