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Entanglement Before Geometry: Random Tensor Networks as the Quantum Completion of the Flux-Gauge Area Law

Mohammad Hannan

Published Jul 20, 2026• 0 citations
Publisher

Abstract

Abstract A confining, non-Abelian gauge connection Uij ∈ SU(N) defined on the links of a dynamical graph has been proposed as the microscopic origin of classical spacetime geometry: the metric is reconstructed from the coset (soldering-form) sector of an enlarged connection, and black-hole and boundary-cut entropy are obtained as the logarithm of a combinatorial microstate count, matched, via the Freedman–Headrick theorem, to the bit-thread (max-flow/min-cut) formulation of holographic entanglement entropy. That matching is exact as a statement about counting, but it is not, by itself, a statement about the entanglement entropy of any specific quantum state: nothing in the combinatorial construction distinguishes a classical ensemble of flux occupation numbers from a genuine quantum wavefunction, so its connection to the entanglement/tensornetwork program of emergent geometry (Van Raamsdonk, Swingle, Ryu–Takayanagi, and the random-tensor-network derivation of Hayden, Qi, Kim, and Preskill) remains an analogy rather than a derivation. This paper closes that gap. We first restate the flux-gauge connection ensemble in full – its partition function, its block-link renormalization-group coarse-graining with closed-form Migdal–Kadanoff coefficients, its emergent metric and Newton’s-constant identification, and its combinatorial boundary-cut entropy – so that the paper is self-contained. We then show, by an exact character-expansion/Peter–Weyl derivation standard in lattice gauge theory but not previously applied to this construction, that the SU(N) link ensemble is a tensor network: each flux-tube segment (link) carries a bond dimension equal to the dimension of the SU(N) representation it transports, and each flux-tube junction (node) carries a gauge-invariant intertwiner tensor, fixed by representation theory rather than assumed. Only a single, narrowly scoped statistical assumption – on the distribution of intertwiners within the multiplicity space that gauge invariance already guarantees – is needed to complete this exact, deterministic tensor network into a genuine random tensor network in the technical sense of Hayden–Qi–Kim–Preskill (HQKP). We further identify the renormalization-group layers of Section 2’s coarse-graining as the isometric maps of a multi-scale entanglement renormalization ansatz (MERA), giving the emergent radial holographic direction a literal tensor-network origin rather than a scaling-dimension analogy. We then investigate, analytically and how the entanglement entropy of such a network reproduces the Ryu–Takayanagi/min-cut law. An exactly solvable limit built from classical (GHZ-type) perfect tensors reproduces the combinatorial law exactly for single-cut (boundary-anchored) regions but is shown to be degenerate for interior, two-sided regions, exposing a genuine gap in the combinatorial entropy argument: classical flux-occupation counting does not, by itself, certify multi-region (entanglement-wedgelevel) quantum entanglement structure. We then construct a genuinely quantum (non-classical, Gaussian-random) tensor-network realization of the link ensemble and confirm numerically, on chains of up to sixteen sites, (i) a clean area-law plateau in which the entanglement entropy of an interior region becomes independent of its size once the region is large enough, in place of continued volume-law growth, and (ii) a linear scaling of the plateau entropy with the logarithm of the bond dimension, S(A) = κNcut(A) logD, fit with R2 = 0.996 and an O(1) efficiency coefficient κ ≈ 0.68 for this finite, non-isometric network. We interpret κ < 1 as a genuine, quantifiable finite-network correction relative to the idealized D → ∞ isometric-projector limit in which HQKP prove κ → 1, and we list the construction of that idealized limit, directly on the SU(N) link ensemble, as the key open numerical task. We close by proposing an explicit correspondence between the tensor-network minimal surface of this paper and the Cartan soldering-form metric of the gauge-connection construction, framed as a falsifiable consistency conjecture rather than an established result.

DOI
10.21203/rs.3.rs-10385698/v1

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Abstract A confining, non-Abelian gauge connection Uij ∈ SU(N) defined on the links of a dynamical graph has been proposed as the microscopic origin of classical spacetime geometry: the metric is reconstructed from the coset (soldering-form) sector of an enlarged connection, and black-hole and boundary-cut entropy are obtained as the logarithm of a combinatorial microstate count, matched, via the Freedman–Headrick theorem, to the bit-thread (max-flow/min-cut) formulation of holographic entanglement entropy. That matching is exact as a statement about counting, but it is not, by itself, a statement about the entanglement entropy of any specific quantum state: nothing in the combinatorial construction distinguishes a classical ensemble of flux occupation numbers from a genuine quantum wavefunction, so its connection to the entanglement/tensornetwork program of emergent geometry (Van Raamsdonk, Swingle, Ryu–Takayanagi, and the random-tensor-network derivation of Hayden, Qi, Kim, and Preskill) remains an analogy rather than a derivation. This paper closes that gap. We first restate the flux-gauge connection ensemble in full – its partition function, its block-link renormalization-group coarse-graining with closed-form Migdal–Kadanoff coefficients, its emergent metric and Newton’s-constant identification, and its combinatorial boundary-cut entropy – so that the paper is self-contained. We then show, by an exact character-expansion/Peter–Weyl derivation standard in lattice gauge theory but not previously applied to this construction, that the SU(N) link ensemble is a tensor network: each flux-tube segment (link) carries a bond dimension equal to the dimension of the SU(N) representation it transports, and each flux-tube junction (node) carries a gauge-invariant intertwiner tensor, fixed by representation theory rather than assumed. Only a single, narrowly scoped statistical assumption – on the distribution of intertwiners within the multiplicity space that gauge invariance already guarantees – is needed to complete this exact, deterministic tensor network into a genuine random tensor network in the technical sense of Hayden–Qi–Kim–Preskill (HQKP). We further identify the renormalization-group layers of Section 2’s coarse-graining as the isometric maps of a multi-scale entanglement renormalization ansatz (MERA), giving the emergent radial holographic direction a literal tensor-network origin rather than a scaling-dimension analogy. We then investigate, analytically and how the entanglement entropy of such a network reproduces the Ryu–Takayanagi/min-cut law. An exactly solvable limit built from classical (GHZ-type) perfect tensors reproduces the combinatorial law exactly for single-cut (boundary-anchored) regions but is shown to be degenerate for interior, two-sided regions, exposing a genuine gap in the combinatorial entropy argument: classical flux-occupation counting does not, by itself, certify multi-region (entanglement-wedgelevel) quantum entanglement structure. We then construct a genuinely quantum (non-classical, Gaussian-random) tensor-network realization of the link ensemble and confirm numerically, on chains of up to sixteen sites, (i) a clean area-law plateau in which the entanglement entropy of an interior region becomes independent of its size once the region is large enough, in place of continued volume-law growth, and (ii) a linear scaling of the plateau entropy with the logarithm of the bond dimension, S(A) = κNcut(A) logD, fit with R2 = 0.996 and an O(1) efficiency coefficient κ ≈ 0.68 for this finite, non-isometric network. We interpret κ < 1 as a genuine, quantifiable finite-network correction relative to the idealized D → ∞ isometric-projector limit in which HQKP prove κ → 1, and we list the construction of that idealized limit, directly on the SU(N) link ensemble, as the key open numerical task. We close by proposing an explicit correspondence between the tensor-network minimal surface of this paper and the Cartan soldering-form metric of the gauge-connection construction, framed as a falsifiable consistency conjecture rather than an established result.
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.
The relation between loop quantum gravity (LQG) and tensor networks is explored from the perspectives of the bulk-boundary duality and holographic entanglement entropy. We find that the LQG spin-network states in a space $\mathrm{\ensuremath{\Sigma}}$ with boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$ is an exact holographic mapping similar to the proposal in [X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282]. The tensor network, understood as the boundary quantum state, is the output of the exact holographic mapping emerging from a coarse-graining procedure of spin networks. Furthermore, when a region AAA and its complement A‾\overline{A}A are specified on the boundary $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$, we show that the boundary entanglement entropy S(A)S(A)S(A) of the emergent tensor network satisfies the Ryu-Takayanagi formula in the semiclassical regime, i.e., S(A)S(A)S(A) is proportional to the minimal area of the bulk surface attached to the boundary of AAA in $\ensuremath{\partial}\mathrm{\ensuremath{\Sigma}}$.
