Post-Newtonian Effective Field Theory Approach to Entanglement Harvesting, Quantum Discord and Bell's Nonlocality Bound Near a Black Hole

TL;DR

This work develops a post-Newtonian effective field theory (PN-EFT) to study how a quantum black hole modifies the reduced states of two static Unruh-DeWitt detectors via mediator fields. By treating the black hole as a tidally deformable thermal body, the authors obtain analytic reduced detector states and express entanglement harvesting, quantum discord, and Bell nonlocality within a unified X-state framework, controlled by leading PN couplings. They find that entanglement harvesting requires mutual detector interactions (Case III) and can exhibit features like entanglement shadows, while quantum discord can persist in the presence of a BH; however, at leading order, the CHSH nonlocality bound is not violated in any configuration. The PN-EFT approach provides a tractable, universal low-energy description and offers a path to include backreaction via higher-order PN corrections, potentially connecting BH quantum information to observable quantum correlations in curved spacetime.

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. Thus, the black hole may convey and retract its quantum information to the nearby quantum probes via the surrounding mediator fields. In this paper, we investigate the effects of a quantum black hole on the reduced states of a pair of static qubit-type Unruh-DeWitt (UDW) detectors acting as a probe, using three complementary quantum information measures: concurrence characterizing entanglement harvesting, quantum discord, and Bell's nonlocality bound. This sheds light on the nature of the quantum state of the black holes. By treating the black hole as a tidally deformable thermal body under the quantum fluctuation of the mediator fields as observed in \cite{Goldberger:2019sya, goldberger2020virtual, biggs2024comparing}, we employ a post-Newtonian effective field theory (PN-EFT) to derive the reduced states of the UDW probes analytically. Based on this, we can easily obtain all three quantum information 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, quantum correlation, and nonlocality bound of the UDW probe systems. Our PN-EFT approach can be extended to include the backreaction on the black holes in future studies by taking the higher-order PN corrections into account.

Figures (12)

• Figure 1: Comparison of the conventional approach (Left) based on quantum fields in curved spacetime and the PN-EFT approach (Right) in studying reduced states of UDW probes near a black hole. In the conventional approach, the black hole is treated as a background spacetime, which thermalizes the mediator field $\Phi$. However, in the PN-EFT approach, the black hole is treated as a point-like object with induced multipole moments and interacts with the point-like UDW probes via PN-type interactions, the higher-order terms of which characterize the back-reaction effects.
• Figure 2: Concurrence $\mathcal{C}$ for $\rho^f_{12}$ of (\ref{['rho_f_no']}): (a) $\mathcal{C}$ vs $T$ for various $r_{12}$: $r_{12}=10$ (solid-blue), $r_{12}=20$ (green-dashed) and $r_{12}=30$ (red-dot-dashed). (b) $\mathcal{C}$ vs $r_{12}$ for various $T$: $T=800$ (solid-blue), $T=850$ (green-dashed) and $T=900$ (red-dot-dashed). In this figure and the other ones shown later on, we fix $g_0=0.01$, $\Omega=0.001$, and $\bar{r}_B=\bar{r}_1=\bar{r}_2=1$, $q_1=q_2=1$.
• Figure 3: Three configurations of UDW probes (small black dots) relative to the black hole (big black dot).
• Figure 4: Concurrence $\mathcal{C}$ of configuration \ref{['fig:BH_UDW_C31']} for $\rho_{12}^f$ of (\ref{['rho_full']}): (a) $\mathcal{C}$ vs $T$ for $r_{1B}=10$ and various $r_{12}$: $r_{12}=20$ (solid-blue), $r_{12}=25$ (green-dashed) and $r_{12}=30$ (red-dot-dashed). (b) $\mathcal{C}$ vs $r_{1B}$ for $r_{12}=25$, and various $T$: $T=900$ (solid-blue), $T=950$ (green-dashed) and $T=1000$ (red-dot-dashed). (c) $\mathcal{C}$ vs $r_{12}$ for $T=100$, and various $r_{1B}$: $r_{1B}=20$ (solid-blue), $r_{1B}=25$ (green-dashed) and $r_{1B}=30$ (red-dot-dashed).
• Figure 5: Concurrence $\mathcal{C}$ of configuration \ref{['fig:BH_UDW_C32']} for $\rho_{12}^f$ of (\ref{['rho_full']}): (a) $\mathcal{C}$ vs $T$ for various $r_{1B}$: $r_{1B}=20$ (solid-blue), $r_{1B}=25$ (green-dashed) and $r_{2B}=30$ (red-dot-dashed). (b) $\mathcal{C}$ vs $r_{1B}$ for various $T$: $T=900$ (solid-blue), $T=950$ (green-dashed) and $T=1000$ (red-dot-dashed).