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arXiv:2610.00478

Algebraic structure in holographic tensor networks

Abhisek Sahu, Jeremy van der Heijden, Mark Van Raamsdonk, Rana Zibakhsh

hep-thgr-qcquant-phSubmitted Sep 30, 2026• Announced Oct 2, 2026• 0 citations• Revised Oct 2, 2026 (v2)
v1 · Sep 30, 2026v2 · Oct 2, 2026
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Abstract

In a previous work 2511.21852, we explored the hypothesis that in general theories of quantum gravity, spacetime regions known as generalized entanglement wedges (introduced by Bousso and Penington) have an associated subalgebra of observables in a semiclassical limit. We defined an entropy associated with these subalgebras that obeys the same monotonicity and strong subadditivity properties as the generalized gravitational entropy of generalized entanglement wedges. Here, we test aspects of these ideas making use of tensor network toy models for holography. In this context, the analog of a generalized entanglement wedge is a region for which no region containing it has a smaller number of boundary legs. In the limit of large bond dimension (analogous to the semiclassical limit), we argue that each such region has a canonically associated subalgebra of the full algebra of observables on the boundary Hilbert space, as well as a family of unitarily related von Neumann factor subalgebras. We investigate in detail how various algebraic operations on these subalgebras (e.g. their intersections, joins, and commutants) are related to corresponding geometrical operations on the tensor network regions. We study to what extent the monotonicity and strong subadditivity properties of generalized entropy for generalized entanglement wedges follow from properties of our algebraic entropy. We also introduce an alternative entropic quantity $S(ω|| ω\circ E_{\mathcal{A}'})$ associated with a subalgebra $\mathcal{A}$ that also obeys monotonicity and strong subadditivity properties with suitable conditions. Finally, we explain how to define a type $II_1$ von Neumann algebra based on an infinite family of random tensor networks with increasing bond dimension such that the regions corresponding to generalized entanglement wedges can be associated with exact subalgebras of this full algebra.

