- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

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

Reflected entropy in an evaporating black hole through a nonisometric map

Bin Chen1,2,3,* and Zhi-jun Yin2,†

  • 1Institute of Fundamental Physics and Quantum Technology, and School of Physical Science and Technology, Ningbo University, Ningbo, Zhejiang 315211, China
  • 2School of Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871, People’s Republic of China
  • 3Center for High Energy Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871, People’s Republic of China

  • *Contact author: chenbin1@nbu.edu.cn
  • †Contact author: yinzhijun@stu.pku.edu.cn
PDF

Phys. Rev. D 112, 046009 – Published 12 August, 2025

DOI: https://doi.org/10.1103/qr71-3r93

Abstract

The black hole information paradox has been an important problem in quantum gravity. In the study of evaporating black hole, it has been proposed that the holographic map between the semiclassical effective description in bulk and the fundamental description in boundary cannot be isometric. In this work, we would like to study the reflected entropy in an evaporating black hole model through a nonisometric holographic map. We assume that the evaporating is slow enough that it makes sense to ascribe a slowly varying temperature to the Hawking radiation. We then introduce a two-sided black hole model to canonically purify the semiclassical state. The holographic map to the fundamental description is nonisometric and defined by a Haar random unitary matrix. We show that the entropy of radiation in the model computed in the fundamental description matches the result read from the quantum extremal surface formula and agrees with the Page curve. Furthermore, through a nonisometric holographic map, we study the reflected entropies between different regions, including the one between the black holes on different sides, the one between the radiations distributed symmetrically but disconnectedly, and the one between the black hole and the radiation on a single side. Our results are consistent with the existing ones based on the effective descriptions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (71)

