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arXiv:1006.1234

Entanglement and final state of a black hole under general unitary transformation

Doyeol Ahn

Published Jun 7, 2010• 1 citation
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Abstract

Final state of a black hole is studied by taking into account a general unitary evolution of a black hole matter state. It is found that the mixedness decreases under general unitary transformation when the initial matter state was in mixed state and the mean fidelity at the evaporation is smaller than the fidelity of the quantum teleportation by the factor of inverse square of the number of states of a black hole. The change of entanglement Alice-Bob pair at evaporation is also studied by calculating the entanglement fidelity and eigenvalues of the partial transposed block density matrix. We assumed that Alice fell onto the black hole when the hole was formed while Bob is maintaining the safe distance from the black hole. It is found that the entanglement fidelity is inversely proportional to the square of the Hilbert space dimension N and the entanglement can survive the evaporation process.

DOI
10.48550/arxiv.1006.1234

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Final state of a black hole is studied by taking into account a general unitary evolution of a black hole matter state. It is found that the mixedness decreases under general unitary transformation when the initial matter state was in mixed state and the mean fidelity at the evaporation is smaller than the fidelity of the quantum teleportation by the factor of inverse square of the number of states of a black hole. The change of entanglement Alice-Bob pair at evaporation is also studied by calculating the entanglement fidelity and eigenvalues of the partial transposed block density matrix. We assumed that Alice fell onto the black hole when the hole was formed while Bob is maintaining the safe distance from the black hole. It is found that the entanglement fidelity is inversely proportional to the square of the Hilbert space dimension N and the entanglement can survive the evaporation process.
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.
We describe a model for unitary black hole evaporation with no information loss in terms of a quantum computation. We assume that there is a fine tuned interaction between the qubits of the black hole Bell states which is the inverse of premeasurement. Evaporation is unitary due to a projection on the black hole state at the end of each computation whereas information comes out of the black hole by teleportation. The model requires only four operations; the qubit interactions, the Hadamard and CNOT gates and the projection which is nonunitary. The model is a concrete quantum computation that realizes the final black hole state idea with some modifications.
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.
Abstract In this paper, we describe the properties of quantum entanglement and teleportation between Alice and Bob who is freely falling toward the Schwarzschild black hole. To this aim, in the flat Minkowski spacetime before a black hole is formed, Alice and Bob share a two-mode entangled coherent state (ECS) generated by Kerr medium and the beam splitter. We will show that the degree of entanglement decreases with increasing the radius of black hole r. Moreover, the fidelity of quantum teleportation via ECS, as a quantum channel, is also reduced because of the Hawking-Unruh effect.
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.