Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Holographic Tensor Networks in Full AdS/CFT

Ning Bao, Geoffrey Penington, Jonathan Sorce, Aron C. Wall

TL;DR

The paper addresses representing holographic CFT states as tensor networks whose geometry tracks discretized bulk AdS spacetimes for static, semiclassical duals. It develops a general procedure using entanglement distillation and the leading behavior of smooth min-/max-entropies to produce boundary tensor networks whose bond dimensions reflect bulk areas via the RT formula, thereby reproducing boundary physics. By invoking the holographic entanglement of purification conjecture, it shows how to subdivide RT surfaces and obtain sub-AdS tensor networks through entanglement wedge cross-sections, and extends the construction to multi-region discretizations. Coupled with companion work, the approach argues that tensor networks provide a precise geometric description of AdS/CFT for static states across a range of discretizations, not merely toy models.

Abstract

We present a general procedure for constructing tensor networks for geometric states in the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence. Given a state in a large-$N$ CFT with a static, semiclassical gravitational dual, our procedure produces a tensor network for the boundary state whose internal geometry matches (a discretization of) the bulk spacetime geometry. By invoking the "holographic entanglement of purification" conjecture, our construction can be made to capture the structure of the bulk spacetime at sub-AdS scales.

Holographic Tensor Networks in Full AdS/CFT

TL;DR

The paper addresses representing holographic CFT states as tensor networks whose geometry tracks discretized bulk AdS spacetimes for static, semiclassical duals. It develops a general procedure using entanglement distillation and the leading behavior of smooth min-/max-entropies to produce boundary tensor networks whose bond dimensions reflect bulk areas via the RT formula, thereby reproducing boundary physics. By invoking the holographic entanglement of purification conjecture, it shows how to subdivide RT surfaces and obtain sub-AdS tensor networks through entanglement wedge cross-sections, and extends the construction to multi-region discretizations. Coupled with companion work, the approach argues that tensor networks provide a precise geometric description of AdS/CFT for static states across a range of discretizations, not merely toy models.

Abstract

We present a general procedure for constructing tensor networks for geometric states in the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence. Given a state in a large- CFT with a static, semiclassical gravitational dual, our procedure produces a tensor network for the boundary state whose internal geometry matches (a discretization of) the bulk spacetime geometry. By invoking the "holographic entanglement of purification" conjecture, our construction can be made to capture the structure of the bulk spacetime at sub-AdS scales.

Paper Structure

This paper contains 6 sections, 15 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Tensor Networks for Holographic States
  3. Tensor Network Preliminaries
  4. Smooth Entropies in Holographic States
  5. Holographic Networks from Entanglement Distillation
  6. Entanglement of Purification and Sub-AdS Structure

Figures (4)

  • Figure 1: The graph corresponding to the tensor network expression in eq.\ref{['eq:simple-network']}.
  • Figure 2: A tensor network for a bipartite discretization of $AdS_3$ by a single Ryu-Takayanagi surface, drawn here as a dashed line in the bulk.
  • Figure 3: The entanglement wedge cross-section $\Sigma_{A_1 : A_2}$ for two neighboring boundary subregions $A_1$ and $A_2$ in vacuum $AdS_3.$
  • Figure 4: (a) A tensor network for the minimally entangled purification of two neighboring boundary regions in vacuum $AdS_3$. (b) The tensor network for the full boundary state formed by adding an isometry to the MEP.