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Post-Newtonian effective field theory approach to entanglement harvesting, quantum discord, and Bell’s nonlocality bound near a black hole

Feng-Li Lin1,* and Sayid Mondal2,†

  • *Contact author: fengli.lin@gmail.com
  • †Contact author: sayid.mondal@gmail.com
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Phys. Rev. D 113, 025016 – Published 28 January, 2026

DOI: https://doi.org/10.1103/ldcl-y2ld

Abstract

Black holes, as characterized by the Hawking effect and Bekenstein-Hawking entropy, can be treated as a compact object carrying nontrivial quantum information obscured behind the event horizon. This quantum information, while hidden behind the event horizon, can be indirectly probed through the black hole’s interactions with surrounding quantum fields. In this paper, we investigate how the quantum nature of a black-hole influences the correlations harvested by a pair of static Unruh-DeWitt (UDW) detectors. To provide a comprehensive analysis, we employ various correlation measures: concurrence, quantum discord, and the violation of Bell’s inequality, thereby shedding light on the quantum nature of the black hole as perceived by local probes. By treating the black hole as a tidally deformable thermal body under the quantum fluctuation of the mediator fields, as observed by Goldberger and Rothstein [J. High Energy Phys. 04 (2020) 056; Phys. Rev. Lett. 125, 211301 (2020)] and Biggs and Maldacena [arXiv:2405.02227], we employ a post-Newtonian effective field theory (PN-EFT) to derive the final states of the two UDW probes analytically. A key advantage of our approach is the ability to analytically derive all these correlation measures without encountering the complicated Matsubara sum of infinite thermal poles, as in the conventional approach based on quantum fields in curved spacetime. By tuning the relative strengths in the action of PN-EFT, we can extract the effects of the black hole on the entanglement harvesting, quantum correlation, and nonlocality bound of the UDW probe systems. Furthermore, our PN-EFT approach can be extended in future studies to include backreaction on black holes by accounting for higher-order PN corrections.

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References (72)

