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Group Field theory and Tensor Networks: towards a Ryu-Takayanagi formula in full quantum gravity

Goffredo Chirco, Daniele Oriti, Zhang, M.

Published Jan 1, 2018• 29 citations
OpenAlex

Abstract

We establish a dictionary between group field theory (thus, spin networks and random tensors) states and generalized random tensor networks. Then, we use this dictionary to compute the R\'{e}nyi entropy of such states and recover the Ryu-Takayanagi formula, in three different cases corresponding to three different truncations/approximations, suggested by the established correspondence.

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We establish a dictionary between group field theory (thus, spin networks and random tensors) states and generalized random tensor networks. Then, we use this dictionary to compute the R'{e}nyi entropy of such states and recover the Ryu-Takayanagi formula, in three different cases corresponding to three different truncations/approximations, suggested by the established correspondence.
We consider the special case of random tensor networks (RTNs) endowed with gauge symmetry constraints on each tensor. We compute the R\'enyi entropy for such states and recover the Ryu-Takayanagi (RT) formula in the large-bond regime. The result provides first of all an interesting new extension of the existing derivations of the RT formula for RTNs. Moreover, this extension of the RTN formalism brings it in direct relation with (tensorial) group field theories (and spin networks), and thus provides new tools for realizing the tensor network/geometry duality in the context of background-independent quantum gravity, and for importing quantum gravity tools into tensor network research.
Abstract We introduce a generalised class of (symmetric) random tensor network states in the framework of group field theory. In this setting, we compute the Rényi entropy for a generic bipartite state via a mapping to the partition function of a topological 3D BF theory, realised as a simple interacting group field theory. The expectation value of the entanglement entropy is calculated by an expansion into stranded Feynman graphs and is shown to be captured by a Ryu–Takayanagi formula. For the simple case of a 3D BF theory, we can prove the linear corrections, given by a polynomial perturbation of the Gaussian measure, to be negligible for a broad class of networks.
This work is meant as a review summary of a series of recent results concerning the derivation of a holographic entanglement entropy formula for generic open spin network states in the group field theory (GFT) approach to quantum gravity. The statistical group-field computation of the Rényi entropy for a bipartite network state for a simple interacting GFT is reviewed, within a recently proposed dictionary between group field theories and random tensor networks, and with an emphasis on the problem of a consistent characterisation of the entanglement entropy in the GFT second quantisation formalism.
Despite the fundamental importance of quantum entanglement in many-body systems, our understanding is mostly limited to bipartite situations. Indeed, even defining appropriate notions of multipartite entanglement is a significant challenge for general quantum systems. In this work, we initiate the study of multipartite entanglement in a rich, yet tractable class of quantum states called stabilizer tensor networks. We demonstrate that, for generic stabilizer tensor networks, the geometry of the tensor network informs the multipartite entanglement structure of the state. In particular, we show that the average number of Greenberger-Horne-Zeilinger (GHZ) triples that can be extracted from a stabilizer tensor network is small, implying that tripartite entanglement is scarce. This, in turn, restricts the higher-partite entanglement structure of the states. Recent research in quantum gravity found that stabilizer tensor networks reproduce important structural features of the AdS/CFT correspondence, including the Ryu-Takayanagi formula for the entanglement entropy and certain quantum error correction properties. Our results imply a new operational interpretation of the monogamy of the Ryu-Takayanagi mutual information and an entropic diagnostic for higher-partite entanglement. Our technical contributions include a spin model for evaluating the average GHZ content of stabilizer tensor networks, as well as a novel formula for the third moment of random stabilizer states, which we expect to find further applications in quantum information.
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.