  1. J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  2. D. N. Page, Information in black hole radiation, Phys. Rev. Lett. 71, 3743 (1993).
  3. D. N. Page, Time dependence of Hawking radiation entropy, J. Cosmol. Astropart. Phys. 09 (2013) 028.
  4. N. Engelhardt and A. C. Wall, Quantum extremal surfaces: Holographic entanglement entropy beyond the classical regime, J. High Energy Phys. 01 (2015) 073.
  5. G. Penington, Entanglement wedge reconstruction and the information paradox, J. High Energy Phys. 09 (2020) 002.
  6. A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, J. High Energy Phys. 12 (2019) 063.
  7. A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, The page curve of Hawking radiation from semiclassical geometry, J. High Energy Phys. 03 (2020) 149.
  8. A. Almheiri, R. Mahajan, and J. Maldacena, Islands outside the horizon, arXiv:1910.11077.
  9. G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, Replica wormholes and the black hole interior, J. High Energy Phys. 03 (2022) 205.
  10. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Replica wormholes and the entropy of Hawking radiation, J. High Energy Phys. 05 (2020) 013.
  11. H. Geng, Replica wormholes and entanglement islands in the Karch-Randall braneworld, J. High Energy Phys. 01 (2025) 063.
  12. T. Banks, M. R. Douglas, G. T. Horowitz, and E. J. Martinec, AdS dynamics from conformal field theory, arXiv:hep-th/9808016.
  13. A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74, 066009 (2006).
  14. I. Heemskerk, D. Marolf, J. Polchinski, and J. Sully, Bulk and transhorizon measurements in AdS/CFT, J. High Energy Phys. 10 (2012) 165.
  15. R. Bousso, B. Freivogel, S. Leichenauer, V. Rosenhaus, and C. Zukowski, Null geodesics, local CFT operators and AdS/CFT for subregions, Phys. Rev. D 88, 064057 (2013).
  16. B. Czech, J. L. Karczmarek, F. Nogueira, and M. Van Raamsdonk, The gravity dual of a density matrix, Classical Quantum Gravity 29, 155009 (2012).
  17. A. C. Wall, Maximin surfaces, and the strong subadditivity of the covariant holographic entanglement entropy, Classical Quantum Gravity 31, 225007 (2014).
  18. M. Headrick, V. E. Hubeny, A. Lawrence, and M. Rangamani, Causality & holographic entanglement entropy, J. High Energy Phys. 12 (2014) 162.
  19. A. Almheiri, X. Dong, and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, J. High Energy Phys. 04 (2015) 163.
  20. F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, J. High Energy Phys. 06 (2015) 149.
  21. X. Dong, D. Harlow, and A. C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Phys. Rev. Lett. 117, 021601 (2016).
  22. D. Harlow, The Ryu–Takayanagi Formula from quantum error correction, Commun. Math. Phys. 354, 865 (2017).
  23. C. Akers, N. Engelhardt, D. Harlow, G. Penington, and S. Vardhan, The black hole interior from non-isometric codes and complexity, J. High Energy Phys. 06 (2024) 155.
  24. Z. Gyongyosi, T. J. Hollowood, S. P. Kumar, A. Legramandi, and N. Talwar, The holographic map of an evaporating black hole, J. High Energy Phys. 07 (2023) 043.
  25. J. M. Maldacena, Eternal black holes in anti-de Sitter, J. High Energy Phys. 04 (2003) 021.
  26. S. D. Mathur, What is the dual of two entangled CFTs?, arXiv:1402.6378.
  27. S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, J. High Energy Phys. 03 (2021) 178.
  28. N. Bao and N. Cheng, Multipartite reflected entropy, J. High Energy Phys. 10 (2019) 102.
  29. H.-S. Jeong, K.-Y. Kim, and M. Nishida, Reflected entropy and entanglement wedge cross section with the first order correction, J. High Energy Phys. 12 (2019) 170.
  30. V. Chandrasekaran, M. Miyaji, and P. Rath, Including contributions from entanglement islands to the reflected entropy, Phys. Rev. D 102, 086009 (2020).
  31. T. Li, J. Chu, and Y. Zhou, Reflected entropy for an evaporating black hole, J. High Energy Phys. 11 (2020) 155.
  32. D. Basu, A. Chandra, V. Raj, and G. Sengupta, Entanglement wedge in flat holography and entanglement negativity, SciPost Phys. Core 5, 013 (2022).
  33. T. Li, M.-K. Yuan, and Y. Zhou, Defect extremal surface for reflected entropy, J. High Energy Phys. 01 (2022) 018.
  34. Y. Ling, P. Liu, Y. Liu, C. Niu, Z.-Y. Xian, and C.-Y. Zhang, Reflected entropy in double holography, J. High Energy Phys. 02 (2022) 037.
  35. C. Akers, T. Faulkner, S. Lin, and P. Rath, The page curve for reflected entropy, J. High Energy Phys. 06 (2022) 089.
  36. D. Basu, H. Parihar, V. Raj, and G. Sengupta, Entanglement negativity, reflected entropy, and anomalous gravitation, Phys. Rev. D 105, 086013 (2022).
  37. J. K. Basak, H. Chourasiya, V. Raj, and G. Sengupta, Reflected entropy in Galilean conformal field theories and flat holography, Eur. Phys. J. C 82, 1169 (2022).
  38. B. Chen, Y. Liu, and B. Yu, Reflected entropy in AdS3/WCFT, J. High Energy Phys. 12 (2022) 008.
  39. M. J. Vasli, M. R. Mohammadi Mozaffar, K. Babaei Velni, and M. Sahraei, Holographic study of reflected entropy in anisotropic theories, Phys. Rev. D 107, 026012 (2023).
