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Hyperinvariant Spin Network States -- An AdS/CFT Model from First Principles

Fynn Otto, Refik Mansuroglu, Norbert Schuch, Otfried Gühne, Hanno Sahlmann

TL;DR

This work embeds holographic tensor-network ideas into loop quantum gravity by constructing Hyperinvariant, SU(2)-invariant Tensors (HITs) that realize a boundary-bulk correspondence within spin-network states. It derives no-go theorems showing that invariant holographic codes and AME states cannot exist under SU(2) symmetry, while providing explicit HIT constructions based on Bell-pair structures that preserve SU(2) invariance and 1-isometry. The authors compute geometric observables using LQG length and area operators, proving that geodesic length scales with boundary graph length via a fixed factor and that surface areas scale with the number of vertices, yielding a quantum-geometric realization of negative curvature in the HIT framework. They further analyze boundary correlations, showing that non-singular decay requires multipartite entanglement that scales with system size, and discuss the implications for emergent spacetime and type-III von Neumann algebras in an inductive limit. Overall, the paper provides a first-principles LQG foundation for HIT-based holography, clarifying both its potentials and fundamental limitations.

Abstract

We study the existence and limitations for hyperinvariant tensor networks incorporating a local SU(2) symmetry. As discrete implementations of the anti de-Sitter/conformal field theory (AdS/CFT) correspondence, such networks have created bridges between the fields of quantum information theory and quantum gravity. Adding SU(2) symmetry to the tensor network allows a direct connection to spin network states, a basis of the kinematic Hilbert space of loop quantum gravity (LQG). We consider a particular situation where the states can be interpreted as kinematic quantum states for three-dimensional quantum gravity. We show that important aspects of the AdS/CFT correspondence are realized in certain quantum states of the gravitational field in LQG, thus justifying, from first principles, a class of models introduced by [F. Pastawski et al., JHEP 06, 149 (2015)]. We provide examples of hyperinvariant tensor networks, but also prove constraints on their existence in the form of no-go theorems that exclude absolutely maximally entangled states as well as general holographic codes from local SU(2)-invariance. We calculate surface areas as expectation values of the LQG area operator and discuss further possible constraints as a consequence of a decay of correlations on the boundary.

Hyperinvariant Spin Network States -- An AdS/CFT Model from First Principles

TL;DR

This work embeds holographic tensor-network ideas into loop quantum gravity by constructing Hyperinvariant, SU(2)-invariant Tensors (HITs) that realize a boundary-bulk correspondence within spin-network states. It derives no-go theorems showing that invariant holographic codes and AME states cannot exist under SU(2) symmetry, while providing explicit HIT constructions based on Bell-pair structures that preserve SU(2) invariance and 1-isometry. The authors compute geometric observables using LQG length and area operators, proving that geodesic length scales with boundary graph length via a fixed factor and that surface areas scale with the number of vertices, yielding a quantum-geometric realization of negative curvature in the HIT framework. They further analyze boundary correlations, showing that non-singular decay requires multipartite entanglement that scales with system size, and discuss the implications for emergent spacetime and type-III von Neumann algebras in an inductive limit. Overall, the paper provides a first-principles LQG foundation for HIT-based holography, clarifying both its potentials and fundamental limitations.

Abstract

We study the existence and limitations for hyperinvariant tensor networks incorporating a local SU(2) symmetry. As discrete implementations of the anti de-Sitter/conformal field theory (AdS/CFT) correspondence, such networks have created bridges between the fields of quantum information theory and quantum gravity. Adding SU(2) symmetry to the tensor network allows a direct connection to spin network states, a basis of the kinematic Hilbert space of loop quantum gravity (LQG). We consider a particular situation where the states can be interpreted as kinematic quantum states for three-dimensional quantum gravity. We show that important aspects of the AdS/CFT correspondence are realized in certain quantum states of the gravitational field in LQG, thus justifying, from first principles, a class of models introduced by [F. Pastawski et al., JHEP 06, 149 (2015)]. We provide examples of hyperinvariant tensor networks, but also prove constraints on their existence in the form of no-go theorems that exclude absolutely maximally entangled states as well as general holographic codes from local SU(2)-invariance. We calculate surface areas as expectation values of the LQG area operator and discuss further possible constraints as a consequence of a decay of correlations on the boundary.

Paper Structure

This paper contains 24 sections, 7 theorems, 68 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Hyperinvariant tensor networks
  3. Construction of hyperinvariant tensor networks
  4. Properties of hyperinvariant tensor networks
  5. LQG states with boundary Qudits
  6. Geometry with Parallel Transport Maps
  7. Spin Network States
  8. Hyperinvariant Loop Quantum Gravitational States
  9. Length and Area Operators
  10. Non-existence of U(1)-invariant two-uniform states
  11. No Invariant AME States
  12. No invariant Evenbly codes
  13. Existence of SU(2)-invariant Hyperinvariant tensor network states
  14. Hyperbolic Geometry from Loop Quantum Gravity
  15. Geodesic Length
  16. ...and 9 more sections

Key Result

Lemma 4.1

Let $\lvert \psi \rangle\in\bigotimes_{i=1}^n \mathcal{H}_{d_i}$ ($n\geq 4$) be a $\operatorname{U}(1)$-invariant state with respect to the representations $R^{(i)}$. Then, for every party $k \in \{ 1, ..., n \}$ with $R^{(k)}(g)[\cdot]R^{(k)}(g)^\dagger\neq \operatorname{id}(\cdot)$ there is a part

Figures (6)

  • Figure 1: Schematic view of the construction methods using a (7,3) tiling of the Poincaré disc. A hyperinvariant tensor network constructed from tensors $A$ on all vertices and $B$ on all edges.
  • Figure 2: Causal cone and entanglement wedge for a boundary region $\mathcal{A}$ in a $(7,3)$-tiling. In this case, the causal cone and the entanglement wedge coincide.
  • Figure 3: Pictorial representation of a spin network state with matter degrees of freedom. The spin quantum numbers, $j_i$, determine the SU(2) representation of the holonomy along the corresponding edge. The vertices depict intertwiner and the matter fields, $\theta^\mu$, couple free indices in an SU(2)-invariant fashion.
  • Figure 4: Equivalence defect of entanglement and casual cone. The strip $\sigma_\mathcal{A}$ is composed of the $A$ tensors represented by the black dots and $B$ tensors in green, for instance as in Eq. \ref{['eq:LR_state4']}.
  • Figure 5: Hyperbolic arrangement of "flat" triangles in a single layer. The triangles from which the dodecagon is built and which are defined by the tensors $A$ are all identical. Although, since we draw the dodecagon in a flat projection, the triangles appear different in size.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Definition 2.1: Hyperinvariant tensor networks
  • Definition 2.2: $k$-isometric tensors and $k$-uniform quantum states
  • Definition 2.3: Causal cone evenbly2017hyperinvariant
  • Definition 2.4: Entanglement wedge Pastawski_2015
  • Definition 3.1: Holonomy
  • Definition 3.2: Intertwiner
  • Definition 3.3: Spin Network States
  • Definition 3.4: Length Operator
  • Definition 3.5: Area Operator
  • Lemma 4.1
  • ...and 14 more