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arXiv:0904.4818 · Physical Review A

Multipartite bound entanglement and multisetting Bell inequalities

Dong Pyo Chi, Kabgyun Jeong, Tae‐Wan Kim, Kyungjin Lee, Soojoon Lee

Published Sep 28, 2009• 3 citations

Abstract

D\"ur [Phys. Rev. Lett. 87, 230402 (2001)] constructed $N$-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 $N$-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 $M$ settings of the measuring apparatus for sufficiently large $M$, we prove that there exists an $N$-qubit bound entangled state violating the $M$-setting Bell inequality if and only if $N\ensuremath{\ge}4$.

DOI: 10.1103/physreva.80.032330

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D"ur [Phys. Rev. Lett. 87, 230402 (2001)] constructed

N

-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

N

-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

M

settings of the measuring apparatus for sufficiently large

M

, we prove that there exists an

N

-qubit bound entangled state violating the

M

-setting Bell inequality if and only if

N\ensuremath{\ge}4

.

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.

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

n

-partite entanglement in a noisy

n

-qubit Greenberger-Horne-Zeilinger (GHZ) state for visibilities as low as

2/3

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

n

odd the above value cannot be beaten with

n

-party-correlation Bell inequalities.

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.

We consider a Bell inequality for a continuous range of settings of the apparatus at each site. This ``functional'' Bell inequality gives a better range of violation for generalized Greenberger, Horne, and Zeilinger states. Also a family of N-qubit bound entangled states violate this inequality for

N>5.

It is shown that Smolin four-qubit bound entangled states [J. A. Smolin, Phys. Rev. A 63, 032306 (2001)] can maximally violate the simple two-setting Bell inequality similar to the standard Clauser-Horne-Shimony-Holt (CHSH) inequality. The simplicity of the setting and the robustness of the entanglement make it promising for current experimental technology. On the other hand, the entanglement does not allow for secure key distillation, so neither entanglement nor maximal violation of Bell inequalities implies directly the presence of a quantum secure key. As a result, one concludes that two tasks---reducing of communication complexity and cryptography---are not (even qualitatively) equivalent in a quantum multipartite scenario.