  40. Y. Lu and J. Lin, The Markov gap in the presence of islands, J. High Energy Phys. 03 (2023) 043.
  41. M. Afrasiar, J. K. Basak, A. Chandra, and G. Sengupta, Reflected entropy for communicating black holes. Part I. Karch-Randall braneworlds, J. High Energy Phys. 02 (2023) 203.
  42. M. Afrasiar, J. K. Basak, A. Chandra, and G. Sengupta, Reflected entropy for communicating black holes II: Planck braneworlds, Eur. Phys. J. C 84, 250 (2024).
  43. D. Basu, H. Chourasiya, V. Raj, and G. Sengupta, Reflected entropy in a BCFT on a black hole background, J. High Energy Phys. 05 (2024) 054.
  44. J. K. Basak, D. Basu, V. Malvimat, H. Parihar, and G. Sengupta, Holographic reflected entropy and islands in interface CFTs, J. High Energy Phys. 05 (2024) 143.
  45. D. Basu and V. Raj, Reflected entropy and timelike entanglement in TT−-deformed CFT2s, Phys. Rev. D 110, 046009 (2024).
  46. B. Ahn, S.-E. Bak, K.-Y. Kim, and M. Nishida, Renyi reflected entropy and entanglement wedge cross section with cosmic branes in AdS/BCFT, Phys. Rev. D 111, 126006 (2025).
  47. M.-K. Yuan, M. Li, and Y. Zhou, Reflected multi-entropy and its holographic dual, arXiv:2410.08546.
  48. C. Akers, T. Faulkner, S. Lin, and P. Rath, Reflected entropy in random tensor networks, J. High Energy Phys. 05 (2022) 162.
  49. C. Akers, T. Faulkner, S. Lin, and P. Rath, Reflected entropy in random tensor networks. Part II. A topological index from canonical purification, J. High Energy Phys. 01 (2023) 067.
  50. B. Czech, S. Shuai, and H. Tang, Entropies and reflected entropies in the Hayden-Preskill protocol, J. High Energy Phys. 02 (2024) 040.
  51. C. Akers, T. Faulkner, S. Lin, and P. Rath, Reflected entropy in random tensor networks III: Triway cuts, J. High Energy Phys. 12 (2024) 209.
  52. P. Bueno and H. Casini, Reflected entropy, symmetries and free fermions, J. High Energy Phys. 05 (2020) 103.
  53. P. Bueno and H. Casini, Reflected entropy for free scalars, J. High Energy Phys. 11 (2020) 148.
  54. S. Dutta, T. Faulkner, and S. Lin, The reflected entanglement spectrum for free fermions, J. High Energy Phys. 02 (2023) 223.
  55. J. K. Basak, D. Giataganas, S. Mondal, and W.-Y. Wen, Reflected entropy and Markov gap in non-inertial frames, Phys. Rev. D 108, 125009 (2023).
  56. C. Berthiere, H. Chen, Y. Liu, and B. Chen, Topological reflected entropy in Chern-Simons theories, Phys. Rev. B 103, 035149 (2021).
  57. Y. Liu, R. Sohal, J. Kudler-Flam, and S. Ryu, Multipartitioning topological phases by vertex states and quantum entanglement, Phys. Rev. B 105, 115107 (2022).
  58. R. Sohal and S. Ryu, Entanglement in tripartitions of topological orders: A diagrammatic approach, Phys. Rev. B 108, 045104 (2023).
  59. Y. Kusuki and K. Tamaoka, Dynamics of entanglement wedge cross section from conformal field theories, Phys. Lett. B 814, 136105 (2021).
  60. J. Kudler-Flam, Y. Kusuki, and S. Ryu, Correlation measures and the entanglement wedge cross-section after quantum quenches in two-dimensional conformal field theories, J. High Energy Phys. 04 (2020) 074.
  61. M. Moosa, Time dependence of reflected entropy in rational and holographic conformal field theories, J. High Energy Phys. 05 (2020) 082.
  62. J. Kudler-Flam, Y. Kusuki, and S. Ryu, The quasi-particle picture and its breakdown after local quenches: mutual information, negativity, and reflected entropy, J. High Energy Phys. 03 (2021) 146.
  63. C. Berthiere and G. Parez, Reflected entropy and computable cross-norm negativity: Free theories and symmetry resolution, Phys. Rev. D 108, 054508 (2023).
  64. C. Berthiere, B. Chen, and H. Chen, Reflected entropy and Markov gap in Lifshitz theories, J. High Energy Phys. 09 (2023) 160.
  65. C. Akers and P. Rath, Entanglement wedge cross sections require tripartite entanglement, J. High Energy Phys. 04 (2020) 208.
  66. P. Hayden, O. Parrikar, and J. Sorce, The Markov gap for geometric reflected entropy, J. High Energy Phys. 10 (2021) 047.
  67. P. Hayden, M. Lemm, and J. Sorce, Reflected entropy: Not a correlation measure, Phys. Rev. A 107, L050401 (2023).
  68. S. D. Mathur, The information paradox: A pedagogical introduction, Classical Quantum Gravity 26, 224001 (2009).
  69. B. Collins, Moments and cumulants of polynomial random variables on unitarygroups, the Itzykson-Zuber integral, and free probability, Int. Math. Res. Not. 2003, 953 (2003).
  70. B. Collins and P. Śniady, Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Commun. Math. Phys. 264, 773 (2006).
  71. J. Chu, R. Qi, and Y. Zhou, Generalizations of reflected entropy and the holographic dual, J. High Energy Phys. 03 (2020) 151.

Outline

Information

Phys. Rev. D 112, 046009– Published 12 August, 2025
  • Received 3 June 2025
  • Accepted 23 July 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. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation