Multipartite entanglement in the quantum tetrahedron

TL;DR

This paper investigates the multipartite entanglement properties of quantum tetrahedron states in loop quantum gravity, revealing distinct entanglement distributions compared to generic tensors and complex dependencies on geometric data.

Contribution

It introduces the use of entropic fill to analyze entanglement in intertwiners, highlighting differences from generic tensors and exploring geometric influences.

Findings

Entanglement distributions differ significantly between intertwiners and generic tensors.

Peak entanglement is at the highest for generic intertwiners, lowest for generic tensors.

Average entanglement is higher in arbitrary tensors than in intertwiners.

Abstract

The space $\mathrm{Inv}(j_1,j_2,j_3,j_4)$ of SU(2)-invariant four-valent tensors, also known as intertwiners, can be understood as the quantum states of a tetrahedron in Euclidean space with fixed areas. In loop quantum gravity, they are states of the smallest "atom of space" with non-zero volume. At the same time they correspond to four-party tensor product states invariant under global rotations. We consider the multipartite entanglement of states in $\mathrm{Inv}(j_1,j_2,j_3,j_4)$ using the recently proposed entropic fill. Numerically evaluating entropic fill in the case of equal spins between $1/2$ and $11$, we find that the distributions of entanglement are very different for intertwiners as compared to generic tensors, and for coherent intertwiners as compared to generic ones. The peak in the distribution seems to be at the highest entanglement for generic intertwiners and at the lowest for generic tensors, but in terms of average entanglement, the roles are switched: average entanglement is highest in arbitrary tensors and lower in intertwiners, at least in the regime of large $j$. We also find that entanglement depends on the geometric data of coherent intertwiners in a complicated way.