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  • Letter
  • Open Access

Nonlocality induces isometry and factorisation in holography

Souvik Banerjee*, Johanna Erdmenger†, and Jonathan Karl‡

  • *Contact author: souvik.banerjee@uni-wuerzburg.de
  • †Contact author: johanna.erdmenger@uni-wuerzburg.de
  • ‡Contact author: jonathan.karl@uni-wuerzburg.de
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Phys. Rev. D 112, L021902 – Published 15 July, 2025

DOI: https://doi.org/10.1103/tm45-3lz8

Abstract

In AdS/CFT holography, two manifestations of the black hole information paradox are given by the nonisometric nature of the bulk-boundary map and by the factorisation puzzle. By considering time-shifted microstates of the eternal black hole, we demonstrate that both puzzles may be simultaneously resolved by taking into account nonlocal quantum corrections that correspond to wormholes arising from state averaging. This is achieved by showing, using a resolvent technique, that the resulting Hilbert space for an eternal black hole is finite-dimensional with a discrete energy spectrum. The latter gives rise to a transition to a type I von Neumann algebra.

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References (63)

  1. Gerard ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308, 284 (1993), arXiv:gr-qc/9310026.
  2. Leonard Susskind, The world as a hologram, J. Math. Phys. (N.Y.) 36, 6377 (1995).
  3. Idse Heemskerk, Joao Penedones, Joseph Polchinski, and James Sully, Holography from conformal field theory, J. High Energy Phys. 10 (2009) 079.
  4. Sheer El-Showk and Kyriakos Papadodimas, Emergent spacetime and holographic CFTs, J. High Energy Phys. 10 (2012) 106.

  5. Holographic CFTs are characterized by large central charges and sparse spectrum of operators with small conformal dimensions. Furthermore, correlators of these low-lying operators in any holographic CFT factorize into products of two-point functions, in other words connected higher point correlators of these operators vanish [4].

  6. Juan Martin Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  7. Edward Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  8. S. S. Gubser, Igor R. Klebanov, and Alexander M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428, 105 (1998).
  9. Chris Akers, Netta Engelhardt, Daniel Harlow, Geoff Penington, and Shreya Vardhan, The black hole interior from non-isometric codes and complexity, J. High Energy Phys. 06 (2024) 155.
  10. Thomas Faulkner and Min Li, Asymptotically isometric codes for holography, arXiv:2211.12439.
  11. Oliver DeWolfe and Kenneth Higginbotham, Non-isometric codes for the black hole interior from fundamental and effective dynamics, J. High Energy Phys. 09 (2023) 068.
  12. Stefano Antonini, Vijay Balasubramanian, Ning Bao, ChunJun Cao, and Wissam Chemissany, Non-isometry, state-dependence and holography, arXiv:2411.07296.
  13. Daniel Harlow and Daniel Jafferis, The factorization problem in Jackiw-Teitelboim gravity, J. High Energy Phys. 02 (2020) 177.
  14. Daniel Louis Jafferis and David K. Kolchmeyer, Entanglement entropy in Jackiw-Teitelboim gravity, arXiv:1911.10663.

  15. We follow the naming convention suggested by Henry Maxfield in order to differentiate this puzzle from the “factorization puzzle” (with a z), which concerns the structure of multi-boundary partition functions. Therefore, Hilbert space factorisation is written with an s.

