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arXiv:quant-ph/0206031 · Physical Review A

Multipartite bound entanglement and three-setting Bell inequalities

Dagomir Kaszlikowski, L. C. Kwek, Jing‐Ling Chen, C. H. Oh

Published Nov 18, 2002• 23 citations
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Abstract

It was shown by Dur [Phys. Rev. Lett. $87,$ 230402 (2001)] that N $(N>~4)$ qubits described by a certain one-parameter family $\mathcal{F}$ of bound entangled states violate the Mermin-Klyshko inequality for $N>~8.$ In this paper we prove that the states from the family $\mathcal{F}$ violate Bell inequalities derived by \ifmmode \dot{Z}\else \.{Z}\fi{}ukowski and Kaszlikowski [Phys. Rev. A $56,$ R1682 (1997)], in which each observer measures three noncommuting sets of orthogonal projectors, for $N>~7.$ We also derive a simple one-parameter family of entanglement witnesses that detect entanglement for all the states belonging to $\mathcal{F}.$ It is possible that these entanglement witnesses could be generated by some Bell inequalities.

DOI
10.1103/physreva.66.052309

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D\"ur [Phys. Rev. Lett. 87, 230402 (2001)] constructed NNN-qubit bound entangled states which violate a Bell inequality for $N\ensuremath{\ge}8$, and his result was recently improved by showing that there exists an NNN-qubit bound entangled state violating the same Bell inequality if and only if $N\ensuremath{\ge}6$ [Phys. Rev. A 79, 032309 (2009)]. On the other hand, it has been also shown that the states which D\"ur considered violate Bell inequalities different from the inequality for $N\ensuremath{\ge}6$. In this paper, by employing different forms of Bell inequalities, in particular, a specific form of Bell inequalities with MMM settings of the measuring apparatus for sufficiently large MMM, we prove that there exists an NNN-qubit bound entangled state violating the MMM-setting Bell inequality if and only if $N\ensuremath{\ge}4$.
We study the relation between violation of Bell inequalities and distillability properties of quantum states. Recently, Dür [Phys. Rev. Lett. 87, 230402 (2001)] has shown that there are some multiparticle bound entangled states which are nonseparable and nondistillable, that violate a Bell inequality. We prove that for all the states violating this inequality there exists at least one splitting of the parties into two groups such that some pure-state entanglement can be distilled, obtaining a connection between Bell inequalities and bipartite distillable entanglement.
Bell's inequalities are important to our understanding of quantum foundations and critical to several quantum technologies. A recent work [E. Wolfe and S. F. Yelin, Phys. Rev. A 86, 012123 (2012)] derived three parametrized families of two-particle, two-setting Bell inequalities. These inequalities are important as they theoretically explore a larger volume of allowed quantum correlations over local hidden-variable models than previous results [A. Cabello, Phys. Rev. A 72, 012113 (2005)] by exploiting marginal, or single particle measurements. In this work we subject those predictions to experimental test using nonmaximally entangled photon pairs to optimize the expected violation. We find excellent agreement with the upper bounds predicted by quantum mechanics with violations of the limits imposed by local hidden-variable models as large as almost 30$\ensuremath{\sigma}$ for the optimal parameters and a significant violation over a wide range of parameters.
In a recent paper, Bancal et al. [Phys. Rev. Lett. 106, 250404 (2011)] put forward the concept of device-independent witnesses of genuine multipartite entanglement. These witnesses are capable of verifying genuine multipartite entanglement produced in a laboratory without resorting to any knowledge of the dimension of the state space or of the specific form of the measurement operators. As a by-product they found a multiparty three-setting Bell inequality which makes it possible to detect genuine nnn-partite entanglement in a noisy nnn-qubit Greenberger-Horne-Zeilinger (GHZ) state for visibilities as low as 2/32/32/3 in a device-independent way. In this paper, we generalize this inequality to an arbitrary number of settings, demonstrating a threshold visibility of $2/\ensuremath{\pi}~0.6366$ for number of settings going to infinity. We also present a pseudotelepathy Bell inequality achieving the same threshold value. We argue that our device-independent witnesses are optimal in the sense that for nnn odd the above value cannot be beaten with nnn-party-correlation Bell inequalities.
We introduce inequalities for multipartite entanglement, derived from the geometry of spin vectors. The criteria are constructed iteratively from cross and dot products between the spins of individual subsystems, each of which may have arbitrary dimension. For qubit ensembles the maximum violation for our inequalities is larger than that for the Mermin-Klyshko Bell inequalities, and the maximally violating states are different from Greenberger-Horne-Zeilinger states. Our inequalities are violated by certain bound entangled states for which no Bell-type violation has yet been found.
Local measurements acting on entangled quantum states give rise to a rich correlation structure in the multipartite scenario. We introduce a versatile technique to build families of Bell inequalities witnessing different notions of multipartite nonlocality for any number of parties. The idea behind our method is simple: a known Bell inequality satisfying certain constraints, for example the Clauser–Horne–Shimony–Holt inequality, serves as the seed to build new families of inequalities for more parties. The constructed inequalities have a clear operational meaning, capturing an essential feature of multipartite correlations: their violation implies that numerous subgroups of parties violate the inequality chosen as seed. The more multipartite nonlocal the correlations, the more subgroups can violate the seed. We illustrate our construction using different seeds and designing Bell inequalities to detect k -way nonlocal multipartite correlations, in particular, genuine multipartite nonlocal correlations—the strongest notion of multipartite nonlocality. For one of our inequalities we prove analytically that a large class of pure states that are genuine multipartite entangled (GME) exhibit genuine multipartite nonlocality for any number of parties, even for some states that are almost product. We also provide numerical evidence that this family is violated by all GME pure states of three and four qubits. Our results make us conjecture that this family of Bell inequalities can be used to prove the equivalence between genuine multipartite pure-state entanglement and nonlocality for any number of parties.