Multipartite entanglement in the quantum tetrahedron

TL;DR

This work investigates genuine multipartite entanglement in SU(2)-invariant four-valent intertwiners, which model quantum tetrahedra with fixed face areas and serve as the smallest nonzero-volume quanta in loop quantum gravity. It applies the entropic fill measure to four-qudit states and compares entanglement distributions among four ensembles: arbitrary tensors, invariant intertwiners, and coherent intertwiners (with/without closure), across local dimensions up to several units of $j$. The results reveal pronounced differences in distributions: average entanglement is highest for arbitrary tensors, while intertwiners tend toward maximal one-to-one entropies; coherent intertwiners show geometry-dependent patterns with sharp peaks, and regular tetrahedral configurations can maximize the entropic fill. The study demonstrates that entropic fill extends to four-qudit systems and uncovers rich structure linking quantum geometry to multipartite entanglement, motivating further exploration of coherent-intertwiner patterns and their relation to quantum gravity dynamics.

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.

Figures (8)

• Figure 1: The inscribed sphere touches each face at a point, defining $12$ triangles on the faces. Pairs of triangles sharing an edge have the same area $\sigma_{ij}$. The vertices are numbered so that $\sigma_{ij}$ is the area of each of the two triangles sharing the edge between the faces opposite vertices $i$ and $j$. This figure has been re-used without modification from EntropicFill under the license https://creativecommons.org/licenses/by/4.0/.
• Figure 2: Sampled distributions of entropic fill for arbitrary states, invariant intertwiners, and coherent intertwiners with and without closure ($5\times 10^5$ for each). We show $1000$ bins in the interesting parts near maximal entropic fill (note the bounds on the $x$-axis for each plot) for $j\in\{0.5,1.5,3\}$. Below these regions, the bins have comparably neglectable counts. Average entropic fill of arbitrary states and general intertwiners increases more rapidly with increasing $j$ than both types of coherent intertwiners, explaining the apparent change in relative counts from one plot to the next.
• Figure 3: Numerically computed means of entropic fill for arbitrary states, invariant intertwiners, and coherent intertwiners with and without closure (sample size $5\times 10^5$ for each), for $j$ ranging from $1/2$ to $11$. We show absolute entropic fill on the left, and the distance from maximal entropic fill on the right (logarithmic scale). All visible differences are well within statistical certainty, the standard errors of the means being at least an order of magnitude smaller at this sample size (except for the two coherent intertwiner categories between $j=6.5$ and $j=7.5$ and the two other categories near $j=10$). The mean entanglement increases fastest for arbitrary states and general intertwiners, out-performing coherent intertwiners. Within the coherent category, the closure condition leads to slightly higher entropic fill for spin values up to about $j=7$.
• Figure 4: Numerically computed ($300\times 300$) two-to-two entropies and (logarithmic inverse distance from maximal) entropic fill on the configuration space of coherent intertwiners with closure for $j=1/2$; the $x$-axis encodes the angle $\theta$ between $\vec{n}_1$ and $\vec{n}_2$, while the $y$-axis represents the angle on the circle of possible endpoints for $\vec{n}_3$ given $\vec{n}_1$ and ${\vec{n}_2}$. The regular tetrahedron configurations appear as spots of maximal entropic fill. Note that not every area element in these plots is equally likely for a random coherent intertwiner with closure; in fact, $\theta$ near $\pi$ is more likely than near zero.
• Figure 5: Idem Fig. \ref{['fig:CohClosure_FullConfigSpace_j0.5']} with $j=3/2$.