Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence

  • Fernando PastawskiEmail author
  • Beni Yoshida
  • Daniel Harlow
  • John Preskill
Open Access
Regular Article - Theoretical Physics


We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1].


AdS-CFT Correspondence Lattice Integrable Models 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Fernando Pastawski
    • 1
    Email author
  • Beni Yoshida
    • 1
  • Daniel Harlow
    • 2
  • John Preskill
    • 1
  1. 1.Institute for Quantum Information & Matter and Walter Burke Institute for Theoretical Physics, California Institute of TechnologyPasadenaUSA
  2. 2.Princeton Center for Theoretical SciencePrinceton UniversityPrincetonUSA

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