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arXiv:2411.09409 · Physical review. D/Physical review. D.

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

Feng-Li Lin, Sayid Mondal

Published Jan 7, 2026• 1 citation
arXivHTMLPDF

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 [; ] and Biggs and Maldacena [], 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.

DOI
10.1103/ldcl-y2ld

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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 [; ] and Biggs and Maldacena [], 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.
A bstract The effect of black holes on entanglement harvesting has been of considerable interest over the past decade. Research involving stationary Unruh-DeWitt (UDW) detectors near a (2+1)-dimensional Bañados-Teitelboim-Zanelli (BTZ) black hole has uncovered phenomena such as entanglement shadows, entanglement amplification through black hole rotation, and differences between bipartite and tripartite entanglement. For a (1+1)-dimensional Schwarzschild black hole, it has been shown that two infalling UDW detectors can harvest entanglement from the scalar quantum vacuum even when separated by an event horizon. In this paper, we calculate the mutual information between two UDW detectors coupled to a massless quantum scalar field, with the detectors starting at rest and falling radially into a non-rotating (2+1)-dimensional BTZ black hole. The trajectory of the detectors includes regions where both detectors are switched on outside of the horizon; where one detector is switched on inside of the horizon while the other switches on outside; and where both detectors switch on inside of the horizon. We investigate different black hole masses, detector energy gaps, widths and temporal separations of the detector switching functions, and field boundary conditions. We find that black holes — even the simplest kind having constant curvature — significantly affect the correlation properties of quantum fields in the vacuum state. These correlations, both outside and inside the horizon, can be mapped out by infalling detectors.
One consequence of the cosmic censorship conjecture is that any topological structure will ultimately collapse to within the horizons of a set of black holes, and as a result, an external classical observer will be unable to probe it. However, a single two-level quantum system [Unruh–DeWitt (UDW) detector] that remains outside of the horizon has been shown to distinguish between a black hole and its associated geon counterpart via its different response rates. Here, we extend this investigation of the quantum vacuum outside of an ℝP2 geon by considering the entanglement structure of the vacuum state of a quantum scalar field in this spacetime, and how this differs from its Banados–Teitelboim–Zanelli (BTZ) black hole counterpart. Employing the entanglement harvesting protocol, where field entanglement is swapped to a pair of UDW detectors, we find that the classically hidden topology of the geon can have an appreciable difference in the amount of entanglement harvested in the two spacetimes for sufficiently small mass. In this regime, we find that detectors with a small energy gap harvest more entanglement in the BTZ spacetime; however, as the energy gap increases, the detectors harvest more entanglement in a geon spacetime. The energy gap at the crossover is dependent on the black hole mass, occurring at lower values for lower masses. This also impacts the size of the entanglement shadow, the region near the horizon where the detectors cannot harvest entanglement. Small gap detectors experience a larger entanglement shadow in a geon spacetime, whereas for large gap detectors, the shadow is larger in a BTZ spacetime.
We carry out the first investigation of the entanglement and mutual information harvesting protocols for detectors freely falling into a black hole. Working in (1+11+11+1)-dimensional Schwarzschild black hole spacetime, we consider two pointlike Unruh-DeWitt detectors in different combinations of free-falling and static trajectories. Employing a generalization of relative velocity suitable for curved spacetimes, we find that the amount of correlations extracted from the black hole vacuum, at least outside the near-horizon regime, is largely kinematic in origin (i.e., it is mostly due to the relative velocities of the detectors). Second, correlations can be harvested purely from the black hole vacuum even when the detectors are causally disconnected by the event horizon. Finally, we show that the previously known `entanglement shadow' near the horizon is indeed absent for the case of two free-falling-detectors, since their relative gravitational redshift remains finite as the horizon is crossed, in accordance with the equivalence principle.
The aim of this work is present the phenomenon denoted entanglement harvesting.We begin by introducing entanglement historically.Following, we go beyond the one particle theory in flat spacetime and introduce Quantum Field Theory in Curved Spacetime, showing two famous consequences: the Unruh effect and the Hawking radiation.Finally, we analyze entanglement harvesting for two Unruh-deWitt detectors.In the fisrt example, we see that there is a "sudden death" point of entanglement harvesting when the detectors are near the BTZ black hole event horizon, due to redshift effect and Hawking radiation.Then, we compare the phenomenon for different scenarios, and find out that it is sensitive to the structure of spacetime.Finally, we see how detectors' parameters affect it and find out that the smoothness of the switching of the detectors' coupling to the field is extremely relevant.We also see how the parameters can be used to optimize entanglement harvested.
A bstract The CPT invariant vacuum states of a scalar field in de Sitter space, called α -vacua, are not unique. We explore the α -vacua from the quantum information perspective by a pair of static Unruh-DeWitt (UDW) detectors coupled to a scalar field with either monopole or dipole coupling, which are in time-like zero separation or space-like antipodal separation. The analytical form of the reduced final state of the UDW detector is derived. We study the entanglement harvesting and quantum discord of the reduced state, which characterize the quantum entanglement and quantum correlation of the underlying α -vacua, respectively. Our results imply that the quantum entanglement gravitated by de Sitter gravity behaves quite differently for time-like and space-like separations. It experiences “sudden death” for the former and grows for the latter as the measuring time or the value of α increases. This demonstrates the nonlocal nature of quantum entanglement. For the quantum discord, we find no “sudden death” behavior, and it experiences superhorizon suppression, which explains the superhorizon decoherence in the inflationary universe scenario. Overall, the time-like or space-like quantum entanglement and correlation behave differently on their dependence of α , measuring time and spectral gaps, with details discussed in this work.
We explore entanglement harvesting using two Unruh-DeWitt (UDW) detectors linearly coupled to the scalar density of a massless spin-1/21/21/2 field in 1+11+11+1 Schwarzschild spacetime. We consider different vacua, including the Boulware, Hartle-Hawking-Israel (HHI), and Unruh vacua, and investigate various configurations of detector trajectories. We find that the transition rate of the static UDW detector exhibits the expected Planckian behavior in the HHI state, while the Unruh state leads to the Helmholtz free energy density of a fermionic thermal bath. We demonstrate that the near-horizon entanglement properties for static detectors in the HHI state have similar behavior to those in Minkowski vacua for uniformly accelerated detectors in Rindler spacetime. We further consider a different interaction Hamiltonian which breaks local Lorentz symmetry and find that the transition rate of the static detector still exhibits Planckian behavior in the HHI state, while in the Unruh state, it leads to the Helmholtz free energy density of a bosonic or fermionic thermal bath corresponding to the static or conformal 2-bein in interaction, respectively. We observe that the anti-Hawking effect enhances the entanglement between the two detectors while the gravitational redshift and Hawking radiation decrease it. In particular, due to the presence of the anti-Hawking effect, the mutual information and concurrence near the event horizon can be non-zero even for static detectors with static 2-bein, which is in contrast with the case of the scalar field. Conclusions are discussed.