  1. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975).
  2. J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973).
  3. J. D. Bekenstein, Generalized second law of thermodynamics in black hole physics, Phys. Rev. D 9, 3292 (1974).
  4. S. W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14, 2460 (1976).
  5. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975).
  6. D. N. Page, Information in black hole radiation, Phys. Rev. Lett. 71, 3743 (1993).
  7. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93, 035002 (2021).
  8. S. D. Mathur, The information paradox: A pedagogical introduction, Classical Quantum Gravity 26, 224001 (2009).
  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. W. G. Unruh, Notes on black-hole evaporation, Phys. Rev. D 14, 870 (1976).
  12. B. S. DeWitt, Quantum gravity: The new synthesis, in General Relativity: An Einstein Centenary Survey (Cambridge University Press, London, 1980), pp. 680–745.
  13. A. Biggs and J. Maldacena, Comparing the decoherence effects due to black holes versus ordinary matter, arXiv:2405.02227.
  14. D. L. Danielson, G. Satishchandran, and R. M. Wald, Gravitationally mediated entanglement: Newtonian field versus gravitons, Phys. Rev. D 105, 086001 (2022).
  15. D. L. Danielson, G. Satishchandran, and R. M. Wald, Killing horizons decohere quantum superpositions, Phys. Rev. D 108, 025007 (2023).
  16. D. L. Danielson, G. Satishchandran, and R. M. Wald, Black holes decohere quantum superpositions, Int. J. Mod. Phys. D 31, 2241003 (2022).
  17. S. E. Gralla and H. Wei, Decoherence from horizons: General formulation and rotating black holes, Phys. Rev. D 109, 065031 (2024).
  18. J. Wilson-Gerow, A. Dugad, and Y. Chen, Decoherence by warm horizons, Phys. Rev. D 110, 045002 (2024).
  19. G. Salton, R. B. Mann, and N. C. Menicucci, Acceleration-assisted entanglement harvesting and rangefinding, New J. Phys. 17, 035001 (2015).
  20. E. Martin-Martinez and B. C. Sanders, Precise space–time positioning for entanglement harvesting, New J. Phys. 18, 043031 (2016).
  21. E. Martín-Martínez, A. R. H. Smith, and D. R. Terno, Spacetime structure and vacuum entanglement, Phys. Rev. D 93, 044001 (2016).
  22. L. J. Henderson, R. A. Hennigar, R. B. Mann, A. R. H. Smith, and J. Zhang, Harvesting entanglement from the black hole vacuum, Classical Quantum Gravity 35, 21LT02 (2018).
  23. S. Kukita and Y. Nambu, Harvesting large scale entanglement in de Sitter space with multiple detectors, Entropy 19, 449 (2017).
  24. T. R. Perche, B. Ragula, and E. Martín-Martínez, Harvesting entanglement from the gravitational vacuum, Phys. Rev. D 108, 085025 (2023).
  25. K. Gallock-Yoshimura, E. Tjoa, and R. B. Mann, Harvesting entanglement with detectors freely falling into a black hole, Phys. Rev. D 104, 025001 (2021).
  26. D. Mendez-Avalos, L. J. Henderson, K. Gallock-Yoshimura, and R. B. Mann, Entanglement harvesting of three Unruh-DeWitt detectors, Gen. Relativ. Gravit. 54, 87 (2022).
  27. F.-L. Lin and S. Mondal, Entanglement harvesting and quantum discord of alpha vacua in de Sitter space, J. High Energy Phys. 08 (2024) 159.
  28. W. D. Goldberger and I. Z. Rothstein, An effective field theory of gravity for extended objects, Phys. Rev. D 73, 104029 (2006).
  29. W. D. Goldberger, Les Houches lectures on effective field theories and gravitational radiation, in Les Houches Summer School—Session 86: Particle Physics and Cosmology: The Fabric of Spacetime (Elsevier, San Diego, CA, 2007).
  30. R. A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rep. 633, 1 (2016).
  31. W. Goldberger and I. Z. Rothstein, Virtual Hawking radiation, Phys. Rev. Lett. 125, 211301 (2020).
  32. W. D. Goldberger and I. Z. Rothstein, An effective field theory of quantum mechanical black hole horizons, J. High Energy Phys. 04 (2020) 056.
  33. T. Damour and A. Nagar, Relativistic tidal properties of neutron stars, Phys. Rev. D 80, 084035 (2009).
  34. T. Binnington and E. Poisson, Relativistic theory of tidal Love numbers, Phys. Rev. D 80, 084018 (2009).
  35. B. Kol and M. Smolkin, Black hole stereotyping: Induced gravito-static polarization, J. High Energy Phys. 02 (2012) 010.
  36. P. Charalambous, S. Dubovsky, and M. M. Ivanov, On the vanishing of Love numbers for Kerr black holes, J. High Energy Phys. 05 (2021) 038.
  37. L. Hui, A. Joyce, R. Penco, L. Santoni, and A. R. Solomon, Ladder symmetries of black holes. Implications for love numbers and no-hair theorems, J. Cosmol. Astropart. Phys. 01 (2022) 032.
