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 Lin et al. [ Classical and Quantum Gravity 43, no. 5 (2026): 055014], 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.