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arXiv:2601.14964 · Physical Review A

Multipartite entanglement in the quantum tetrahedron

Robert Amelung, Hanno Sahlmann

quant-phgr-qcSubmitted Jan 21, 2026• Announced Jan 22, 2026• 0 citations
v1 · Jan 21, 2026v2 · Oct 8, 2026
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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.

Journal reference
Phys. Rev. A 114, 022454 (2026)
DOI
10.1103/l1t8-84dk
Comments
13 pages, 24 figures

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The space Inv(j1,j2,j3,j4)\mathrm{Inv}(j_1,j_2,j_3,j_4)Inv(j1​,j2​,j3​,j4​) 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(j1,j2,j3,j4)\mathrm{Inv}(j_1,j_2,j_3,j_4)Inv(j1​,j2​,j3​,j4​) using the recently proposed entropic fill.
Numerically evaluating entropic fill in the case of equal spins between 1/21/21/2 and 111111, 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 jjj. We also find that entanglement depends on the geometric data of coherent intertwiners in a complicated way.
In this paper, we carry out the entanglement calculations on the coherent intertwiners. We first consider the entanglement introduced by the group averaging of the tensor-product-type intertwiner on a 4-valents vertex. Then, we calculate explicitly the entanglement carried by the gauge-invariant coherent intertwiner with four legs. Our numerical results show that the entanglement can be controlled by the local semiclassical geometry described by the coherent intertwiner. We also extend our analytical calculation to the coherent intertwiners with an arbitrary number of legs. Especially, we apply the previous results to the entanglement of the spin-network state labeled by coherent intertwiners, with the network puncturing a boundary by its vertices. We show that the entanglement of such spin-network state is not only determined by the area of the boundary but also carries a quantum correction controlled by the semiclassical geometry associated to the vertices puncturing the boundary.
Despite the fundamental importance of quantum entanglement in many-body systems, our understanding is mostly limited to bipartite situations. Indeed, even defining appropriate notions of multipartite entanglement is a significant challenge for general quantum systems. In this work, we initiate the study of multipartite entanglement in a rich, yet tractable class of quantum states called stabilizer tensor networks. We demonstrate that, for generic stabilizer tensor networks, the geometry of the tensor network informs the multipartite entanglement structure of the state. In particular, we show that the average number of Greenberger-Horne-Zeilinger (GHZ) triples that can be extracted from a stabilizer tensor network is small, implying that tripartite entanglement is scarce. This, in turn, restricts the higher-partite entanglement structure of the states. Recent research in quantum gravity found that stabilizer tensor networks reproduce important structural features of the AdS/CFT correspondence, including the Ryu-Takayanagi formula for the entanglement entropy and certain quantum error correction properties. Our results imply a new operational interpretation of the monogamy of the Ryu-Takayanagi mutual information and an entropic diagnostic for higher-partite entanglement. Our technical contributions include a spin model for evaluating the average GHZ content of stabilizer tensor networks, as well as a novel formula for the third moment of random stabilizer states, which we expect to find further applications in quantum information.
Genuine multipartite entanglement is crucial for quantum information and related technologies, but quantifying it has been a long-standing challenge. Most proposed measures do not meet the "genuine" requirement, making them unsuitable for many applications. In this work, we propose a journey toward addressing this issue by introducing an unexpected relation between multipartite entanglement and hypervolume of geometric simplices, leading to a tetrahedron measure of quadripartite entanglement. By comparing the entanglement ranking of two highly entangled four-qubit states, we show that the tetrahedron measure relies on the degree of permutation invariance among parties within the quantum system. We demonstrate potential future applications of our measure in the context of quantum information scrambling within many-body systems.
Organising the space of entanglement structures of a multipartite quantum system is a much more challenging task than its bipartite version: while the local unitary (LU) orbit of a bipartite pure state can be conveniently characterized by its entanglement spectrum, invariants of multipartite entanglement structures are comparatively difficult to define and work with. The root cause of this difference is that the bipartite problem can be reduced to the analysis of matrix invariants, while its multipartite version is governed by a much richer space of tensor invariants. The present work explores the latter through the lens of so-called trace-invariants, which are in one-to-one correspondence with combinatorial objects known as colored graphs. We first explain why trace-invariant evaluations can serve as labels of LU-orbits of multipartite pure states, how this strategy extends to random states, and how the effect of local operations (LO) can be analyzed through such data. We then focus on entanglement classification within an (infinite-dimensional) subspace of reference states, whose basic building blocks are GHZ states of various dimensions. We show that relatively simple subclasses of trace-invariants are sufficient to separate the LU-orbits of reference states, and enable a complete (resp. an incomplete) characterization of their relations in the LO (resp. LOCC) resource theory of entanglement. Finally, we investigate how a (still infinite) subclass of reference states of local dimension N can be efficiently distinguished at leading and subleading orders in an asymptotic large-N expansion (among themselves, or from Haar-random states). This analysis relies crucially on combinatorial quantities associated to colored graphs, some of which have already played instrumental roles in the recent literature on random tensors. Results of broader relevance are reported along the way.
Three-dimensional Heisenberg model in the form of a tetrahedron lattice is investigated. The concurrence and multipartite entanglement are calculated through 2-concurrence C and 4-concurrence C 4 . The concurrence C and multipartite entanglement C 4 depend on different coupling strengths J i and are decreased when the temperature T is increased. For a symmetric tetrahedron lattice, the concurrence C is symmetric about J 1 when J 2 is negative while the multipartite entanglement C 4 is symmetric about J 1 when J 2 <2. For a regular tetrahedron lattice, the concurrence C of ground state is 1/3 for ferromagnetic case while C = 0 for antiferromagnetic case. However, there is no multipartite entanglement since C 4 = 0 in a regular tetrahedron lattice. The external magnetic field B can increase the maximum value of the concurrence C B and induce two or three peaks in C B . There is a peak in the multipartite entanglement C 4 B when C 4 B is varied as a function of the temperature T . This peak is mainly induced by the magnetic field B .
In the context of quantum gravity, we clarify entanglement calculations on spin networks: we distinguish the gauge-invariant entanglement between intertwiners located at the nodes and the entanglement between spin states located on the network's links. We compute explicitly these two notions of entanglement between neighboring nodes and show that they are always related to the typical ln(2j+1)\mathrm{ln}(2j+1)ln(2j+1) term depending on the spin jjj living on the link between them. This ln(2j+1)\mathrm{ln}(2j+1)ln(2j+1) contribution comes from looking at nongauge-invariant states; thus we interpret it as gauge breaking and unphysical. In particular, this confirms that pure spin network basis states do not carry any physical entanglement, so that true entanglement and correlations in loop quantum gravity comes from spin or intertwiner superpositions.