Abstract
The Ryu-Takayanagi formula directly connects quantum entanglement and geometry. Yet the assumption of static geometry lead to an exponentially small mutual information between far-separated disjoint regions, which does not hold in many systems such as free fermion conformal field theories. In this paper, we proposed a microscopic model by superimposing entanglement features of an ensemble of random tensor networks of different bond dimensions, which can be mapped to a statistical gravity model consisting of a massive scalar field on a fluctuating background geometry. We propose a machine-learning algorithm that recovers the underlying geometry fluctuation from multi-region entanglement entropy data by modeling the bulk geometry distribution via a generative neural network. To demonstrate its effectiveness, we tested the model on a free fermion system and showed mutual information can be mediated effectively by geometric fluctuation. Remarkably, locality emerged from the learned distribution of bulk geometries, pointing to a local statistical gravity theory in the holographic bulk.
Highlights
The holographic duality [1–5] is a duality between boundary d-dimensional quantum field theories and bulk (d + 1)dimensional gravitational theories in asymptotically anti-de Sitter (AdS) space
Multi-region entanglement entropies further encode the correlation among multiple extremal surfaces, which could reveal how the bulk geometry fluctuations around its classical background
We present a machine-learning approach to extract the holography statistical gravity theory from the data of multiregion entanglement entropy in a quantum many-body system
Summary
The holographic duality [1–5] is a duality between boundary d-dimensional quantum field theories and bulk (d + 1)dimensional gravitational theories in asymptotically anti-de Sitter (AdS) space. It provides an appealing explanation for the emergence of spacetime geometry from quantum entanglement [6–16]. Progress has been made to reconstruct the bulk geometry from the boundary data in terms of geodesic lengths [19–22], extremal areas [23–25] or entanglement entropies [26,27]. A majority of the effort has been focused on reconstructing a classical geometry from single-region entanglement entropies (or independent extremal surfaces). Multi-region entanglement entropies further encode the correlation among multiple extremal surfaces, which could reveal how the bulk geometry fluctuations around its classical background (assuming a semiclassical description of the bulk gravity). We will explore the possibility to extract information about fluctuating holographic bulk geometries from multiregion entanglement entropies of a quantum system using generative models in machine learning
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