  16. Juan Maldacena and Leonard Susskind, Cool horizons for entangled black holes, Fortschr. Phys. 61, 781 (2013).
  17. Mark Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Relativ. Gravit. 42, 2323 (2010).
  18. Juan Martin Maldacena, Eternal black holes in anti-de Sitter, J. High Energy Phys. 04 (2003) 021.
  19. Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras (Springer, New York, 1996).
  20. Edward Witten, Gravity and the crossed product, J. High Energy Phys. 10 (2022) 008.
  21. Kyriakos Papadodimas and Suvrat Raju, Local operators in the eternal black hole, Phys. Rev. Lett. 115, 211601 (2015).
  22. Kyriakos Papadodimas and Suvrat Raju, Remarks on the necessity and implications of state-dependence in the black hole interior, Phys. Rev. D 93, 084049 (2016).
  23. Herman Verlinde, ER=EPR revisited: On the entropy of an Einstein-Rosen bridge, arXiv:2003.13117.
  24. Flavio S. Nogueira, Souvik Banerjee, Moritz Dorband, René Meyer, Jeroen van den Brink, and Johanna Erdmenger, Geometric phases distinguish entangled states in wormhole quantum mechanics, Phys. Rev. D 105, L081903 (2022).
  25. Souvik Banerjee, Moritz Dorband, Johanna Erdmenger, René Meyer, and Anna-Lena Weigel, Berry phases, wormholes and factorization in AdS/CFT, J. High Energy Phys. 08 (2022) 162.
  26. Souvik Banerjee, Pablo Basteiro, Rathindra Nath Das, and Moritz Dorband, Geometric quantum discord signals non-factorization, J. High Energy Phys. 08 (2023) 104.
  27. Souvik Banerjee, Moritz Dorband, Johanna Erdmenger, and Anna-Lena Weigel, Geometric phases characterise operator algebras and missing information, J. High Energy Phys. 10 (2023) 026.

  28. We refer to the semiclassical description, together with perturbative corrections as the effective description, while the full quantum gravity theory is the fundamental description, including non-perturbative corrections. The fundamental description is also sometimes termed as the boundary description.

  29. This method was also used to give a microscopic interpretation of Bekenstein-Hawking entropy by considering a different family of microstates [30, 36, 37, 38, 39], and it has been discussed in the context of time-shifted TFD states in [31].

  30. V. Balasubramanian, B. Craps, J. Hernandez, M. Khramtsov, and M. Knysh, Counting microstates of out-of-equilibrium black hole fluctuations, arXiv:2412.06884.
  31. J. M. Magan and Q. Wu, Two types of quantum chaos: Testing the limits of the Bohigas-Giannoni-Schmit conjecture, arXiv:2411.08186.
  32. Predrag Cvitanović, Planar perturbation expansion, Phys. Lett. 99B, 49 (1981).
  33. Roland Speicher, Free probability theory, arXiv:0911.0087.
  34. Geoff Penington, Stephen H. Shenker, Douglas Stanford, and Zhenbin Yang, Replica wormholes and the black hole interior, J. High Energy Phys. 03 (2022) 205.
  35. Adam R. Brown, Hrant Gharibyan, Geoff Penington, and Leonard Susskind, The python’s Lunch: Geometric obstructions to decoding Hawking radiation, J. High Energy Phys. 08 (2020) 121.
  36. Hao Geng and Yikun Jiang, Microscopic origin of the entropy of single-sided black holes, J. High Energy Phys. 04 (2025) 133.
  37. Vijay Balasubramanian, Albion Lawrence, Javier M. Magan, and Martin Sasieta, Microscopic origin of the entropy of black holes in general relativity, Phys. Rev. X 14, 011024 (2024).
  38. Vijay Balasubramanian, Albion Lawrence, Javier M. Magan, and Martin Sasieta, Microscopic origin of the entropy of astrophysical black holes, Phys. Rev. Lett. 132, 141501 (2024).
  39. Ana Climent, Roberto Emparan, Javier M. Magan, Martin Sasieta, and Alejandro Vilar López, Universal construction of black hole microstates, Phys. Rev. D 109, 086024 (2024).
  40. Jan Boruch, Luca V. Iliesiu, Guanda Lin, and Cynthia Yan, How the Hilbert space of two-sided black holes factorises, J. High Energy Phys. 06 (2025) 092.
  41. Vijay Balasubramanian, Ben Craps, Juan Hernandez, Mikhail Khramtsov, and Maria Knysh, Factorization of the Hilbert space of eternal black holes in general relativity, J. High Energy Phys. 01 (2025) 046.
  42. Pan Li, Notes on the factorisation of the Hilbert space for two-sided black holes in higher dimensions, J. High Energy Phys. 02 (2025) 060.
  43. H. Roos, Independence of local algebras in quantum field theory, Commun. Math. Phys. 16, 238 (1970).
  44. Detlev Buchholz, Product states for local algebras, Commun. Math. Phys. 36, 287 (1974).
  45. Christopher J. Fewster, The split property for quantum field theories in flat and curved spacetimes, in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (Springer, New York, 2016), Vol. 86, pp. 153–175.
  46. G. W. Gibbons and S. W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15, 2752 (1977).
  47. Suvrat Raju, Failure of the split property in gravity and the information paradox, Classical Quantum Gravity 39, 064002 (2022).