Comments
57 pages + appendices, 7 figures; reference added

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In a previous work 2511.21852, we explored the hypothesis that in general theories of quantum gravity, spacetime regions known as generalized entanglement wedges (introduced by Bousso and Penington) have an associated subalgebra of observables in a semiclassical limit. We defined an entropy associated with these subalgebras that obeys the same monotonicity and strong subadditivity properties as the generalized gravitational entropy of generalized entanglement wedges. Here, we test aspects of these ideas making use of tensor network toy models for holography. In this context, the analog of a generalized entanglement wedge is a region for which no region containing it has a smaller number of boundary legs. In the limit of large bond dimension (analogous to the semiclassical limit), we argue that each such region has a canonically associated subalgebra of the full algebra of observables on the boundary Hilbert space, as well as a family of unitarily related von Neumann factor subalgebras. We investigate in detail how various algebraic operations on these subalgebras (e.g. their intersections, joins, and commutants) are related to corresponding geometrical operations on the tensor network regions. We study to what extent the monotonicity and strong subadditivity properties of generalized entropy for generalized entanglement wedges follow from properties of our algebraic entropy. We also introduce an alternative entropic quantity S(ω∣∣ω∘EA′)S(ω|| ω\circ E_{\mathcal{A}'})S(ω∣∣ω∘EA′​) associated with a subalgebra A\mathcal{A}A that also obeys monotonicity and strong subadditivity properties with suitable conditions. Finally, we explain how to define a type II1II_1II1​ von Neumann algebra based on an infinite family of random tensor networks with increasing bond dimension such that the regions corresponding to generalized entanglement wedges can be associated with exact subalgebras of this full algebra.
Generalized entanglement wedges (GWs), proposed by Bousso and Penington for gravitating regions in general spacetimes, share many properties of ordinary entanglement wedges, including nesting, monotonicity and strong subadditivity of their generalized entropies. In AdS/CFT, the corresponding properties of entanglement wedges are consequences of the structure of the dual CFT. In this paper we ask whether the properties of GWs have a similar origin. We explore this question in vacuum AdS3_33​ at leading order in GGG. We first show that every GW on a static slice is a union of geodesically convex regions. Each such region is an intersection of ordinary entanglement wedges, so every GW is built from unions and intersections of entanglement wedges. We then use the Crofton formula of integral geometry to write the perimeter of a convex region as the measure of the set of geodesics that intersect it. In AdS3_33​ every geodesic is the Ryu--Takayanagi surface of a boundary interval, so this measure can be written in terms of the von Neumann entropies of CFT intervals. Strong subadditivity of the CFT entropy guarantees that the measure is positive. Monotonicity and strong subadditivity of GWs then follow from simple inequalities between sets of geodesics. The strong subadditivity of GWs is therefore a direct consequence of strong subadditivity on the boundary. Our results further suggest that GWs correspond to subsystems of the CFT that are more general than boundary subregions.
An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III _{1} 1 , in which entropy cannot be well-defined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type II _∞ ∞ factor, in which traces and hence von Neumann entropy can be well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, formally paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag’s assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type II factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.
A bstract The concept of the generalized entanglement wedge was recently proposed by Bousso and Penington, which states that any bulk gravitational region a possesses an associated generalized entanglement wedge E ( a ) ⊃ a on a static Cauchy surface M in general gravitational spacetimes, where E ( a ) may contain an entanglement island I ( a ). It suggests that the fine-grained entropy for bulk region a is given by the generalized entropy S gen ( E ( a )). Motivated by this proposal, we extend the quantum bit thread description to general gravitational spacetimes, no longer limited to the AdS spacetime. By utilizing the convex optimization techniques, a dual flow description for the generalized entropy S gen ( E ( a )) of a bulk gravitational region a is established on the static Cauchy surface M , such that S gen ( E ( a )) is equal to the maximum flux of any flow that starts from the boundary ∂M and ends at bulk region a , or equivalently, the maximum number of bit threads that connect the boundary ∂M to the bulk region a . In addition, the nesting property of flows is also proved. Thus the basic properties of the entropy for bulk regions, i.e. the monotonicity, subadditivity, Araki-Lieb inequality and strong subadditivity, can be verified from flow perspectives by using properties of flows, such as the nesting property. Moreover, in max thread configurations, we find that there exists some lower bounds on the bulk entanglement entropy of matter fields in the region E ( a ) \ a , particularly on an entanglement island region I ( a ) ⊂ ( E ( a ) \ a ), as required by the existence of a nontrivial generalized entanglement wedge. Our quantum bit thread formulation may provide a way to investigate more fine-grained entanglement structures in general spacetimes.
Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.
We introduce a simple geometrical construction similar to covariant holographic entanglement entropy but with the addition of a new term proportional to boundary region volume. This new procedure has properties strongly resembling classical Shannon entropy of probability distributions rather than von Neumann entropy, so we call the quantity holographic Shannon entropy. The holographic Shannon entropy of a region AAA is divergent in AdS/CFT, but upon regulation, it appears to be equal to the outer entropy of the entanglement wedge of AAA with the entanglement wedge of the complement of AAA held fixed. The construction is unambiguous when applied to compact surfaces with convex shape in general spacetimes obeying the null curvature condition (of which AdS with a regulator is a special case). In this context we prove that holographic Shannon entropy obeys monotonicity, a key property of Shannon entropy, as well as all known balanced inequalities of dynamical holographic entanglement entropy. In the static case, we explain why there must exist some classical probability distribution on random variables locally distributed on the boundary with the property that the Shannon entropies of all marginals are exactly reproduced by the holographic Shannon entropy formula.
We study the holographic properties of a class of quantum geometry states characterized by a superposition of discrete geometric data, in the form of generalized tensor networks. This class specifically includes spin networks, the kinematic states of lattice gauge theory, and discrete quantum gravity. We employ an algebraic, operatorial definition of holography based on quantum information channels, an approach which is particularly valuable in settings, such as the one we consider, where the relevant Hilbert space of states does not factorize into subsystem Hilbert spaces due to gauge invariance. We apply random tensor network techniques (successfully used in the AdS/CFT context) to analyze information transport properties of the bulk-to-boundary and boundary-to-boundary maps associated with this superposition of quantum geometries and produce typicality results about the average over the geometric data coloring the fixed graph structure. In this context, one naturally obtains a nontrivial area operator encoding the dominant contribution to entropy calculations. Among our main results is the requirement that one can only isometrically map a bulk region onto boundaries with fixed total area. We furthermore inquire about similar state-induced mappings between segments of the boundary and discuss related conditions for isometric behavior. These generalizations make further steps toward quantum gravity implementations of tensor network holography.