  38. O. Combaluzier-Szteinsznaider, L. Hui, L. Santoni, A. R. Solomon, and S. S. C. Wong, Symmetries of vanishing nonlinear Love numbers of Schwarzschild black holes, J. High Energy Phys. 03 (2025) 124.
  39. A. Kehagias and A. Riotto, Black holes in a gravitational field: The non-linear static Love number of Schwarzschild black holes vanishes, J. Cosmol. Astropart. Phys. 05 (2025) 039.
  40. M. M. Ivanov and Z. Zhou, Revisiting the matching of black hole tidal responses: A systematic study of relativistic and logarithmic corrections, Phys. Rev. D 107, 084030 (2023).
  41. M. M. Ivanov and Z. Zhou, Vanishing of black hole tidal Love numbers from scattering amplitudes, Phys. Rev. Lett. 130, 091403 (2023).
  42. M. Perry and M. J. Rodriguez, Dynamical Love numbers for Kerr black holes, arXiv:2310.03660.
  43. H. Ollivier and W. H. Zurek, Quantum discord: A measure of the quantumness of correlations, Phys. Rev. Lett. 88, 017901 (2001).
  44. L. Henderson and V. Vedral, Classical, quantum and total correlations, J. Phys. A 34, 6899 (2001).
  45. R. Horodecki, P. Horodecki, and M. Horodecki, Violating bell inequality by mixed spin-12 states: Necessary and sufficient condition, Phys. Lett. A 200, 340 (1995).
  46. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
  47. H. Buhrman, R. Cleve, S. Massar., and R. de Wolf, Nonlocality and communication complexity, Rev. Mod. Phys. 82, 665 (2010).
  48. A. Einstein, L. Infeld, and B. Hoffmann, The gravitational equations and the problem of motion, Ann. Math. 39, 65 (1938).
  49. R. Kubo, Statistical mechanical theory of irreversible processes. 1. General theory and simple applications in magnetic and conduction problems, J. Phys. Soc. Jpn. 12, 570 (1957).
  50. R. Kubo, M. Yokota, and S. Nakajima, Statistical-mechanical theory of irreversible processes. II. Response to thermal disturbance, J. Phys. Soc. Jpn. 12, 1203 (1957).
  51. P. C. Martin and J. Schwinger, Theory of many-particle systems. I, Phys. Rev. 115, 1342 (1959).
  52. R. Fabbri, Scattering and absorption of electromagnetic waves by a Schwarzschild black hole, Phys. Rev. D 12, 933 (1975).
  53. D. N. Page, Particle Emission Rates from a black hole: Massless particles from an uncharged, nonrotating hole, Phys. Rev. D 13, 198 (1976).
  54. F.-L. Lin and S. Mondal, Can Newtonian gravity produce quantum entanglement?, arXiv:2510.23584.
  55. T. Yu and J. H. Eberly, Evolution from entanglement to decoherence of bipartite mixed “x” states, Quantum Inf. Comput. 7, 459 (2007).
  56. W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80, 2245 (1998).
  57. J.-i. Koga, K. Maeda, and G. Kimura, Entanglement extracted from vacuum into accelerated Unruh-DeWitt detectors and energy conservation, Phys. Rev. D 100, 065013 (2019).
  58. M. Ali, A. R. P. Rau, and G. Alber, Quantum discord for two-qubit x states, Phys. Rev. A 81, 042105 (2010).
  59. M. A. Yurischev, On the quantum discord of general x states, Quantum Inf. Process. 14, 3399 (2015).
  60. Y. Huang, Quantum discord for two-qubit x states: Analytical formula with very small worst-case error, Phys. Rev. A 88, 014302 (2013).
  61. A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 47, 777 (1935).
  62. E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001 (2019).
  63. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70, 1895 (1993).
  64. C. H. Bennett and S. J. Wiesner, Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states, Phys. Rev. Lett. 69, 2881 (1992).
  65. B. S. Cirel’son, Quantum generalizations of Bell’s inequality, Lett. Math. Phys. 4, 93 (1980).
  66. L. A. Khalfin and B. S. Tsirelson, Quantum/classical correspondence in the light of Bell’s inequalities, Found. Phys. 22, 879 (1992).
  67. D. Rohrlich and S. Popescu, Nonlocality as an axiom for quantum theory, in 60 Years of EPR: Workshop on the Foundations of Quantum Mechanics (In Honor of Nathan Rosen) (1995).
  68. V. Capasso, D. Fortunato, and F. Selleri, Sensitive observables of quantum mechanics, Int. J. Theor. Phys. 7, 319 (1973).
  69. N. Gisin, Bell’s inequality holds for all non-product states, Phys. Lett. A 154, 201 (1991).
  70. D. Home and F. Selleri, Bell’s theorem and the EPR paradox, Riv. Nuovo Cimento 14, 1 (1991).
  71. R. F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model, Phys. Rev. A 40, 4277 (1989).
  72. F. Verstraete and M. M. Wolf, Entanglement versus Bell violations and their behavior under local filtering operations, Phys. Rev. Lett. 89, 170401 (2002).

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Phys. Rev. D 113, 025016– Published 28 January, 2026
  • Received 25 November 2024
  • Accepted 7 January 2026
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