Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Timelike entanglement entropy revisited

Xin Jiang* and Haitang Yang†

  • *Contact author: domoki@stu.scu.edu.cn
  • †Contact author: hyanga@scu.edu.cn

Phys. Rev. D 113, 106021 – Published 19 May, 2026

DOI: https://doi.org/10.1103/rl9b-373v

Abstract

We present an operator-algebraic definition for timelike entanglement entropy in quantum field theory under a few mild postulates. This rigorously defined timelike entanglement entropy is real-valued due to the timelike tube theorem. We further demonstrate why the timelike entanglement entropy should be real-valued from both path integral argument and holography perspective.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (61)

  1. Peng Wang, Houwen Wu, and Haitang Yang, Fix the dual geometries of TT¯ deformed CFT2 and highly excited states of CFT2, Eur. Phys. J. C 80, 1117 (2020).
  2. Bowei Liu, Hao Chen, and Biao Lian, Entanglement entropy of free fermions in timelike slices, Phys. Rev. B 110, 144306 (2024).
  3. Kazuki Doi, Jonathan Harper, Ali Mollabashi, Tadashi Takayanagi, and Yusuke Taki, Pseudoentropy in dS/CFT and timelike entanglement entropy, Phys. Rev. Lett. 130, 031601 (2023).
  4. Kazuki Doi, Jonathan Harper, Ali Mollabashi, Tadashi Takayanagi, and Yusuke Taki, Timelike entanglement entropy, J. High Energy Phys. 05 (2023) 052.
  5. Alexey Milekhin, Zofia Adamska, and John Preskill, Observable and computable entanglement in time, arXiv:2502.12240.
  6. Edward Witten, Algebras, regions, and observers, Proc. Symp. Pure Math. 107, 247 (2024).
  7. Hans-Jürgen Borchers, Field operators as C∞ functions in spacelike directions, Nuovo Cimento (1955–1965) 33, 1600 (1964).
  8. Rudolf Haag and Daniel Kastler, An algebraic approach to quantum field theory, J. Math. Phys. (N.Y.) 5, 848 (1964).
  9. Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras, Theoretical and Mathematical Physics (Springer, Berlin, 1996), 10.1007/978-3-642-61458-3.
  10. Horacio Casini, Marina Huerta, Javier M. Magan, and Diego Pontello, Entropic order parameters for the phases of QFT, J. High Energy Phys. 04 (2021) 277.
  11. Horacio Casini and Javier M. Magan, On completeness and generalized symmetries in quantum field theory, Mod. Phys. Lett. A 36, 2130025 (2021).
  12. Sergio Doplicher and Roberto Longo, Standard and split inclusions of von Neumann algebras, Inventiones Mathematicae 75, 493 (1984).
  13. Roberto Longo and Feng Xu, Von Neumann entropy in QFT, Commun. Math. Phys. 381, 1031 (2021).
  14. Heide Narnhofer, Entanglement, split and nuclearity in quantum field theory, Rep. Math. Phys. 50, 111 (2002).
  15. Horacio Casini and Marina Huerta, Reduced density matrix and internal dynamics for multicomponent regions, Classical Quantum Gravity 26, 185005 (2009).
  16. Yul Otani and Yoh Tanimoto, Towards entanglement entropy with uv cutoff in conformal nets, Ann. Henri Poincare 19, 1817 (2018).
  17. Stefan Hollands and Ko Sanders, Entanglement Measures and Their Properties in Quantum Field Theory, volume 34 of SpringerBriefs in Mathematical Physics (Springer, Cham, 2018), 10.1007/978-3-319-94902-4.
  18. Edward Witten, APS medal for exceptional achievement in research: Invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90, 045003 (2018).
  19. Souvik Dutta and Thomas Faulkner, A canonical purification for the entanglement wedge cross-section, J. High Energy Phys. 03 (2021) 178.
  20. Jonah Kudler-Flam, Samuel Leutheusser, Adel A. Rahman, Gautam Satishchandran, and Antony J. Speranza, Covariant regulator for entanglement entropy: Proofs of the Bekenstein bound and the quantum null energy condition, Phys. Rev. D 111, 105001 (2025).
  21. Detlev Buchholz and Eyvind H. Wichmann, Causal independence and the energy level density of states in local quantum field theory, Commun. Math. Phys. 106, 321 (1986).
  22. Detlev Buchholz, Claudio D’Antoni, and Klaus Fredenhagen, The universal structure of local algebras, Commun. Math. Phys. 111, 123 (1987).
  23. Detlev Buchholz, Claudio D’Antoni, and Roberto Longo, Nuclear maps and modular structures. I. General properties, J. Funct. Anal. 88, 233 (1990).
  24. Detlev Buchholz, Claudio D’Antoni, and Roberto Longo, Nuclear maps and modular structures. II. Applications to quantum field theory, Commun. Math. Phys. 129, 115 (1990).
  25. Detlev Buchholz, Claudio D’Antoni, and Roberto Longo, Nuclearity and thermal states in conformal field theory, Commun. Math. Phys. 270, 267 (2007).
