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Toward random tensor networks and holographic codes in CFT

Jeevan Chandra, Thomas Hartman

TL;DR

This work connects holographic quantum error-correcting codes to exact CFT data by constructing (pseudo)random tensor networks from OPE data under eigenstate thermalization, thereby discretizing the bulk radial direction. It demonstrates an isometric-to-coisometric transition across horizons, linked to the breakdown of the Virasoro identity block in complex interior states, and provides a CFT-based mechanism for bulk reconstruction beyond toy qubit models. By formulating Virasoro OPE blocks and a large-$c$ ensemble of random OPE coefficients, the authors realize holographic codes directly within 2d CFT and derive that bond dimensions on internal links equal $e^{S}$ with $S=A/4$ at minimal surfaces. The results offer a quantitative bridge between CFT data, random tensor networks, and bulk AdS$_3$/CFT$_2$ gravity, with implications for black hole interiors, singularities, and extensions to higher dimensions. Overall, the paper advances a concrete, CFT-centric realization of holographic tensor networks and deepens understanding of horizon-associated isometric properties and bulk reconstruction.

Abstract

In holographic CFTs satisfying eigenstate thermalization, there is a regime where the operator product expansion can be approximated by a random tensor network. The geometry of the tensor network corresponds to a spatial slice in the holographic dual, with the tensors discretizing the radial direction. In spherically symmetric states in any dimension and more general states in 2d CFT, this leads to a holographic error-correcting code, defined in terms of OPE data, that can be systematically corrected beyond the random tensor approximation. The code is shown to be isometric for light operators outside the horizon, and non-isometric inside, as expected from general arguments about bulk reconstruction. The transition at the horizon occurs due to a subtle breakdown of the Virasoro identity block approximation in states with a complex interior.

Toward random tensor networks and holographic codes in CFT

TL;DR

This work connects holographic quantum error-correcting codes to exact CFT data by constructing (pseudo)random tensor networks from OPE data under eigenstate thermalization, thereby discretizing the bulk radial direction. It demonstrates an isometric-to-coisometric transition across horizons, linked to the breakdown of the Virasoro identity block in complex interior states, and provides a CFT-based mechanism for bulk reconstruction beyond toy qubit models. By formulating Virasoro OPE blocks and a large- ensemble of random OPE coefficients, the authors realize holographic codes directly within 2d CFT and derive that bond dimensions on internal links equal with at minimal surfaces. The results offer a quantitative bridge between CFT data, random tensor networks, and bulk AdS/CFT gravity, with implications for black hole interiors, singularities, and extensions to higher dimensions. Overall, the paper advances a concrete, CFT-centric realization of holographic tensor networks and deepens understanding of horizon-associated isometric properties and bulk reconstruction.

Abstract

In holographic CFTs satisfying eigenstate thermalization, there is a regime where the operator product expansion can be approximated by a random tensor network. The geometry of the tensor network corresponds to a spatial slice in the holographic dual, with the tensors discretizing the radial direction. In spherically symmetric states in any dimension and more general states in 2d CFT, this leads to a holographic error-correcting code, defined in terms of OPE data, that can be systematically corrected beyond the random tensor approximation. The code is shown to be isometric for light operators outside the horizon, and non-isometric inside, as expected from general arguments about bulk reconstruction. The transition at the horizon occurs due to a subtle breakdown of the Virasoro identity block approximation in states with a complex interior.
Paper Structure (8 sections, 33 equations, 2 figures)

This paper contains 8 sections, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. An example
  3. Spherically symmetric states
  4. The probe OPE as a random tensor network
  5. Derivation of the energy hierarchy with spherical symmetry
  6. Random tensor networks from 2d CFT
  7. Virasoro OPE blocks
  8. The tensor network

Figures (2)

  • Figure 1: The spatial geometry of a pure-state black hole in AdS$_3$ and the corresponding pseudorandom tensor network. The tensors, which are defined in terms of OPE coefficients (circles) and Virasoro OPE blocks (semicircles), discretize the radial direction in the bulk.
  • Figure 2: The semiclassical CFT state as a tensor network, as in \ref{['psistarexp2']}. Internal lines are labeled by CFT primaries with $h,\bar{h} > \frac{c}{24}$. The red tensors are OPE coefficients and the final tensor on the left is the (mod-squared) OPE block, which is a map from the space of primaries to the physical CFT Hilbert space.