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arXiv:1010.6154 · Physical Review A

Tripartite entanglement in a noninertial frame

Mi-Ra Hwang, DaeKil Park, Eylee Jung

Published Jan 31, 2011• 120 citations
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Abstract

Tripartite entanglement is examined when one of the three parties moves with a uniform acceleration with respect to other parties. As the Unruh effect indicates, tripartite entanglement exhibits a decreasing behavior with increasing acceleration. Unlike bipartite entanglement, however, tripartite entanglement does not completely vanish in the infinite acceleration limit. If the three parties, for example, share the Greenberger-Horne-Zeilinger or $W$ state initially, the corresponding $\ensuremath{\pi}$-tangle, one of the measures of tripartite entanglement, is shown to be $\ensuremath{\pi}/6~0.524$ or $0.176$ in this limit, respectively. This fact indicates that tripartite quantum-information processing may be possible even if one of the parties approaches the Rindler horizon. The physical implications of this striking result are discussed in the context of black-hole physics.

DOI
10.1103/physreva.83.012111

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arXiv:1906.07267•May 27, 2019•1 citation
Tripartite entanglement is examined when one of the three parties moves with a uniform acceleration with respect to other parties. As the Unruh effect indicates, tripartite entanglement exhibits a decreasing behavior with increasing acceleration. Unlike bipartite entanglement, however, tripartite entanglement does not completely vanish in the infinite acceleration limit. If the three parties, for example, share the Greenberger-Horne-Zeilinger or WWW state initially, the corresponding \ensuremathπ\ensuremath{\pi}\ensuremathπ-tangle, one of the measures of tripartite entanglement, is shown to be \ensuremathπ/6 0.524\ensuremath{\pi}/6~0.524\ensuremathπ/6 0.524 or 0.1760.1760.176 in this limit, respectively. This fact indicates that tripartite quantum-information processing may be possible even if one of the parties approaches the Rindler horizon. The physical implications of this striking result are discussed in the context of black-hole physics.
In this work, we generalize the entanglement of three-qbit Bosonic systems beyond the single-mode approximation when one of the observers is accelerated. For this purpose, we review the effects of acceleration on field modes and quantum states. The single-mode approximation and beyond the single-mode approximation methods are introduced. After this brief introduction, the main problem of this paper, tripartite entanglement of bosonic systems in a noninertial frame beyond the single- mode approximation is investigated. The tripartite entangled states have different classes with GHZ and W states being most important. Here, we choose &pi-tangle as a measure of tripartite entanglement. If the three parties share GHZ state, the corresponding &pi-tangle will increase by increasing acceleration for some Unruh modes. This phenomenon, increasing entanglement, has never been observed in the single-mode approximation for bosonic case. Moreover, the &pi-tangle dose not exhibit a monotonic behavior with increasing acceleration. In the infinite acceleration limit, the &pi-tangle goes to different nonzero values for distinct Unruh modes. Unlike GHZ state, the entanglement of the W state shows only monotonically increasing and decreasing behaviors with increasing acceleration. Also, the entanglement for all possible choices of Unruh modes approaches only 0.176 in the high acceleration limit. Therefore, according to the quantum entanglement, there is no distinction between the single-mode approximation and beyond the single-mode approximation methods in this limit.
The three-tangle-dependence of $S_{max} = \max $, where SSS is Svetlichny operator, are explicitly derived when one party moves with an uniform acceleration with respect to other parties in the generalized Greenberger-Horne-Zeilinger and maximally slice states. The πππ-tangle-dependence of SmaxS_{max}Smax​ are also derived implicitly. From the dependence we conjecture that the multipartite entanglement is not the only physical resource for quantum mechanical multipartite non-locality.
Entanglement degradation caused by the Unruh effect is discussed for the tripartite GHZ or W states constructed by modes of a non-interacting quantum field viewed by one inertial observer and two uniformly accelerated observers. For fermionic states, the Unruh effect even for infinite accelerations cannot completely remove the entanglement. However, for the bosonic states, the situation is different and the entanglement vanishes asymptotically. Also, the entanglement is studied for the bipartite subsystems. While for the GHZ states all the bipartite subsystems are identically disentangled, for the W states the bipartite subsystems are somewhat entangled, though, this entanglement can be removed for appropriately accelerated observers. Interestingly, logarithmic negativity as a measure for determining the entanglement of one part of the system relative to the other two parts, is not generally the same for different parts. This means that we encounter tripartite systems where each part is differently entangled to the other two parts.
We consider two entangled accelerating qubits coupled with real scalar fields, each described by the Unruh-Wald model. It is demonstrated that because of the Unruh effect of the fields, the bipartite entanglement between the two qubits suddenly dies when the acceleration of one or more qubits are large enough. We also consider three entangled accelerating qubits in GHZ state and in W state, with equal acceleration-frequency ratio, and found that in either state, the tripartite entanglement suddenly dies at a certain value of acceleration-frequency ratio. The equivalence between the Rindler metric and the Schwarzschild metric in the vicinity of the horizon of a black hole implies that for two entangled qubits outside a black hole, the entanglement suddenly dies when one or both of the qubits are close enough to the horizon, while the three entangled qubits in GHZ or W state, the tripartite entanglement suddenly dies when these qubits are close enough to the horizon.
The bipartite and tripartite entanglement of a 3-qubit fermionic system when one or two subsystems accelerate are investigated. It is shown that all the one-tangles decrease as the acceleration increases. However, unlike the scalar case, here one-tangles NCI(ABI){\mathcal{N}}_{{C}_{I}({\mathrm{AB}}_{I})}NCI​(ABI​)​ and NCI(AB){\mathcal{N}}_{{C}_{I}(\mathrm{AB})}NCI​(AB)​ never reduce to zero for any acceleration. It is found that the system has only tripartite entanglement when either one or two subsystems accelerate, which means that the acceleration does not generate bipartite entanglement and does not affect the entanglement structure of the quantum states in this system. It is of interest to note that the $\ensuremath{\pi}$-tangle of the two-observer-accelerated case decreases much quicker than that of the one-observer-accelerated case and it reduces to a nonzero minimum in the infinite-acceleration limit. Thus we argue that the qutrit systems are better than qubit systems in performing quantum information processing tasks in noninertial systems.
We investigate the entanglement measures of tripartite W-State and GHZ-state in noninertial frame through the coordinate transformation between Minkowski and Rindler. First it is shown that all three qubits undergo in a uniform acceleration aaa of W-State, we find that the one-tangle, two-tangle, and πππ-tangle decrease when the acceleration parameter rrr increases, and the two-tangle cannot arrive to infinity of the acceleration. Next we show that the one qubit goes in a uniform acceleration a1a_{1}a1​ and the other two undergo in a uniform acceleration aaa of GHZ-state, we find that the two-tangle is equal to zero and NBI(AICI)=NCI(AIBI)≠NAI(BICI)N_{B_I (A_I C_I)} = N_{C_I (A_I B_I)}\neq N_{A_I (B_I C_I)}NBI​(AI​CI​)​=NCI​(AI​BI​)​=NAI​(BI​CI​)​, but one-tangle and πππ-tangle never reduce to zero for any acceleration.