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Holographic Tensor Networks as Tessellations of Geometry

Qiang Wen, Mingshuai Xu, Haocheng Zhong

Abstract

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 general geometries using the Crofton formula. In this paper, we assign a quantum state to each vertex in the PEE network and develop two holographic tensor network models: the factorized PEE tensor network, which takes the form of a tensor product of EPR pairs, and the random PEE tensor network. In both models we reproduce the exact Ryu-Takayanagi formula by showing that the minimal number of cuts along a homologous surface in the network exactly computes the area of this surface.

Holographic Tensor Networks as Tessellations of Geometry

Abstract

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 general geometries using the Crofton formula. In this paper, we assign a quantum state to each vertex in the PEE network and develop two holographic tensor network models: the factorized PEE tensor network, which takes the form of a tensor product of EPR pairs, and the random PEE tensor network. In both models we reproduce the exact Ryu-Takayanagi formula by showing that the minimal number of cuts along a homologous surface in the network exactly computes the area of this surface.
Paper Structure (2 sections, 2 theorems, 30 equations, 5 figures)

This paper contains 2 sections, 2 theorems, 30 equations, 5 figures.

Table of Contents

  1. The RT formula in the factorized PEE tensor network
  2. HaPPY-like PEE tensor network

Key Result

Proposition A.2

Given an isometry $T:\left|a_1 a_2\right\rangle \mapsto T_{b a_1 a_2}|b\rangle$ and if $\mathcal{H}_A=\mathcal{H}_{A_1}\otimes\mathcal{H}_{A_2}$, then there exists an isometry (up to a constant) $\widetilde{T}: \mathcal{H}_{A_2} \rightarrow \mathcal{H}_B \otimes \mathcal{H}_{A_1}$ acting as which obeys $\widetilde{T}^{\dagger} \widetilde{T}=\operatorname{dim}\left(A_1\right) I_{A_1}$.

Figures (5)

  • Figure 1: Extracted from Wen:2024uwr. The PEE network as a tessellation of a time slice of AdS$_3$. Here the blue lines represent the PEE threads, the red line is a surface $\Sigma_A$ homologous to the boundary region $A$.
  • Figure 2: A tensor state located at the bulk site $\mathbf{z}$ (hollow white dot in the center). The legs connecting $\mathbf{z}$ is actually continuously distributed, which can be described by a vector field as in Lin:2023rxc. Only six legs are exhibited, while other legs are collectively represented by the blue region. The contraction between tensors at $\mathbf{z}$ and at adjacent site $\mathbf{z'}$ is realized by projecting the total state into a maximally entangled state $| \mathbf{z}\mathbf{z'} \rangle$ along the PEE thread in between.
  • Figure 3: Contraction of bulk tensors along a PEE thread results in a PEE thread with two open legs of the boundary sites tangent to the thread. In the factorized PEE tensor network, the thread represents a two-qudit state $T_{ab}(\mathbf{x},\mathbf{y})| a \rangle _{\mathbf{x}}| b \rangle _{\mathbf{y}}$.
  • Figure 4: (a) The three type of PEE threads, denoted by $\mathcal{C}^{A}$, $\mathcal{C}^{\bar{A}}$ and $\mathcal{C}^{A\bar{A}}$, are labeled by gray, orange and blue solid simi-circles respectively, while the RT surface $\gamma_{A}$ is drawn as the red dashed line; (b) The circuit interpretation of the factorized PEE tensor network. Each PEE thread individually gives a unitary map, and the RT surface $\gamma_{A}$ only cuts the threads in $\mathcal{C}^{A\bar{A}}$ once. The graph can be compared with Fig.19 of Pastawski:2015qua where the "circuit" is built from the net effect of perfect tensors of certain bulk subregions as unitary gates.
  • Figure 5: Three local configurations for the legs emanating from one bulk site (the hollow circle) on the homologous surface $\Sigma_A$ (red dashed line) in the PEE network. Here, $\mathcal{W}_{\Sigma_A}$ is marked as the blue region, and the PEE thread tangent to $\Sigma_A$ at this bulk site is marked as the black dashed line.

Theorems & Definitions (6)

  • Definition A.1
  • Proposition A.2
  • proof
  • Proposition A.3
  • proof
  • Definition B.1