Abstract We introduce a local operational protocol for measuring quantum information scrambling in a curved spacetime background. In this framework, two Unruh-DeWitt (UDW) detectors, coupled to a massless conformally coupled scalar field in the Hartle–Hawking vacuum of the non-rotating ( 2 + 1 ) -dimensional Banados–Teitelboim–Zanelli (BTZ) black hole, serve as local probes. Evaluating the detector out-of-time-ordered correlator requires a fourth-order Dyson series expansion. Within the scrambling regime, this correlator exhibits exponential growth with Lyapunov exponent, exactly saturating the Maldacena–Shenker–Stanford bound at the global Hawking temperature. Governed by the black hole surface gravity, the Lyapunov exponent is universal and independent of detector positions, energy gaps, and coupling strengths. Furthermore, the scrambling amplitude shows a monotonic dependence on the radial positions of the detectors, vanishing at the event horizon and increasing outward. Local two-level probes thus witness the maximal chaos of the BTZ geometry without reconstructing the global quantum field state.
Abstract We introduce a local operational protocol for measuring quantum information scrambling in a curved spacetime background. In this framework, two Unruh-DeWitt (UDW) detectors, coupled to a massless conformally coupled scalar field in the Hartle–Hawking vacuum of the non-rotating ( 2 + 1 ) -dimensional Banados–Teitelboim–Zanelli (BTZ) black hole, serve as local probes. Evaluating the detector out-of-time-ordered correlator requires a fourth-order Dyson series expansion. Within the scrambling regime, this correlator exhibits exponential growth with Lyapunov exponent, exactly saturating the Maldacena–Shenker–Stanford bound at the global Hawking temperature. Governed by the black hole surface gravity, the Lyapunov exponent is universal and independent of detector positions, energy gaps, and coupling strengths. Furthermore, the scrambling amplitude shows a monotonic dependence on the radial positions of the detectors, vanishing at the event horizon and increasing outward. Local two-level probes thus witness the maximal chaos of the BTZ geometry without reconstructing the global quantum field state.
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.
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.
Abstract Out-of-time-order correlators (OTOCs) that capture maximally chaotic properties of a black hole are determined by scattering processes near the horizon. This prompts the question to what extent OTOCs display chaotic behaviour in horizonless microstate geometries. This question is complicated by the fact that Lyapunov growth of OTOCs requires nonzero temperature, whereas constructions of microstate geometries have been mostly restricted to extremal black holes. In this paper, we compute OTOCs for a class of extremal black holes, namely maximally rotating BTZ black holes, and show that on average they display “slow scrambling”, characterized by cubic (rather than exponential) growth. Superposed on this average power-law growth is a sawtooth pattern, whose steep parts correspond to brief periods of Lyapunov growth associated to the nonzero temperature of the right-moving degrees of freedom in a dual conformal field theory. Next we study the extent to which these OTOCs are modified in certain “superstrata”, horizonless microstate geometries corresponding to these black holes. Rather than an infinite throat ending on a horizon, these geometries have a very deep but finite throat ending in a cap. We find that the superstrata display the same slow scrambling as maximally rotating BTZ black holes, except that for large enough time intervals the growth of the OTOC is cut off by effects related to the cap region, some of which we evaluate explicitly.
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.
There is currently significant interest in emulating the essential characteristics of black holes, such as their Hawking radiation or their optimal scrambling behavior, using condensed matter models. In this article, we investigate a chiral spin-chain, whose mean field theory effectively captures the behavior of Dirac fermions in the curved spacetime geometry of a black hole. We find that within the region of the chain that describe the interior of the black hole, strong correlations prevail giving rise to many-body chaotic dynamics. Employing out-of-time-order correlations as a diagnostic tool, we numerically compute the associated Lyapunov exponent. Intriguingly, we observe a linear increase in the Lyapunov exponent with temperature within the black hole's interior at low temperatures, indicative of optimal scrambling behavior. This contrasts with the quadratic temperature dependence exhibited by the spin-chain on the region outside the black hole. Our findings contribute to a deeper understanding of the interplay between black hole geometry and quantum chaos, offering insights into fundamental aspects of quantum gravity.
We explore entanglement harvesting using two Unruh-DeWitt (UDW) detectors linearly coupled to the scalar density of a massless spin-1/2 field in 1+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.