Harvesting correlations from BTZ black hole coupled to a Lorentz-violating vector field

In this paper, we investigate the effects of Lorentz violation on correlations harvesting, specifically focusing on the harvested entanglement and harvested mutual information between two Unruh-DeWitt detectors interacting with a quantum field in the Lorentz-violating BTZ-like black hole spacetime. Our findings reveal that Lorentz symmetry breaking has contrasting impacts on entanglement harvesting and mutual information harvesting in BTZ backgrounds: it enhances mutual information harvesting while suppressing entanglement harvesting. This phenomenon suggests that the increase in total correlations in Lorentz-violating vector field backgrounds with gravitational coupling is predominantly driven by classical components, with quantum correlations contributing less to the overall mutual information. These results indicate that Lorentz violation, as a quantum property of spacetime, may impose intrinsic constraints on the quantum information capacity encoded in spacetime due to competition among quantum degrees of freedom for resources. Furthermore, Lorentz symmetry breaking expands the \textit{entanglement shadow} region, further demonstrating its disruptive effect on quantum correlations.

💡 Research Summary

In this work the authors explore how spontaneous Lorentz‑symmetry breaking, modeled by the Einstein‑Bumblebee theory, influences the harvesting of quantum correlations from a (2+1)‑dimensional BTZ‑like black hole. By introducing a non‑zero vacuum expectation value for the Bumblebee vector field (B_{\mu}) they obtain a modified black‑hole metric that depends on a single Lorentz‑violating parameter (\alpha = \varrho b^{2}). When (\alpha=0) the geometry reduces to the standard BTZ solution; for (\alpha\neq0) the metric components acquire a factor ((1+\alpha)) which changes the red‑shift factor, the Hawking temperature (T_{H}=r_{h}/(2\pi\ell^{2}\sqrt{1+\alpha})), and the proper distance between static observers.

The quantum field considered is a conformally coupled massless scalar. Its Wightman function in the Lorentz‑violating AdS(_3) background is derived analytically, and the BTZ‑like Wightman function is constructed by an image‑sum over angular identifications. The authors adopt Dirichlet boundary conditions at spatial infinity.

Two point‑like Unruh‑DeWitt detectors (Alice and Bob) are placed at fixed radii (r_{A}) and (r_{B}) with (r_{h}<r_{A}<r_{B}). Both detectors share the same energy gap (\Omega), coupling strength (\lambda), and Gaussian switching duration (\sigma). Their interaction Hamiltonian couples the monopole operator to the scalar field along each world‑line. By expanding the time‑ordered evolution operator to second order in (\lambda) the reduced density matrix (\rho_{AB}) of the detectors is obtained. The matrix elements are expressed in terms of the exchange response function (X) and the local response function (C), which are double integrals over the Wightman function weighted by the switching functions and phase factors (e^{\pm i\Omega\tau}).

Numerical evaluation is performed for a range of (\alpha) values (0 to 0.5) while keeping the proper separation (d_{AB}) fixed. The main findings are:

1. Entanglement harvesting – quantified by the negativity – is strongly suppressed as (\alpha) grows. Beyond a critical (\alpha) the negativity vanishes for all separations considered, defining an enlarged “entanglement shadow” where no quantum entanglement can be harvested. This reflects the fact that Lorentz violation modifies the causal structure and reduces the effective interaction strength between the detectors.

2. Mutual information harvesting – measured by (I(A:B)=S_{A}+S_{B}-S_{AB}) – behaves oppositely: it increases with (\alpha). The total correlations (classical plus quantum) become larger even though the genuinely quantum part (entanglement) shrinks. The authors interpret this as the Lorentz‑violating background enhancing classical correlations (e.g., thermal noise associated with the modified Hawking temperature) while diminishing the quantum contribution.

3. The Hawking temperature decreases as (\alpha) increases, which lowers the thermal excitation probability of each detector. However, when the local Kubo‑Martin‑Schwinger temperature is held fixed (by accounting for the red‑shift), the geometry‑induced change in proper distance still leads to a net increase in mutual information.

4. The expansion of the entanglement shadow with (\alpha) suggests that spontaneous Lorentz breaking imposes an intrinsic bound on the quantum‑information capacity of spacetime. The competition between limited quantum resources (entanglement) and abundant classical resources (mutual information) becomes more pronounced in the presence of Lorentz violation.

The paper concludes that Lorentz‑violating vector fields act as a double‑edged sword for quantum information processing in curved spacetime: they restrict the ability to harvest entanglement while simultaneously allowing more total (mostly classical) correlations to be extracted. This insight opens avenues for further investigations, such as extending the analysis to rotating BTZ black holes, higher‑dimensional spacetimes, or alternative boundary conditions, and exploring implications for quantum communication protocols and quantum gravity phenomenology.