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arXiv:2609.10667

Entanglement entropy in topological tensor networks

Vijay Balasubramanian, Charlie Cummings

Published Sep 9, 2026• 0 citations
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Abstract

We derive an entropy formula for recently proposed tensor network models which prepare diffeomorphism invariant states of topological quantum field theories with non-compact and/or continuous gauge groups. We show that our entropy formula generalizes the notion of ``topological entanglement entropy'' to incorporate the infinite number of particle-like excitations in such theories. When our networks are endowed with gauge group $\mathrm{SL}(2,\mathbb{R})$, we can interpret them as models of three-dimensional gravity with small Newton's constant and possibly non-invertible metrics.

DOI
10.48550/arxiv.2609.10667

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We derive an entropy formula for recently proposed tensor network models which prepare diffeomorphism invariant states of topological quantum field theories with non-compact and/or continuous gauge groups. We show that our entropy formula generalizes the notion of ``topological entanglement entropy'' to incorporate the infinite number of particle-like excitations in such theories. When our networks are endowed with gauge group SL(2,R)\mathrm{SL}(2,\mathbb{R})SL(2,R), we can interpret them as models of three-dimensional gravity with small Newton's constant and possibly non-invertible metrics.
We derive an entropy formula for a recently proposed tensor network model of states in 3D gravity which satisfy the constraint equations of diffeomorphism invariance, and including the effects of matter. After accounting for the invertibility of the spacetime metric, we argue that this formula matches the quantum extremal surface formula (for small but finite GNG_NGN​), and is not restricted to time symmetric spacetimes. We derive the result via canonical quantization without relying on path integrals, and discuss an extension to compact bulk subregions.
A bstract Tensor networks prepare states which share many features of states in quantum gravity. However, standard constructions are not diffeomorphism invariant and do not support an algebra of non-commuting area operators. Recently, analogues of both problems were addressed in a tensor network discretization of topological field theories (TFT) with finite or compact gauge groups. Here, we extend this work towards gravity by generalizing to gauge groups that are discrete or continuous, compact or non-compact. Applied to SL(2, ℝ ) × SL(2, ℝ ) Chern-Simons theory, our construction can be interpreted as building states of three dimensional gravity with a negative cosmological constant. Our tensor networks prepare states which satisfy the constraints of Chern-Simons theory. In metric variables, this implies that the states we construct satisfy the Wheeler-DeWitt equation and momentum constraints, and so are diffeomorphism invariant.
Abstract We compute the topological entanglement entropy for a large set of lattice models in d-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy SA in a sub-region A is proportional to log (GSDA~) \left({GSD}_{\tilde{A}}\right) (GSDA~​) GSD A ˜ , where GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ is the ground state degeneracy of a particular restriction of the full model to A. The quantity GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂A. There is also a topological contribution coming from A itself, that may be non-zero when A has a non-trivial homology. We present some examples and discuss how the topology of A affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension d. The result is in agreement with entanglement calculations for known topological models.
We compute topological entanglement entropy for a large set of lattice models in d-dimensions.It is well known that many such quantum systems can be constructed out of lattice gauge models.For dimensionality higher than 2 there are generalizations going beyond gauge theories.They are called higher gauge theories and rely on higher-order generalizations of groups.Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories.In this work, we calculate the bipartition entanglement entropy for this class of models.Our formalism allows us to do most of the calculation for arbitrary dimension d.We show that the entanglement entropy S A in a sub-region A is proportional to log(GSD Ã), where GSD Ã is the ground state degeneracy of a particular restriction of the full model to A. When A has the topology of a d-dimensional ball, the GSD Ã counts the number of edge states.In this case, S A scales with the area of the (d -1)-dimensional boundary of A. The precise formula for the entropy we obtain is in agreement with entanglement calculations for known topological models.
A definition for the entanglement entropy in a gauge theory was given recently in arXiv:1501.02593. Working on a spatial lattice, it involves embedding the physical state in an extended Hilbert space obtained by taking the tensor product of the Hilbert space of states on each link of the lattice. This extended Hilbert space admits a tensor product decomposition by definition and allows a density matrix and entanglement entropy for the set of links of interest to be defined. Here, we continue the study of this extended Hilbert space definition with particular emphasis on the case of Non-Abelian gauge theories. We extend the electric centre definition of Casini, Huerta and Rosabal to the Non-Abelian case and find that it differs in an important term. We also find that the entanglement entropy does not agree with the maximum number of Bell pairs that can be extracted by the processes of entanglement distillation or dilution, and give protocols which achieve the maximum bound. Finally, we compute the topological entanglement entropy which follows from the extended Hilbert space definition and show that it correctly reproduces the total quantum dimension in a class of Toric code models based on Non-Abelian discrete groups.
Entanglement entropy is a valuable tool for characterizing the correlation structure of quantum field theories. When applied to gauge theories, subtleties arise which prevent the factorization of the Hilbert space underlying the notion of entanglement entropy. Borrowing techniques from extended topological field theories, we introduce a new definition of entanglement entropy for both Abelian and non-Abelian gauge theories. Being based on the notion of excitations, it provides a completely relational way of defining regions. Therefore, it naturally applies to background independent theories, e.g. gravity, by circumventing the difficulty of specifying the position of the entangling surface. We relate our construction to earlier proposals and argue that it brings these closer to each other. In particular, it yields the non-Abelian analogue of the ‘magnetic centre choice’, as obtained through an extended-Hilbert-space method, but applied to the recently introduced fusion basis for 3D lattice gauge theories. We point out that the different definitions of entanglement entropy can be related to a choice of (squeezed) vacuum state.