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arXiv:1201.1335 · Journal of Physics A Mathematical and Theoretical

Tripartite entanglement-dependence of tripartite non-locality in non-inertial frames

DaeKil Park

Published Oct 2, 2012• 10 citations
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Abstract

The three-tangle-dependence of $S_{max} = \max $, where $S$ 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 $π$-tangle-dependence of $S_{max}$ 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.

DOI
10.1088/1751-8113/45/41/415308

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The three-tangle-dependence of Smax=max⁡S_{max} = \max Smax​=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.
We analyze the relationship between tripartite entanglement and genuine tripartite nonlocality for three-qubit pure states in the Greenberger-Horne-Zeilinger class. We consider a family of states known as the generalized Greenberger-Horne-Zeilinger states and derive an analytical expression relating the three-tangle, which quantifies tripartite entanglement, to the Svetlichny inequality, which is a Bell-type inequality that is violated only when all three qubits are nonlocally correlated. We show that states with three-tangle less than 1/2 do not violate the Svetlichny inequality. On the other hand, a set of states known as the maximal slice states does violate the Svetlichny inequality, and exactly analogous to the two-qubit case, the amount of violation is directly related to the degree of tripartite entanglement. We discuss further interesting properties of the generalized Greenberger-Horne-Zeilinger and maximal slice states.
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 $\ensuremath{\pi}$-tangle, one of the measures of tripartite entanglement, is shown to be $\ensuremath{\pi}/6~0.524$ 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.
As stated by S. Ghose et al. [Phys. Rev. Lett. 102, 250404 (2009)], there are certain relationships between tripartite entanglement and tripartite nonlocality for three-qubit Greenberger-Horne-Zeilinger (GHZ) class states. In the present work, we have experimentally demonstrated the theoretical results of Ghose et al. by using both three-photon generalized GHZ (GGHZ) states and maximal slice (MS) states with a count of $\ensuremath{\sim}$10/s. From the data, we have verified the agreement of the experimental violation of the Svetlichny inequality with the one predicted by quantum mechanics given the reconstructed density matrix. For the MS states, it is demonstrated that the amount of violation increases linearly following the increase of the degree of tripartite entanglement. In contrast, for GGHZ states, there is a minimal value of the violation when the degree of tripartite entanglement is 1/3. Both of the results are consist with the theoretical prediction.
Entangled states of multiple qubits can violate Bell-type inequalities indicating nonlocal behavior of multiqubit quantum correlations. We analyze the relation between multipartite entanglement and genuine multipartite nonlocality, characterized by Svetlichny inequality violations, for two families of N − qubit states. We show that for the generalized Greenberger-Horne-Zeilinger family of states, Svetlichny inequality is not violated when the n − tangle is less than 1 / 2 for any even number of qubits. On the other hand, the maximal slice states always violate the Svetlichny inequality when n − tangle is nonzero, and the violation increases monotonically with tangle. Our work generalizes the relations between tangle and Svetlichny inequality violations previously derived for three qubits.
The violation of the Svetlichny's inequality (SI) [Phys. Rev. D 35, 3066 (1987)] is sufficient but not necessary for genuine tripartite nonlocal correlations. Here we quantify the relationship between tripartite entanglement and the maximum expectation value of the Svetlichny operator (which is bounded from above by the inequality) for the two inequivalent subclasses of pure three-qubit states: the Greenberger-Horne-Zeilinger (GHZ) class and the WWW class. We show that the maximum for the GHZ-class states reduces to Mermin's inequality [Phys. Rev. Lett. 65, 1838 (1990)] modulo a constant factor, and although it is a function of the three tangle and the residual concurrence, large numbers of states do not violate the inequality. We further show that by design SI is more suitable as a measure of genuine tripartite nonlocality between the three qubits in the WWW-class states, and the maximum is a certain function of the bipartite entanglement (the concurrence) of the three reduced states, and only when their sum attains a certain threshold value do they violate the inequality.
Quantum entanglement and non-locality are two special aspects of quantum correlations. The relationship between multipartite entanglement and non-locality is at the root of the foundations of quantum mechanics but there is still no general quantitative theory. In order to address this issue we analyze the relationship between tripartite non-locality and tripartite entanglement measure, called the three-tangle. We describe the states which give the extremal quantum values of a Bell-type inequality for a given value of the tripartite entanglement. Moreover, we show that such extremal states can be reached if one introduced an appropriate order induced by the three-π entanglement measure. Finally, we derive an analytical expression relating tripartite entanglement to the maximal violations of the Bell-type inequalities.