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arXiv:hep-th/0606028 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Final state boundary condition of the Schwarzschild black hole

Doyeol Ahn

Published Oct 9, 2006• 37 citations
arXivHTMLPDF

Abstract

It is shown that the internal stationary state of the Schwarzschild black hole can be represented by a maximally entangled two-mode squeezed state of collapsing matter and infalling Hawking radiation. The final boundary condition at the singularity is then described by the random unitary transformation acting on the collapsing matter field. The outgoing Hawking radiation is obtained by the final-state projection on the total wave function, which looks like a quantum teleportation process without the classical information transmitted. The black hole evaporation process as seen by the observer outside the black hole is now a unitary process but nonlocal physics is required to transmit the information outside the black hole. It is also shown that the final-state projection by the evaporation process is strongly affected by the quantum state outside the event horizon, which clearly violates the locality principle.

DOI
10.1103/physrevd.74.084010

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It is shown that the internal stationary state of the Schwarzschild black hole can be represented by a maximally entangled two-mode squeezed state of collapsing matter and infalling Hawking radiation. The final boundary condition at the singularity is then described by the random unitary transformation acting on the collapsing matter field. The outgoing Hawking radiation is obtained by the final-state projection on the total wave function, which looks like a quantum teleportation process without the classical information transmitted. The black hole evaporation process as seen by the observer outside the black hole is now a unitary process but nonlocal physics is required to transmit the information outside the black hole. It is also shown that the final-state projection by the evaporation process is strongly affected by the quantum state outside the event horizon, which clearly violates the locality principle.
Mathematical connection between the quantum teleportation, the most unique feature of quantum information processing, and the black hole final state is studied taking into account the nontrivial spacetime geometry. We use the twist operation for the generalized entanglement measurement and the final-state boundary conditions to obtain transfer theorems for the black hole evaporation. This would enable us to put together the universal quantum teleportation and the black hole evaporation in the unified mathematical footing. For a renormalized post selected final state of outgoing Hawking radiation, we found that the information content, quantified by the mixedness of the state, is preserved only in the special case of the final-state boundary condition in the microcanonical form, which resembles a perfect teleportation channel.
Hawking's black hole evaporation process suggests that we may need to choose between quantum unitarity and other basic physical principles such as no-signaling, entanglement monogamy, and the equivalence principle. We here show that the Hawking's quantum model for the black hole evaporation is consistent with the above fundamental principles. Our analysis does not involve exotic new physics, but rather uses standard quantum theory, general relativity, and the Einstein–Hilbert action including matter. We explicitly show that the whole state consisting of matter and radiation (in a joint superposition of different energy states) is pure at any stage of the evaporation process, including the particular case of 0 mass. Moreover, after full evaporation the state for radiation at infinity is pure and in one-to-one correspondence with the initial state forming the black hole. Thus there is no information loss upon full evaporation according to the quantum information theory. The original entanglement of the black hole matter (if any) gets transferred to the outgoing particles via a process similar to entanglement swapping, without violation of causality (as proved explicitly). On the other hand, if the initial state is a tensor product state, the entanglement of Hawking particles, present in the intermediate phase, is broken when the black hole evaporates completely. Therefore, the final state (entangled or tensor product depending on the nature of initial state) after the full black hole evaporation is pure without loss of information.
Hawking's black hole evaporation process suggests that we may need to choose between quantum unitarity and other basic physical principles such as no-signalling, entanglement monogamy, and the equivalence principle. We here provide a quantum model for Hawking pair black hole evaporation within which these principles are all respected. The model does not involve exotic new physics, but rather uses quantum theory and general relativity. The black hole and radiation are in a joint superposition of different energy states at any stage of the evaporation process. In the particular branch where the black hole mass is 0, the radiation state is pure and one-to-one with the initial state forming the black hole. Thus there is no information loss upon full evaporation. The original Hawking's pair entanglement between infalling and outgoing particles gets transferred to outgoing particles via entanglement swapping, without violation of no-signalling or the entanglement's monogamy. The final state after the full black hole evaporation is pure, without loss of information, violation of monogamy, or the equivalence principle.
We quantify the nature of the information loss in a Schwarzschild black hole as a function of its mass. We consider a scalar field in a two-mode squeezed state in the far past, and study the degradation of entanglement and classical correlations from the perspective of observers outside the black hole due to the Hawking radiation. Due to monogamy constraints on the distribution of entanglement, we show that the lost bipartite entanglement is recovered as multipartite entanglement among modes inside and outside the horizon. In the black hole evaporation limit, all bipartite entanglement between modes outside the black hole vanishes while the genuine multipartite entanglement grows infinitely.
In an attempt to restore the unitarity of the evaporation process, Horowitz and Maldacena recently proposed a boundary-condition constraint for the final quantum state of an evaporating black hole at its singularity. Gottesman and Preskill have argued that the proposed constraint must lead to nonlinear evolution of the initial (collapsing) quantum state. Here we show that in fact this evolution allows signaling, making it detectable outside the event horizon with entangled-probe experiments of the kind we proposed recently. As a result the Horowitz-Maldacena proposal may be subject to terrestrial tests.