Valence-Bond-Solid state entanglement in a 2-D Cayley tree

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The Valence-Bond-Solid (VBS) states are in general ground states for certain gapped models. We consider the entanglement of VBS states on a two-dimensional Cayley tree. We show that the entropy of the reduced density operator does not depend on the whole size of the Cayley tree. We also show that asymptotically, the entropy is liearly proportional to the number of singlet states cut by the reduced density operator of the VBS state.