# Note on quantum entanglement and quantum geometry

## TL;DR

This paper explores the connection between quantum entanglement in boundary states and quantum geometry in the bulk within spin networks, suggesting that entanglement relates to geometric properties like volume and orientation.

## Contribution

It proposes a conjecture linking non-perfect $SU(2)$-invariant tensors to emergent space and demonstrates this connection for specific spins in spin networks.

## Key findings

- Maximally entangled $SU(2)$-invariant tensors correspond to eigenstates of volume square.
- Entanglement reflects the orientation of quantum tetrahedral geometry.
- Results are specific to spins j=1/2 and j=1 in the studied spin network configurations.

## Abstract

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.