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Physical Review A

Asymptotic relation between Bell-inequality violations and entanglement distillability

Younghun Kwon

Published Nov 30, 2010• 1 citation
Publisher

Abstract

We investigate the asymptotic relation between violations of the Mermin-Belinskii-Klyshko inequality and the entanglement distillability of multipartite entangled states, as the number of parties increases. We in particular consider noisy multiqubit GHZ and so-called D\"ur states in the Mermin-Belinskii-Klyshko inequality, and show that, in the asymptotic limit of the number of parties, the violation of the inequality implies the distillability in almost all bipartitions.

DOI
10.1103/physreva.82.054104

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We investigate the asymptotic relation between violations of the Mermin-Belinskii-Klyshko inequality and the entanglement distillability of multipartite entangled states, as the number of parties increases. We in particular consider noisy multiqubit GHZ and so-called D"ur states in the Mermin-Belinskii-Klyshko inequality, and show that, in the asymptotic limit of the number of parties, the violation of the inequality implies the distillability in almost all bipartitions.
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.
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.
Multiparty quantum cryptography based on distributed entanglement will find its natural application in the upcoming quantum networks.The security of many multipartite device-independent (DI) protocols, such as DI conference-key agreement, relies on bounding the von Neumann entropy of the parties' outcomes conditioned on the eavesdropper's information, given the violation of a multipartite Bell inequality.We consider three parties testing the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality and certify the privacy of their outcomes by bounding the conditional entropy of a single party's outcome and the joint conditional entropy of two parties' outcomes.From the former bound, we show that genuine multipartite entanglement is necessary to certify the privacy of a party's outcome, while the latter significantly improves previous results.We obtain the entropy bounds thanks to two general results of independent interest.The first one drastically simplifies the quantum setup of an N -partite Bell scenario.The second one provides an upper bound on the violation of the MABK inequality by an arbitrary N -qubit state, as a function of the state's parameters.
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.
It was shown by Dur [Phys. Rev. Lett. 87,87,87, 230402 (2001)] that N (N> 4)(N>~4)(N> 4) qubits described by a certain one-parameter family F\mathcal{F}F of bound entangled states violate the Mermin-Klyshko inequality for N> 8.N>~8.N> 8. In this paper we prove that the states from the family F\mathcal{F}F violate Bell inequalities derived by \ifmmode \dot{Z}\else \.{Z}\fi{}ukowski and Kaszlikowski [Phys. Rev. A 56,56,56, R1682 (1997)], in which each observer measures three noncommuting sets of orthogonal projectors, for N> 7.N>~7.N> 7. We also derive a simple one-parameter family of entanglement witnesses that detect entanglement for all the states belonging to F.\mathcal{F}.F. It is possible that these entanglement witnesses could be generated by some Bell inequalities.