  48. In general the microstate of a thermodynamical system is defined as a pure state |ψ⟩, which is indistinguishable from the equilibrium state represented by a thermal density matrix ρ, when probed by simple operators [36, 39], in the sense that ⟨ψ|O1…On|ψ⟩=Tr(ρO1…On).

  49. In the present example of an eternal black hole, the phases arising from the global time evolution can be measured if two observers in a gedanken experimental set-up perform a non-local measurement by jumping into the black hole from different sides. They would detect a time-shift while comparing their respective times at their meeting point in the bulk, provided their watches were initially synchronized at asymptotic infinity [23].

  50. See Supplemental Material at http://link.aps.org/supplemental/10.1103/tm45-3lz8 and Refs. [51–53] therein for a path integral definition of the black hole microstates, as well as their overlaps (10), derivation of (18) using the resolvent method, and summary on the type classification of von Neumann algebras.
  51. J. B. Hartle and S. W. Hawking, Path-integral derivation of black-hole radiance, Phys. Rev. D 13, 2188 (1976).
  52. Daniel Harlow, Jerusalem lectures on black holes and quantum information, Rev. Mod. Phys. 88, 015002 (2016).
  53. B. Eynard, T. Kimura, and S. Ribault, Random matrices, arXiv:1510.04430
  54. Herman Verlinde, Wormholes in quantum mechanics, arXiv:2105.02129.
  55. Venkatesa Chandrasekaran, Geoff Penington, and Edward Witten, Large N algebras and generalized entropy, J. High Energy Phys. 04 (2023) 009.

  56. The rank of the Gram matrix may in principle directly be determined by either performing a Gram-Schmidt orthogonalization procedure, or equivalently by diagonalizing G. Since we have to find dΩ in the limit Ω→∞, this requires diagonalizing a matrix of infinite size, which is very hard. The resolvent method provides a simpler way to compute the eigenvalue density of the Gram matrix.

  57. Raphael Bousso, The holographic principle, Rev. Mod. Phys. 74, 825 (2002).
  58. Keiichiro Furuya, Nima Lashkari, Mudassir Moosa, and Shoy Ouseph, Information loss, mixing and emergent type III1 factors, J. High Energy Phys. 08 (2023) 111.
  59. Edward Witten, Why does quantum field theory in curved spacetime make sense? And what happens to the algebraof observables in the thermodynamic limit?, arXiv:2112.11614.
  60. Samuel Leutheusser and Hong Liu, Causal connectability between quantum systems and the black hole interior in holographic duality, Phys. Rev. D 108, 086019 (2023).
  61. Samuel Leutheusser and Hong Liu, Emergent times in holographic duality, Phys. Rev. D 108, 086020 (2023).

  62. A similar structure of the boundary Hilbert space was studied [63], where general states with a gravity dual were considered.

  63. S. Leutheusser and H. Liu, Subalgebra-subregion duality: Emergence of space and time in holography, arXiv:2212.13266

Outline

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Phys. Rev. D 112, L021902– Published 15 July, 2025
  • Received 18 December 2024
  • Accepted 17 June 2025
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