Machine Learning Statistical Gravity from Multi-region Entanglement Entropy
The holographic duality 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, in particular via the Ryu-Takayanagi (RT) formula which assumes the gravity theory is in the classical limit.Yet the assumption of classical geometry has lead to exponentially small mutual information between disjoint sub-regions, which is not true in many system such as free fermion. In this work, we study a generalized Random Tensor Network (RTN) model with fluctuating bond dimensions, which is mapped to a statistical gravity model consisting a massive scalar field on a fluctuating background geometry. A concrete algorithm is constructed to recover the underlying geometry fluctuation from multi-region entanglement entropy data by modelling its distribution as a generative neural network. To demonstrate its effectiveness, we train the model using entanglement entropy of a free fermion system and showed mutual information can be mediated effectively by geometric fluctuation. Remarkably, locality emerges from the learned geometric distribution.