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arXiv:0707.2809 · Physical Review A

Multipartite positive-partial-transpose inequalities exponentially stronger than local reality inequalities

Koji Nagata

Published Aug 7, 2007• 3 citations
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Abstract

We show that positivity of every partial transpose of $N$-partite quantum states implies inequalities on Bell correlations which are stronger than standard Bell inequalities by a factor of ${2}^{(N\ensuremath{-}1)∕2}$. A violation of the inequality implies that the system is in a bipartite distillable entangled state. It turns out that a family of $N$-qubit bound entangled states proposed by D\"ur [Phys. Rev. Lett. 87, 230402 (2001)] violates the inequality for $N\ensuremath{\geqslant}4$.

DOI
10.1103/physreva.76.024101

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We show that positivity of every partial transpose of NNN-partite quantum states implies inequalities on Bell correlations which are stronger than standard Bell inequalities by a factor of 2(N\ensuremath−1)∕2{2}^{(N\ensuremath{-}1)∕2}2(N\ensuremath−1)∕2. A violation of the inequality implies that the system is in a bipartite distillable entangled state. It turns out that a family of NNN-qubit bound entangled states proposed by D"ur [Phys. Rev. Lett. 87, 230402 (2001)] violates the inequality for N\ensuremath⩾4N\ensuremath{\geqslant}4N\ensuremath⩾4.
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.
I derive separability inequalities for Bell correlations of observables in arbitrary pure or mixed NNN Qudit states in DND^NDN-dimensional state space. I find states (a continuum of states if D>3D>3D>3) including maximally entangled states which violate these inequalities by a factor 2N−12^{N-1}2N−1 ; local reality Bell inequalities are much weaker, their maximum violation being by a factor 2(N−1)/22^{(N-1)/2}2(N−1)/2. The separability inequalities allow tests of entanglement of unknown states using only the measured correlations .
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 $N\ensuremath{\ge}4$.
A family of Bell-type inequalities is present, which are constructed directly from the "standard" Bell inequalities involving two dichotomic observables per site. It is shown that the inequalities are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits. Remarkably, our new inequalities can provide stronger non-locality tests in a sense that the local reality inequalities are exponentially stronger than the corresponding multipartite separability inequalities. This reveals that the exponential violation of local realism by separable states is an interesting consequence of quantum fluctuation of multipartite systems.