Discrete Gravity on Random Tensor Network and Holographic Rényi Entropy

Abstract

In this paper we apply the discrete gravity and Regge calculus to tensor networks and Anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We construct the boundary many-body quantum state $|\Psi ⟩$ using random tensor networks as the holographic mapping, applied to the Wheeler-deWitt wave function of bulk Euclidean discrete gravity in 3 dimensions. The entanglement Rényi entropy of $|\Psi ⟩$ is shown to holographically relate to the on-shell action of Einstein gravity on a branch cover bulk manifold. The resulting Rényi entropy $S_n$ of $|\Psi ⟩$ approximates with high precision the Rényi entropy of ground state in 2-dimensional conformal field theory (CFT). In particular it reproduces the correct $n$ dependence. Our results develop the framework of realizing the AdS${}_3$/CFT${}_2$ correspondence on random tensor networks, and provide a new proposal to approximate CFT ground state.

8+2 pages, 10 figures, presentation improved, references added

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Cited by in corpus (10)

• Flat entanglement spectra in fixed-area states of quantum gravity
• Holographic Renyi Entropy from Quantum Error Correction
• Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT
• Machine Learning Spatial Geometry from Entanglement Features
• Entanglement Features of Random Hamiltonian Dynamics
• Ryu-Takayanagi Formula for Symmetric Random Tensor Networks
• Holographic duality between local Hamiltonians from random tensor networks
• Quantum error correction and entanglement spectrum in tensor networks
• Tensor chain and constraints in tensor networks
• Random Invariant Tensors