Multipartite entanglement in the quantum tetrahedron

Abstract

The space 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 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 consider the dependence of entanglement on the geometric data of coherent intertwiners. Coherent intertwiners labeled by a symmetric tetrahedron have maximal entanglement, while minimal entanglement implies a coherent state labeled by a degenerate tetrahedron. Away from these cases, entanglement depends on the geometry in an intricate way.

17 pages, 10 figures. v2: same as published version, up to layout and minor language editing