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Cited by 57Black Holes and Theoretical PhysicsQuantum many-body systemsAdvanced Operator Algebra Research

Entanglement wedge reconstruction using the Petz map

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Chen, Chi-Fang, Penington, Geoffrey, Salton, Grant·2020·Journal of High Energy Physics
DOI 10.1007/jhep01(2020)168
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Abstract

Abstract At the heart of recent progress in AdS/CFT is the question of subregion duality, or entanglement wedge reconstruction: which part(s) of the boundary CFT are dual to a given subregion of the bulk? This question can be answered by appealing to the quantum error correcting properties of holography, and it was recently shown that robust bulk (entanglement wedge) reconstruction can be achieved using a universal recovery channel known as the twirled Petz map. In short, one can use the twirled Petz map to recover bulk data from a subset of the boundary. However, this map involves an averaging procedure over bulk and boundary modular time, and hence it can be somewhat intractable to evaluate in practice. We show that a much simpler channel, the Petz map, is sufficient for entanglement wedge reconstruction for any code space of fixed finite dimension — no twirling is required. Moreover, the error in the reconstruction will always be non-perturbatively small. From a quantum information perspective, we prove a general theorem extending the use of the Petz map as a general-purpose recovery channel to subsystem and operator algebra quantum error correction.

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  • Let M_a be a subalgebra on the code space C with dimension d_code, and N be any noise map. Suppose there exists an optimal recovery map R_opt such that (R_opt • N)(ρ) - ρ_a < δ for all ρ ∈ B(C) and its projection ρ_a onto M_a. Then the Petz map R_τ,N defined using the maximally mixed state τ on the codespace satisfies (R_τ,N • N)(ρ)|_a - ρ_a ≤ d_code √(8δ).

    “Let M a be a subalgebra on the code space C with dimension d code , and N be any noise map.Suppose there exists an optimal recovery map R opt such that (R opt • N )(ρ) -ρ a 1 < δ, for all ρ ∈ B(C) and its projection ρ a onto M a .Then, the Petz map R τ,N (.) ≡ 1 d code N † N (τ ) -1/2 (.)N (τ ) -1/2 , defined using the maximally mixed state τ on the codespace, satisfies, (R τ,N ) • N )(ρ)| a -ρ a”

    From Holographic spacetime, black holes and quantum error correcting codes: a review (2022) · cites this paper as [187]

  • The standard form of the Petz map in Eq. (68) suffices to achieve approximate bulk reconstruction.

    “Moving away from the twirled Petz map which involves an averaging over the modular time, it was argued in [187] that the standard form of the Petz map in Eq. ( 68) suffices to achieve approximate bulk reconstruction.”

    From Holographic spacetime, black holes and quantum error correcting codes: a review (2022) · cites this paper as [187]

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Authors
Chen, C.-F., Penington, G., & Salton, G.
Affiliations
California Institute of Technology (US)
Corresponding author
Chen, Chi-Fang
ORCID
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2020-01-28
Volume
2020
Issue
1
ISSN
1029-8479 (ISSN-L)
DOI
10.1007/jhep01(2020)168
OpenAlex
W2929692894
Cited by
54
References
27
Citations by year
2026: 3 · 2025: 10 · 2024: 7 · 2023: 9 · 2022: 9 · 2021: 9 · 2020: 7
FWCI
3.76
Type
journal article
Language
en
Open access
Licence
Locations
  • Journal of High Energy Physics (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion)
  • CaltechAUTHORS (California Institute of Technology) (submittedVersion, cc-by)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • U.S. Department of Energy (DESC0018407)
  • Simons Foundation
  • California Institute of Technology
  • Institute for Quantum Information and Matter, California Institute of Technology
  • Air Force Office of Scientific Research (FA9550-, FA9550-16-1-0082, FA9550-16, -16-1-, FA9550-16-1)
  • SLAC National Accelerator Laboratory
Topics
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
  • Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
Keywords
entanglement wedge reconstruction, AdS/CFT correspondence, quantum error correction, holography, quantum information

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