Pseudoentangled states are defined by their ability to hide their entanglement structure: they are indistinguishable from random states to any observer with polynomial resources, yet can have much less entanglement than random states. Existing constructions of pseudoentanglement based on phase and/or subset states are limited in the entanglement structures they can hide, e.g., the states may have low entanglement on a single cut, on all cuts at once, or on local cuts in one dimension. Here, we introduce new constructions of pseudoentangled states based on (pseudo)random tensor networks that afford much more flexibility in the achievable entanglement structures. We illustrate our construction with the simplest example of a matrix product state, realizable as a staircase circuit of pseudorandom unitary gates, which exhibits pseudo-area-law scaling of entanglement in one dimension. We then generalize our construction to arbitrary tensor network structures that admit an isometric realization. A notable application of this result is the construction of pseudoentangled "holographic" states whose entanglement entropy obeys a Ryu-Takayanagi "minimum-cut" formula, answering a question posed in Aaronson et al. [arXiv:2211.00747].
The entanglement entropy of a random tensor network (RTN) is studied using tools from free probability theory. Random tensor networks are simple toy models that help in understanding the entanglement behavior of a boundary region in the anti-de Sitter/conformal field theory (AdS/CFT) context. These can be regarded as specific probabilistic models for tensors with particular geometry dictated by a graph (or network) structure. First, we introduce a model of RTN obtained by contracting maximally entangled states (corresponding to the edges of the graph) on the tensor product of Gaussian tensors (corresponding to the vertices of the graph). The entanglement spectrum of the resulting random state is analyzed along a given bipartition of the local Hilbert spaces. The limiting eigenvalue distribution of the reduced density operator of the RTN state is provided in the limit of large local dimension. This limiting value is described through a maximum flow optimization problem in a new graph corresponding to the geometry of the RTN and the given bipartition. In the case of series-parallel graphs, an explicit formula for the limiting eigenvalue distribution is provided using classical and free multiplicative convolutions. The physical implications of these results are discussed, allowing the analysis to move beyond the semiclassical regime without any cut assumption, specifically in terms of finite corrections to the average entanglement entropy of the RTN.
Holographic tensor networks serve as toy models for the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, capturing many of its essential features in a concrete manner. However, existing holographic tensor network models remain far from a complete theory of quantum gravity. A key obstacle is their discrete structure, which only approximates the semi-classical geometry of gravity in a qualitative sense. In \cite{Lin:2024dho}, it was shown that a network of partial-entanglement-entropy (PEE) threads, which are bulk geodesics with a specific density distribution, generates a perfect tessellation of AdS space. Moreover, such PEE-network tessellations can be constructed for more highly symmetric geometries using the Crofton formula. In this paper, we assign a quantum state to each vertex in the PEE network and develop several holographic tensor network models: (1) the factorized PEE tensor network, which takes the form of a tensor product of EPR pairs; (2) the HaPPY-like PEE tensor network constructed from perfect tensors; and (3) the random PEE tensor network. In all these models, we reproduce the exact Ryu-Takayanagi formula by showing that the minimal number of cuts along a homologous surface in the network exactly equals the area of that surface.
In tensor networks, a geometric operation of pushing a bond cut surface toward a minimal surface corresponds to entanglement distillation. Cutting bonds defines a reduced transition matrix on the bond cut surface and the associated quantum state naturally emerges from it. We justify this picture quantitatively by evaluating the trace distance between the maximally entangled states and the states on bond cut surfaces in the multiscale entanglement renormalization ansatz (MERA) and matrix product states in a canonical form. Our numerical result for the random MERA is in a reasonable agreement with our proposal. The result sheds new light on a deeper understanding of the Ryu-Takayanagi formula for entanglement entropy in holography and the emergence of geometry from the entanglement structure.