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

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.