  26. Vincenzo Morinelli, Yoh Tanimoto, and Mihály Weiner, Conformal covariance and the split property, Commun. Math. Phys. 357, 379 (2018).
  27. Fikret Ceyhan and Thomas Faulkner, Bounds on CFT correlations from the thermal partition function, arXiv:2510.24042.
  28. Rainer Verch, Nuclearity, split property and duality for the Klein-Gordon field in curved space-time, Lett. Math. Phys. 29, 297 (1993).
  29. Claudio D’Antoni and Stefan Hollands, Nuclearity, local quasiequivalence and split property for dirac quantum fields in curved spacetime, Commun. Math. Phys. 261, 133 (2006).
  30. Christopher J. Fewster, The split property for locally covariant quantum field theories in curved spacetime, Lett. Math. Phys. 105, 1633 (2015).
  31. Xuchen Cao, Thomas Faulkner, and Zhencheng Wang, Gravitational algebras with two areas, arXiv:2512.04435.
  32. Hans-Jürgen Borchers, Über die vollständigkeit lorentzinvarianter felder in einer zeitartigen röhre, Nuovo Cimento (1955–1965) 19, 787 (1961).
  33. Huzihiro Araki, A generalization of borchers theorem, Helvetica Physica Acta (Switzerland), 36, 1963, https://api.semanticscholar.org/CorpusID:117239128.
  34. R. Haag and B. Schroer, Postulates of quantum field theory, J. Math. Phys. (N.Y.) 3, 248 (1962).
  35. Curtis G. Callan, Jr. and Frank Wilczek, On geometric entropy, Phys. Lett. B 333, 55 (1994).
  36. Shinsei Ryu and Tadashi Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006).
  37. Shinsei Ryu and Tadashi Takayanagi, Aspects of holographic entanglement entropy, J. High Energy Phys. 08 (2006) 045.
  38. Pasquale Calabrese and John L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.
  39. Pasquale Calabrese and John L. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42, 504005 (2009).
  40. John L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. 270B, 186 (1986).
  41. John L. Cardy, Boundary conformal field theory, arXiv:hep-th/0411189.
  42. John Cardy and Erik Tonni, Entanglement Hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. (2016) 123103.
  43. J. David Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity, Commun. Math. Phys. 104, 207 (1986).
  44. Xin Jiang, Peng Wang, Houwen Wu, and Haitang Yang, Alternative to purification in conformal field theory, Phys. Rev. D 111, L021902 (2025).
  45. Xin Jiang, Peng Wang, Houwen Wu, and Haitang Yang, How Einstein’s equations emerge from CFT2, Phys. Rev. D 112, 8 (2025).
  46. Xin Jiang, Peng Wang, Houwen Wu, and Haitang Yang, Mixed state entanglement entropy in CFT, J. High Energy Phys. 09 (2025) 133.
  47. Xin Jiang and Haitang Yang, Entanglement entropy of conformal field theory in all dimensions, J. High Energy Phys. 01 (2026) 015.
  48. Pasquale Calabrese, John Cardy, and Erik Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. (2009) P11001.
  49. Pasquale Calabrese, John Cardy, and Erik Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. (2011) P01021.
  50. Joseph J. Bisognano and Eyvind H. Wichmann, On the duality condition for a hermitian scalar field, J. Math. Phys. (N.Y.) 16, 985 (1975).
  51. D. Buchholz, On the Structure of Local Quantum Fields with Nontrivial Interaction (Teubner, Leipzig, 1977), pp. 146–153.
  52. S. Jay Olson and Timothy C. Ralph, Entanglement between the future and past in the quantum vacuum, Phys. Rev. Lett. 106, 110404 (2011).
  53. Samuel Leutheusser and Hong Liu, Causal connectability between quantum systems and the black hole interior in holographic duality, Phys. Rev. D 108, 086019 (2023).
  54. Samuel Leutheusser and Hong Liu, Emergent times in holographic duality, Phys. Rev. D 108, 086020 (2023).
  55. Edward Witten, Gravity and the crossed product, J. High Energy Phys. 10 (2022) 008.
  56. Geoff Penington and Edward Witten, Algebras and states in JT gravity, arXiv:2301.07257.
  57. Venkatesa Chandrasekaran, Roberto Longo, Geoff Penington, and Edward Witten, An algebra of observables for de Sitter space, J. High Energy Phys. 02 (2023) 082.
  58. Edward Witten, A background-independent algebra in quantum gravity, J. High Energy Phys. 03 (2024) 077.
  59. Alexander Strohmaier, On the local structure of the Klein-Gordon field on curved spacetimes, Lett. Math. Phys. 54, 249 (2000).
  60. Alexander Strohmaier and Edward Witten, Analytic states in quantum field theory on curved spacetimes, Ann. Henri Poincare 25, 4543 (2024).
  61. Alexander Strohmaier and Edward Witten, The timelike tube theorem in curved spacetime, Commun. Math. Phys. 405, 153 (2024).

Outline

Information

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation