- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

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

Temporal Entanglement from Holographic Entanglement Entropy

Michal P. Heller1,2,*, Fabio Ori1,†, and Alexandre Serantes1,‡

  • 1Department of Physics and Astronomy, Ghent University, 9000 Ghent, Belgium
  • 2Institute of Theoretical Physics and Mark Kac Center for Complex Systems Research, Jagiellonian University, 30-348 Cracow, Poland

  • *Contact author: michal.p.heller@ugent.be
  • †Contact author: fabio.ori@ugent.be
  • ‡Contact author: alexandre.serantesrubianes@ugent.be
PDF

Phys. Rev. X 15, 041022 – Published 6 November, 2025

DOI: https://doi.org/10.1103/qlsv-gp22

Abstract

Recently, several notions of entanglement in time have emerged as a novel frontier in quantum many-body physics, quantum field theory, and gravity. We propose a systematic prescription to characterize temporal entanglement in relativistic quantum field theory in a general state for an arbitrary subregion on a flat, constant time slice in a flat spacetime. Our prescription starts with the standard entanglement entropy of a spatial subregion and amounts to transporting the unchanged subregion to boosted time slices all the way across the light cone when it becomes, in general, a complex characterization of the corresponding temporal subregion. For holographic quantum field theories, our prescription amounts to an analytic continuation of all codimension-two bulk extremal surfaces satisfying the homology constraint and picking the one with the smallest real value of the area as the leading saddle point. We implement this prescription for holographic conformal field theories in thermal states on both a two-dimensional Lorentzian cylinder and three-dimensional Minkowski space, and we show that it leads to results with self-consistent physical properties of temporal entanglement.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Entanglement is one of the most fascinating features of quantum physics—it describes how parts of a system can remain deeply connected even when far apart. Scientists have long understood how to describe this connection in space, by dividing a system into spatial regions and measuring how much information they share. But what about connections that unfold in time, rather than space? In this study, we develop a consistent method for describing and calculating temporal entanglement, especially in the context of holographic theories that link gravity and quantum physics through the AdS/CFT correspondence.

To obtain a consistent definition of entanglement entropy in time, we start from the usual definition of spatial entanglement and gradually transform the region of interest so that it stretches across time. More specifically, in the holographic picture—where the entropy of quantum systems is represented by surfaces in a higher-dimensional space—we track the surfaces that define entanglement as they evolve from the spacelike case into the timelike one. When several possible surfaces appear, our method univocally identifies the one with the smallest real area as the correct physical choice.

Our approach eliminates the puzzling ambiguity about how to describe temporal entanglement when multiple surfaces appear. It also ensures that the new definition remains consistent with what we already know from the better-understood case of entanglement in space. This new perspective opens the door to studying how information is shared not just across space, but across time—offering new insights into how the notion of entanglement can be extended to the time direction.

Article Text

References (51)

  1. R. Orus, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. (Amsterdam) 349, 117 (2014).
  2. J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  3. B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (Springer, New York, 2019).
  4. T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, Thermalization and prethermalization in isolated quantum systems: A theoretical overview, J. Phys. B 51, 112001 (2018).
  5. J. Berges, M. P. Heller, A. Mazeliauskas, and R. Venugopalan, QCD thermalization: Ab initio approaches and interdisciplinary connections, Rev. Mod. Phys. 93, 035003 (2021).
  6. H. Casini and M. Huerta, Lectures on entanglement in quantum field theory, Proc. Sci., TASI2021 (2023) 002.
  7. T. Nishioka, S. Ryu, and T. Takayanagi, Holographic entanglement entropy: An overview, J. Phys. A 42, 504008 (2009).
  8. M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, (Springer, New York, 2017), Vol. 931.
  9. B. Chen, B. Czech, and Z.-z. Wang, Quantum information in holographic duality, Rep. Prog. Phys. 85, 046001 (2022).
  10. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93, 035002 (2021).
  11. M. C. Bañuls, M. B. Hastings, F. Verstraete, and J. I. Cirac, Matrix product states for dynamical simulation of infinite chains, Phys. Rev. Lett. 102, 240603 (2009).
  12. M. B. Hastings and R. Mahajan, Connecting entanglement in time and space: Improving the folding algorithm, Phys. Rev. A 91, 032306 (2015).
  13. M. Frías-Pérez and M. C. Bañuls, Light cone tensor network and time evolution, Phys. Rev. B 106, 115117 (2022).
  14. A. Lerose, M. Sonner, and D. A. Abanin, Overcoming the entanglement barrier in quantum many-body dynamics via space-time duality, Phys. Rev. B 107, L060305 (2023).
  15. G. Giudice, G. Giudici, M. Sonner, J. Thoenniss, A. Lerose, D. A. Abanin, and L. Piroli, Temporal entanglement, quasiparticles, and the role of interactions, Phys. Rev. Lett. 128, 220401 (2022).
  16. S. Carignano, C. R. Marimón, and L. Tagliacozzo, Temporal entropy and the complexity of computing the expectation value of local operators after a quench, Phys. Rev. Res. 6, 033021 (2024).
  17. K. Doi, J. Harper, A. Mollabashi, T. Takayanagi, and Y. Taki, Pseudoentropy in dS/CFT and timelike entanglement entropy, Phys. Rev. Lett. 130, 031601 (2023).
  18. K. Doi, J. Harper, A. Mollabashi, T. Takayanagi, and Y. Taki, Timelike entanglement entropy, J. High Energy Phys. 05 (2023) 052.
  19. K. Narayan, De Sitter space, extremal surfaces, and time entanglement, Phys. Rev. D 107, 126004 (2023).
  20. C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B424, 443 (1994).
  21. P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002
  22. S. Carignano and L. Tagliacozzo, Loschmidt echo, emerging dual unitarity and scaling of generalized temporal entropies after quenches to the critical point, Quantum 9, 1859 (2025).
  23. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006).
  24. V. E. Hubeny, M. Rangamani, and T. Takayanagi, A covariant holographic entanglement entropy proposal, J. High Energy Phys. 07 (2007) 062.
  25. H. Casini, M. Huerta, and R. C. Myers, Towards a derivation of holographic entanglement entropy, J. High Energy Phys. 05 (2011) 036.
  26. A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, J. High Energy Phys. 08 (2013) 090.
  27. X. Dong, A. Lewkowycz, and M. Rangamani, Deriving covariant holographic entanglement, J. High Energy Phys. 11 (2016) 028.
  28. M. P. Heller, F. Ori, and A. Serantes, Geometric interpretation of timelike entanglement entropy, Phys. Rev. Lett. 134, 131601 (2025).
  29. L. Fidkowski, V. Hubeny, M. Kleban, and S. Shenker, The black hole singularity in AdS/CFT, J. High Energy Phys. 02 (2004) 014.
  30. V. Balasubramanian, A. Bernamonti, B. Craps, V. Keränen, E. Keski-Vakkuri, B. Müller, L. Thorlacius, and J. Vanhoof, Thermalization of the spectral function in strongly coupled two dimensional conformal field theories, J. High Energy Phys. 04 (2013) 069.
  31. N. Čeplak, H. Liu, A. Parnachev, and S. Valach, Black hole singularity from OPE, J. High Energy Phys. 10 (2024) 105.
  32. M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82, 126010 (2010).
  33. F. M. Haehl, T. Hartman, D. Marolf, H. Maxfield, and M. Rangamani, Topological aspects of generalized gravitational entropy, J. High Energy Phys. 05 (2015) 023.
  34. L. Susskind and E. Witten, The holographic bound in anti-de Sitter space, arXiv:hep-th/9805114.
  35. Y. Kusuki, T. Takayanagi, and K. Umemoto, Holographic entanglement entropy on generic time slices, J. High Energy Phys. 06 (2017) 021.
  36. B. Liu, H. Chen, and B. Lian, Entanglement entropy of free fermions in timelike slices, Phys. Rev. B 110, 144306 (2024).
  37. J. Erdmenger and N. Miekley, Non-local observables at finite temperature in AdS/CFT, J. High Energy Phys. 03 (2018) 034.
  38. T. Hartman and J. Maldacena, Time evolution of entanglement entropy from black hole interiors, J. High Energy Phys. 05 (2013) 014.
  39. M. Banados, C. Teitelboim, and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69, 1849 (1992).
  40. V. Balasubramanian, B. D. Chowdhury, B. Czech, and J. de Boer, Entwinement and the emergence of spacetime, J. High Energy Phys. 01 (2015) 048.
  41. V. E. Hubeny, M. Rangamani, and E. Tonni, Global properties of causal wedges in asymptotically AdS spacetimes, J. High Energy Phys. 10 (2013) 059.
  42. V. E. Hubeny, H. Liu, and M. Rangamani, Bulk-cone singularities & signatures of horizon formation in AdS/CFT, J. High Energy Phys. 01 (2007) 009.
  43. M. Gary, S. B. Giddings, and J. Penedones, Local bulk S-matrix elements and CFT singularities, Phys. Rev. D 80, 085005 (2009).
  44. J. Maldacena, D. Simmons-Duffin, and A. Zhiboedov, Looking for a bulk point, J. High Energy Phys. 01 (2017) 013.
  45. M. Dodelson, C. Iossa, R. Karlsson, A. Lupsasca, and A. Zhiboedov, Black hole bulk-cone singularities, J. High Energy Phys. 07 (2024) 046.
  46. B. Czech, L. Lamprou, S. McCandlish, and J. Sully, Integral geometry and holography, J. High Energy Phys. 10 (2015) 175.
  47. J. de Boer, M. P. Heller, R. C. Myers, and Y. Neiman, Holographic de Sitter geometry from entanglement in conformal field theory, Phys. Rev. Lett. 116, 061602 (2016).
  48. B. Czech, L. Lamprou, S. McCandlish, B. Mosk, and J. Sully, A stereoscopic look into the bulk, J. High Energy Phys. 07 (2016) 129.
  49. J. de Boer, F. M. Haehl, M. P. Heller, and R. C. Myers, Entanglement, holography and causal diamonds, J. High Energy Phys. 08 (2016) 162.
  50. A. Milekhin, Z. Adamska, and J. Preskill, Observable and computable entanglement in time, arXiv:2502.12240.
  51. C. Nunez and D. Roychowdhury, Interpolating between space-like and time-like entanglement via holography, arXiv:2507.17805.

Outline

Information

Phys. Rev. X 15, 041022– Published 6 November, 2025
  • Received 22 August 2025
  • Accepted 30 September 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation