Entanglement Links Matter’s Structure to Fundamental Theory

Researchers have long sought to understand the deep connection between entanglement and topology in quantum systems.

Michele Del Zotto from the Mathematics Institute, Uppsala University, Department of Physics and Astronomy, Uppsala University, Center for Geometry and Physics, Uppsala University, working with Abhijit Gadde from the Department of Theoretical Physics at the Tata Institute for Fundamental Research and Pavel Putrov from The Abdus Salam International Centre for Theoretical Physics, now propose a precise relationship between multipartite entanglement and topological quantum field theory (TQFT). Their work demonstrates that genuine multipartite entanglement, characterised by a dimensional manifold, directly corresponds to the partition function of the low-energy TQFT residing on that manifold. Significantly, the team’s conjecture suggests that the ground state wavefunction itself fully determines the tensor category describing the TQFT, and they validate this for general (2+1)-dimensional Levin-Wen string-net models, offering a potential pathway to characterise topological phases of matter through entanglement measurements. For decades, linking quantum entanglement with the underlying structure of physical space has remained an elusive goal. Now, a compelling connection between many-particle entanglement and topological quantum field theories offers a fresh perspective on how order emerges in complex materials, suggesting that entanglement patterns directly encode the fundamental properties of these exotic states of matter. Scientists are increasingly focused on understanding phases of matter described by topological quantum field theories (TQFTs). These gapped phases exhibit properties dictated by a TQFT, a mathematical framework describing the physics without reference to specific details of the underlying system. Research establishes a connection between multipartite entanglement within the ground state of a gapped theory and the partition function of the low energy TQFT on a corresponding manifold. This the ground state wavefunction, specifically its entanglement properties, can determine the complete topological description of the long-distance physics. By characterising a TQFT solely from the ground state has remained a significant challenge, though quantum-information tools, such as entanglement entropy and modular Hamiltonians, have previously aided in extracting TQFT data. A systematic way to fully define the TQFT from the wavefunction was lacking. Recent advances in quantifying multipartite entanglement have led to the development of multi-invariants, local-unitary invariants effective for characterising entanglement in complex quantum states. Observations have shown certain multi-invariants reproduce the TQFT partition function on specific geometric spaces, known as Lens spaces. Scientists conjecture a precise relationship between genuine multipartite entanglement, labelled by a d-dimensional manifold, and the partition function of the low energy TQFT defined on that manifold. For a three-dimensional case, this implies the ground state wavefunction contains enough information to define the modular tensor category describing the TQFT. Multi-invariants are constructed as polynomials related to the wavefunction and are represented graphically as edge-coloured graphs — by extracting a specific component of entanglement, termed the “signal”, from these multi-invariants, a connection to geometric manifolds becomes apparent. A “graph-encoded manifold” (GEM) is created by mapping the simplices of a triangulated manifold to vertices and edges. Providing a direct link between geometry and entanglement. This allows for the formulation of a conjecture relating the multi-invariant signal to the TQFT partition function, and with a correction factor dependent on the triangulation and the method used to extract the signal. Entanglement’s quantification links ground state wavefunctions to topological quantum field theories Genuine multipartite entanglement, quantified via a dimensional manifold, reveals a direct connection to the partition function of a low-energy topological field theory (TQFT) on that manifold. Work detailed here establishes this relationship for general (2+1)-dimensional Levin-Wen string-net models, verifying a long-standing conjecture linking entanglement to topological order. Calculations demonstrate that for these configurations, the ground state wavefunction completely determines the tensor category describing the low-energy TQFT. To explore this connection further necessitates understanding the behaviour of edge modes where regions intersect in the ground state. Investigations into chiral topological theories, such as those describing fractional quantum Hall phases or Kitaev’s honeycomb model. Provide a valuable testing ground for this conjecture. Prior work computed the chiral central charge for these theories. Here, this project aims to solidify those results within the broader framework established here. At the core of The assessment lies the examination of geometric multi-invariants derived from multipartite entanglement. Specifically, the genus of a two-dimensional surface was extracted from tripartite stabilizer states, even those with non-local stabilizer generators. To extend this approach to higher-partite states presents a compelling avenue for future research. Since the size of a region becomes undefined for non-local states, enforcing the conjecture without taking limits is essential for meaningful analysis. Pachner-type moves, dipole cancellations and additions. Relating different geometric entanglement measures (GEMs) of a topological manifold appear to play a significant role in characterising states satisfying this conjecture. Beyond the standard (d+1)-partite entanglement, considering scenarios with q > D+1 regions opens possibilities for capturing additional aspects of the TQFT — these higher-partite entanglement measures may reveal further details about the underlying topological structure. By dividing space into regions, researchers can compute genuine q-partite entanglement, potentially uncovering connections to extended TQFT data. Such as boundaries, corners, and topological defects. Also, the framework naturally extends to theories containing fermions, described by spin-TQFT, where observables depend on the spin structure of the manifold. Offering a richer field for exploration. Multi-invariant construction and topological quantum field theory foundations In turn, a detailed examination of multi-invariants forms the basis of this effort, beginning with a review of local-unitary invariants of multipartite quantum states. These multi-invariants, polynomial functions of a quantum state, remain unchanged under local unitary transformations applied to individual subsystems. Construction of signals denoting genuine multi-partite entanglement proceeds from a general multi-invariant, associating it with a triangulated manifold to describe its geometry. Such an approach, termed a graph-encoded manifold or GEM in combinatorial topology, allows manifold representation through graph structures. Then, the necessary foundations of topological quantum field theory are reviewed, starting with the universal construction and proceeding to the Turaev-Viro state-sum model. The modern treatment detailed in existing literature guides this review, providing a framework for understanding the mathematical tools employed. At the same time, this theoretical groundwork prepares for the computation of Z(M∆; |ψ⟩H), specifically for four-region decompositions within a Levin-Wen string-net model — by treating the ground state as a vector within a factorized Hilbert space necessitates extending this space by relaxing the local Gauss law. Here, a common practice in many physical examples. In turn, this extension yields a multi-invariant that closely corresponds to the partition function of the low-energy TQFT on the manifold M. Though initial calculations include unwanted factors. Such extra contributions originate from bipartite quantum states localized at the boundaries between regions, and a prescription is devised to remove them, isolating the pure TQFT partition function. Through understanding the construction of these multi-invariants requires considering n replicas of the quantum state and its dual, with the degree of homogeneity, n, representing the replica number. Once a factorized basis is chosen, the state is expressed as a sum over basis vectors, with coefficients transforming as fundamental or anti-fundamental representations under local unitary transformations. Instead, the multi-invariant is explicitly defined through a contraction of these indices, labelled by a permutation tuple indicating how fundamental and anti-fundamental indices are paired across replicas. Entanglement mapping unlocks topological quantum field theory’s potential for material characterisation Once considered a purely mathematical curiosity, topological quantum field theory is edging closer to becoming a practical tool for materials science. This effort establishes a compelling link between the subtle entanglement within a material’s quantum state and the properties predicted by these abstract theories. For years, a major obstacle in this field has been translating the elegant equations of TQFT into measurable characteristics of real-world systems. This project offers a potential pathway. Rather than relying on complex simulations, it suggests that detailed analysis of a material’s ground state, the lowest energy configuration, could reveal its underlying topological order. The implications extend beyond simply identifying exotic materials, as connecting entanglement patterns to TQFT provides scientists with a new language for describing and potentially controlling these systems. Unlike conventional materials where properties depend on surface details, topological materials are defined by their bulk behaviour, offering inherent stability against imperfections.

The team demonstrate this connection using specific mathematical models, but the broader question remains of how widely applicable this principle is across different material classes. A significant limitation lies in the difficulty of accurately measuring the many-body entanglement required to confirm these predictions. The conjecture focuses on ‘gapped’ phases of matter. Leaving open the question of how it applies to gapless systems which are also of considerable interest. Extracting the full entanglement structure of a complex material is a formidable experimental challenge. Future work will likely focus on developing more accessible experimental probes of topological entanglement, and exploring how this framework can be extended to describe more complex, interacting systems. This progress holds the potential to design materials with entirely new functionalities, protected by the fundamental laws of topology. 👉 More information 🗞 From Multipartite Entanglement to TQFT 🧠 ArXiv: https://arxiv.org/abs/2602.16770 Tags: