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arXiv:quant-ph/0108029 · Physical Review Letters

Distillability, Bell Inequalities, and Multiparticle Bound Entanglement

Antonio Acín

Published Dec 20, 2001• 56 citations
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Abstract

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.

DOI
10.1103/physrevlett.88.027901

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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 study the explicit relation between violation of Bell inequalities and bipartite distillability of multiqubit states. It has been shown that even though for $N\ensuremath{\ge}8$ there exist NNN-qubit bound entangled states which violates a Bell inequality [W. D\"ur, Rev. Lett. 87, 230402 (2001)], for all the states violating the inequality there exists at least one splitting of the parties into two groups such that pure-state entanglement can be distilled [A. Ac\'{\i}n, Rev. Lett. 88, 027901 (2002)]. We here prove that for all NNN-qubit states violating the inequality the number of distillable bipartite splits increases exponentially with NNN, and hence the probability that a randomly chosen bipartite split is distillable approaches 1 exponentially with NNN, as NNN tends to infinity. We also show that there exists at least one NNN-qubit bound entangled state violating the inequality if and only if $N\ensuremath{\ge}6$.
A multipartite quantum state violates a Bell inequality asymptotically if, after jointly processing by general local operations an arbitrarily large number of copies of it, the result violates the inequality. In the bipartite case we show that asymptotic violation of the Clauser-Horne-Shimony-Holt inequality is equivalent to distillability. Hence, bound entangled states do not violate it. In the multipartite case we consider the complete set of full-correlation Bell inequalities with two dichotomic observables per site. We show that asymptotic violation of any of these inequalities by a multipartite state implies that pure-state entanglement can be distilled from it, although the corresponding distillation protocol may require that some of the parties join into several groups. We also obtain the extreme points of the set of distributions generated by measuring N quantum systems with two dichotomic observables per site.
The relation between Bell's inequalities with two two-outcome measurements per site and distillability is analyzed in systems of an arbitrary number of quantum bits. We observe that the violation of any of these inequalities by a quantum state implies that pure-state entanglement can be distilled from it. The corresponding distillation protocol may require that some of the parties join into several groups. We show that there exists a link between the amount of the Bell inequality violation and the size of the groups they have to form for distillation. Thus, a strong violation is always sufficient for full N-partite distillability. This result also allows for a security proof of multipartite quantum key distribution protocols.
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 $N\ensuremath{\ge}4$.