Note on quantum entanglement and quantum geometry

In this note we present preliminary study on the relation between the quantum entanglement of boundary states and the quantum geometry in the bulk in the framework of spin networks. We conjecture that the emergence of space with non-zero volume reflects the non-perfectness of the $SU(2)$-invariant tensors. Specifically, we consider four-valent vertex with identical spins in spin networks. It turns out that when $j = 1/2$ and $j = 1$, the maximally entangled $SU(2)$-invariant tensors on the boundary correspond to the eigenstates of the volume square operator in the bulk, which indicates that the quantum geometry of tetrahedron has a definite orientation.