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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

How the Hawking effect affects multipartite entanglement of Dirac particles in the background of a Schwarzschild black hole

Shuai Xu, Xue‐Ke Song, Jiadong Shi, Liu Ye

Published Mar 18, 2014• 77 citations
Publisher

Abstract

In this paper, the effect of Hawking radiation on the multipartite entanglement of Dirac particles in the background of a Schwarzschild black hole is investigated. It is shown that all the 1-tangles decrease as the Hawking temperature grows, and the Coffman--Kundu--Wootters monogamy inequality is always saturated in this system. It is also shown that the entanglement's status of subsystems, which have the same quantum state, depend on its detector's position, and this proposition can be generalized to a multipartite system. Then, we find that the Hawking effect does not change the entanglement structure of the quantum state in Schwarzschild space-time. In addition, we discuss the distributions of quantum information and find that not only is the entanglement redistributed, but also the mutual information is distributed to the physical inaccessible region. At last, we calculate the mutual information of a $N$-qubit multipartite system, ${|\mathrm{\ensuremath{\Psi}}⟩}_{123\dots{}N}={(\ensuremath{\beta}{|0⟩}^{\ensuremath{\bigotimes}N}+\sqrt{1\ensuremath{-}{\ensuremath{\beta}}^{2}}{|1⟩}^{\ensuremath{\bigotimes}N})}_{123\dots{}N}$, and draw a conclusion that when the black hole approximates to evaporate completely the mutual information of this system will be close to $(N\ensuremath{-}1)/N$ of the original if a particle freely falls in toward a Schwarzschild black hole and locates near the event horizon, and the other particles remain in the asymptotically flat region.

DOI
10.1103/physrevd.89.065022

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In this paper, the effect of Hawking radiation on the multipartite entanglement of Dirac particles in the background of a Schwarzschild black hole is investigated. It is shown that all the 1-tangles decrease as the Hawking temperature grows, and the Coffman--Kundu--Wootters monogamy inequality is always saturated in this system. It is also shown that the entanglement's status of subsystems, which have the same quantum state, depend on its detector's position, and this proposition can be generalized to a multipartite system. Then, we find that the Hawking effect does not change the entanglement structure of the quantum state in Schwarzschild space-time. In addition, we discuss the distributions of quantum information and find that not only is the entanglement redistributed, but also the mutual information is distributed to the physical inaccessible region. At last, we calculate the mutual information of a NNN-qubit multipartite system, ∣\ensuremathΨ⟩123…N=(\ensuremathβ∣0⟩\ensuremath⨂N+1\ensuremath−\ensuremathβ2∣1⟩\ensuremath⨂N)123…N{|\mathrm{\ensuremath{\Psi}}⟩}_{123\dots{}N}={(\ensuremath{\beta}{|0⟩}^{\ensuremath{\bigotimes}N}+\sqrt{1\ensuremath{-}{\ensuremath{\beta}}^{2}}{|1⟩}^{\ensuremath{\bigotimes}N})}_{123\dots{}N}∣\ensuremathΨ⟩123…N​=(\ensuremathβ∣0⟩\ensuremath⨂N+1\ensuremath−\ensuremathβ2​∣1⟩\ensuremath⨂N)123…N​, and draw a conclusion that when the black hole approximates to evaporate completely the mutual information of this system will be close to (N\ensuremath−1)/N(N\ensuremath{-}1)/N(N\ensuremath−1)/N of the original if a particle freely falls in toward a Schwarzschild black hole and locates near the event horizon, and the other particles remain in the asymptotically flat region.
In this Letter, we analytically explore the effect of the Hawking radiation on the quantum correlation and Bell non-locality for Dirac particles in the background of Schwarzschild black hole. It is shown that when the Hawking effect is almost nonexistent, corresponding to the case of an almost extreme black hole, the quantum properties of physically accessible state are the same for the initial situation. For finite Hawking temperature T, the accessible quantum correlation monotonously decreases along with increasing T owing to the thermal fields generated by the Hawking effect, and the accessible quantum non-locality will be disappeared when the Hawking temperature is more than a fixed value which increases with the parameter r of Werner state growing. Then we analyze the redistribution of quantum correlation, and find that for the case of the Hawking temperature being infinite, corresponding to the case of the black hole evaporating completely, the quantum correlation of physically accessible state is equal to the one of the inaccessible states. Moreover, due to the Pauli exclusion principle and the differences between Fermi–Dirac and Bose–Einstein statistics, for the Dirac fields the accessible classical correlation decreases with increase of the Hawking temperature, which is different for the scalar fields. For Bell non-locality, we also find that the quantum non-locality is always extinct for physically inaccessible states, and the strength of the non-locality decreases with enlarging intensity of Hawking effect when the non-locality is existent in physically accessible state.
Abstract In this paper, we analytically explore the effect of Hawking radiation on quantum Fisher information (QFI) for Dirac particles in the background of Schwarzschild black hole. What surprise us is that the Hawking radiation could not disturb QFI with respect to the state parameter ( ). Additionally, we also find in the case of an almost extreme black hole, or when the black hole is approximate to evaporating completely, QFI with respect to the Hawking temperature T ( ) always being zero indicates that the parameter estimation precision about the Hawking temperature is minimum in the two situations above. In particular, the maximal is obtained when the Hawking temperature is a fixed value for every state parameter . Furthermore, we investigate how the Hawking effect influences the distribution of the QFI over subsystems in physical accessible regions and find that QFI can be either subadditive or superadditive, depending on the specific parameter under the Hawking effect. At last, we also study the QFI of other possible bipartite divisions, which are physically inaccessible, and the nontrivial results reveal that QFI is averagely distributed in different physical regions.
We investigate the Hawking effect on entangled fields. By considering a scalar field which is in a two-mode squeezed state from the point of view of freely falling (Kruskal) observers crossing the horizon of a Schwarzschild black hole, we study the degradation of quantum and classical correlations in the state from the perspective of physical (Schwarzschild) observers confined outside the horizon. Due to monogamy constraints on the entanglement distribution, we show that the lost bipartite entanglement is recovered as multipartite entanglement among modes inside and outside the horizon. In the limit of a small-mass black hole, no bipartite entanglement is detected outside the horizon, while the genuine multipartite entanglement interlinking the inner and outer regions grows infinitely.
Abstract Complex quantum information tasks in a gravitational background require multipartite entanglement for effective processing. Therefore, it is necessary to investigate the properties of multipartite entanglement in a relativistic setting. In this paper, we study genuine N-partite entanglement of massless Dirac fields in the Schwarzschild-de Sitter (SdS) spacetime, characterized by the presence of a black hole event horizon (BEH) and a cosmological event horizon (CEH). We obtain the general analytical expression of genuine N-partite entanglement shared by n observers near BEH and m ( n+m=Nn+m=Nn+m=N n + m = N ) observers near CEH. It is shown that genuine N-partite entanglement monotonically decreases with the decrease of the mass of the black hole, suggesting that the Hawking effect of the black hole destroys quantum entanglement. It is interesting to note that genuine N-partite entanglement is a non-monotonic function of the cosmological constant, meaning that the Hawking effect of the expanding universe can enhance quantum entanglement. This result contrasts with multipartite entanglement in single-event horizon spacetime, offering a new perspective on the Hawking effect in multi-event horizon spacetime.
The genuine entanglement of Dirac fields for an N-partite system is investigated in Schwarzschild spacetime and the analysis is carried out using the single-mode approximation. Due to the Hawking effect, quantum entanglement is divided into two parts physically accessible and inaccessible entanglement. We obtain a general analytic expression of genuine N-partite entanglement that includes all accessible and inaccessible entanglement in a Schwarzschild black hole. Unlike bosonic entanglement, the accessible N-partite entanglement of Dirac fields monotonically decreases to a nonzero value with the Hawking temperature. Interestingly, the inaccessible N-partite entanglement is a monotonic or non-monotonic function of the Hawking temperature, depending on the ratio between accessible and inaccessible modes, in contrast to bipartite or tripartite entanglement that is only a monotonic function of the Hawking temperature. Finally, we obtain two restrictive relationships for the quantum information of the black hole. This conclusion provides a new understanding of Hawking effect of the black hole.