Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT

TL;DR

The paper develops a constructive program to encode AdS/CFT geometry in tensor networks by distilling boundary entanglement into bond-dimension data tied to Ryu-Takayanagi surfaces, achieving sub-AdS granularity through holographic entanglement of purification and its generalizations. It provides rigorous tree-network constructions from one-shot entanglement distillation, demonstrates bulk-to-boundary error-correction properties, and introduces loop networks to localize information on RT surfaces. A no-go theorem shows that fully isometric bulk-to-boundary maps cannot exist for networks with intersecting RT surfaces, guiding the development of subleading 'moral' isometries and sub-AdS iterations. The work also explores quantum geometry via superpositions of tensor networks to model bulk fluctuations and discusses implications for entanglement shadows, bit threads, and dynamics, highlighting both the potential and the limitations of representing continuous spacetime with entanglement-only data.

Abstract

We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-$N$ CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For 'tree' tensor networks (i.e., those that are constructed by discretizing spacetime with non-intersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the 'holographic entanglement of purification' conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.

Figures (16)

• Figure 1: A simple network for the state given in equation (\ref{['simple_network']}). Outward-pointing arrows denote up-indices, inward-pointing arrow denote down-indices, and arrows connecting tensors denote contractions.
• Figure 2: Tensor networks for equations (\ref{['simple_contraction']}) and (\ref{['simple_PEP']}). (a) A tensor network on a bipartite system with a single contraction. (b) A PEP network for the same state created by replacing the contraction with a maximally entangled state.
• Figure 3: A bulk discretization of vacuum $AdS_3$ by non-intersecting RT surfaces. Blue curves represent extremal surfaces in the bulk, red lines are edges in the dual graph, and black dots are vertices in the dual graph. In the resulting tensor network, the dangling edges passing through the boundary of the spacetime will correspond to uncontracted (physical) Hilbert space indices.
• Figure 4: A tensor network for a bipartite discretization of $AdS_3$ by a single Ryu-Takayanagi surface. $\phi^{\gamma \overline{\gamma}}$ is a maximally entangled state on a Hilbert space of dimension $e^{O(S(A))}$, while $\sigma^{f \overline{f}}$ is a (generally not maximally entangled) state on a Hilbert space of dimension $e^{O\left(\sqrt{S(A)}\right)}$. The tensors $V$ and $W$ embed these states isometrically into the boundary.
• Figure 5: The bulk discretization of vacuum $AdS_3$ that was originally sketched in Figure \ref{['fig:tree_discretization']} has here been given a root-leaf orientation on its dual graph by choosing an arbitrary boundary edge as the "root" (represented here by a white circle).

Theorems & Definitions (1)

• Theorem 1