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arXiv:0712.3643

Black Hole Information in a Detector (Atom) - Field Analog

B. L. Hu, Shih-Yuin Lin

Published Dec 21, 2007• 0 citations
arXivHTMLPDF

Abstract

This is a synopsis of our recent work on quantum entanglement, recoherence and information flow between an uniformly accelerated detector and a massless quantum scalar field. The availability of exact solutions to this model enables us to explore the black hole information issue with some quantifiable results and new insights. To the extent this model can be used as an analog to the system of a black hole interacting with a quantum field, our result seems to suggest in the prevalent non-Markovian regime, assuming unitarity for the combined system, that black hole information is not lost but transferred to the quantum field degrees of freedom. This combined system will evolve into a highly entangled state between a remnant of large area (in Bekenstein's black hole atom analog) without any information of its initial state, while the quantum field is imbued with complex information content not-so-easily retrievable by a local observer.

DOI
10.48550/arxiv.0712.3643

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This is a synopsis of our recent work on quantum entanglement, recoherence and information flow between an uniformly accelerated detector and a massless quantum scalar field. The availability of exact solutions to this model enables us to explore the black hole information issue with some quantifiable results and new insights. To the extent this model can be used as an analog to the system of a black hole interacting with a quantum field, our result seems to suggest in the prevalent non-Markovian regime, assuming unitarity for the combined system, that black hole information is not lost but transferred to the quantum field degrees of freedom. This combined system will evolve into a highly entangled state between a remnant of large area (in Bekenstein's black hole atom analog) without any information of its initial state, while the quantum field is imbued with complex information content not-so-easily retrievable by a local observer.
We study an exactly solvable model where a uniformly accelerated detector is linearly coupled to a massless scalar field initially in the Minkowski vacuum. Using the exact correlation functions obtained in Lin and Hu (2006 Phys. Rev. D 73 124018, 2007 Phys. Rev. D 76 064008) we show that as soon as the coupling is switched on one can see information flowing from the detector to the field and propagating with the radiation into null infinity. By expressing the reduced density matrix of the detector in terms of the two-point functions, we calculate the purity function in the detector and study the evolution of quantum entanglement between the detector and the field. Only in the ultraweak coupling regime could some degree of recoherence in the detector appear at late times, but never in full restoration. We explicitly show that under the most general conditions the detector never recovers its quantum coherence and the entanglement between the detector and the field remains large at late times. To the extent that this model can be used as an analog to the system of a black hole interacting with a quantum field, our result seems to suggest in the prevalent non-Markovian regime, assuming unitarity for the combined system, that black hole information is not lost but transferred to the quantum field degrees of freedom. Our combined system will evolve into a highly entangled state between a remnant of large area (in Bekenstein's black hole atom analog) without any information of its initial state, and the quantum field, now imbued with complex information content not-so-easily retrievable by a local observer.
A unitary effective field model of the black hole evaporation is proposed to satisfy almost the four postulates of the black hole complementarity (BHC). In this model, we enlarge a black hole-scalar field system by adding an extra radiation detector that couples with the scalar field. After performing a partial trace over the scalar field space, we obtain an effective entanglement between the black hole and the detector (or radiation in it). As the whole system evolves, the S-matrix formula can be constructed formally step by step. Without local quantum measurements, the paradoxes of the information loss and AMPS’s firewall can be resolved. However, the information can be lost due to quantum decoherence, as long as some local measurement has been performed on the detector to acquire the information of the radiation in it. But unlike Hawking’s completely thermal spectrum, some residual correlations can be found in the radiations. All these considerations can be simplified in a qubit model that provides a modified quantum teleportation to transfer the information via an EPR pairs.
It has been known for decades that black holes behave as thermodynamic objects and, as such, have an entropy, which was acquired in exact form as the Bekenstein-Hawking entropy. However, the origin of this entropy, and in particular its proportionality to the area of the horizon, remained unexplained. Motivated by this mystery, several authors examined the possibility of entanglement entropy as a source of black hole entropy, by considering a scalar ?eld in the ground state and\ntracing out the degrees of freedom inside an imaginary sphere, analogous to the interior of the black hole. The resulting entanglement entropy between the two regions turned out to scale with the area of their mutual boundary. In the simplest model the scalar ?eld is represented by a system of coupled harmonic oscillators. After an introduction to the phenomenon of quantum entanglement and the associated entanglement entropy, we will review the problem of two coupled oscillators in the ground state and derive the entanglement entropy using an approach in Fock space, as opposed to the standard method involving integrals in positon space. This will also\nallow us to look at some excited states of the system. Subsequently we will try to extend this approach to a system of N coupled harmonic oscillators and derive the entanglement entropy between an inner and outer region of oscillators.
We analyze a system consisting of an oscillator coupled to a field. With the field traced out as an environment, the oscillator loses coherence on a very short decoherence time scale; but, on a much longer relaxation time scale, it evolves predictably into a unique, pure (ground) state. This example of recoherence has interesting implications both for the interpretation of quantum theory and for the loss of information during black hole evaporation---the so-called black hole information paradox. We examine these implications by investigating the intermediate and final states of the quantum field, treated as an open system coupled to an unobserved oscillator.
The complete evaporation of black holes, as a natural endpoint of Hawking radiation, gives rise to the black hole information paradox, which fundamentally challenges the principles of unitarity and information conservation in quantum mechanics. Although the AdS/CFT correspondence indicates that information is preserved during black hole evaporation, the precise mechanism by which it is recovered from the Hawking radiation remains an open question. To explore a potential resolution, we investigate information transfer in a spin-chain model featuring an effective horizon realized via position-dependent couplings within an XY spin chain. We derive and demonstrate Page curve-like behavior, and analyze the transmission of quantum resources, such as entanglement and coherence, across the effective horizon. Our results show that quantum resources initially localized within an interior subsystem can be transferred to the exterior via particle radiation through the horizon. This study provides a novel perspective from quantum simulation on how information may escape from black holes, thereby contributing to the further understanding of the black hole information paradox.
The black hole information paradox is a contradiction between fundamental principles which has puzzled physicists for over forty years. The crux of the problem lies in an assumption about the structure of entanglement across the event horizon, namely, that the Hilbert space factorizes. While valid in quantum mechanics, this fails drastically in quantum field theory, and hence a deeper understanding of entanglement is required if further progress is to be made. Recently, ideas from algebraic quantum field theory have provided new insight into this issue, and show promise for elucidating the connection between entropy and horizons that underlies black hole thermodynamics.