Holographic duality from random tensor networks

Abstract

Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.

Quantum Entanglement Growth under Random Unitary Dynamics

“…(19). Examples of nonunitary tensor networks in which the minimal cut bound becomes exact are also known [52], including a large-bonddimension limit similar to that we discuss here [53].…”

“…This very general bound has been related to the Ryu-Takayanagi formula for entanglement in holographic conformal field theories [50][51][52][53]74], and has also been applied to unitary networks as a heuristic picture for entanglement growth [10].…”

“…It also allows us to generalize from 1D to higher dimensions. The picture is reminiscent of the Ryu-Takayanagi prescription for calculating the entanglement entropy of conformal field theories in the AdS=CFT correspondence, which makes use of a minimal surface in the bulk space [50], and analogous results for certain tensor network states [51][52][53]. Here, however, the cut lives in space-time rather than in space, and in a noisy system its shape is random rather than deterministic.…”

Exploring the tensor networks/AdS correspondence

“…The important ingredient that goes into the proposal is perfect tensors, which have the magic property of being a unitary map between any choice of a set of half of the indices to the other half. This proposal is further developed in [32,33] so that there is a map between every state in the physical theory and a bulk tensor network, relaxing the restriction to the "code-subspace" in [31]. Perfect tensors were shown [33] to emerge rather typically for tensors with large bond dimension.…”

Holographic fluctuations and the principle of minimal complexity

“…Since these foundational works there has been a huge amount of effort exploring such AdS/CFT dualities. Most recently, quantum information ideas have been exploited to provide microscopic toy models to understand quantum gravity [7] and bulk/boundary correspondences [4][5][6].…”