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arXiv:quant-ph/0102024 · Physical Review A

All-multipartite Bell-correlation inequalities for two dichotomic observables per site

Reinhard F. Werner, Michael M. Wolf

Published Aug 17, 2001• 497 citations
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Abstract

We construct a set of ${2}^{{2}^{n}}$ independent Bell-correlation inequalities for n-partite systems with two dichotomic observables each, which is complete in the sense that the inequalities are satisfied if and only if the correlations considered allow a local classical model. All these inequalities can be summarized in a single, albeit nonlinear inequality. We show that quantum correlations satisfy this condition provided the state has positive partial transpose with respect to any grouping of the n systems into two subsystems. We also provide an efficient algorithm for finding the maximal quantum-mechanical violation of each inequality, and show that the maximum is always attained for the generalized GHZ state.

DOI
10.1103/physreva.64.032112

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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.
Consider a scenario where NNN separated quantum systems are measured, each with one among two possible dichotomic observables. Assume that the NNN events corresponding to the choice and performance of the measurement in each site are space-like separated. In the present paper, the correlations among the measurement outcomes that arise in this scenario are analyzed. It is shown that all extreme points of this convex set are attainable by measuring NNN-qubit pure-states with projective observables. This result allows the possibility of using known algorithms in order decide whether some correlations are achievable within quantum mechanics or not. It is also proven that if an NNN-partite state ρρρ violates a given Bell inequality, then, ρρρ can be transformed by stochastic local operations into an NNN-qubit state that violates the same Bell inequality by an equal or larger amount.
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 ${2}^{(N\ensuremath{-}1)∕2}$. 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 $N\ensuremath{\geqslant}4$.
We introduce a single general representation incorporating in a unique manner all Bell-type inequalities for a multipartite correlation scenario with an arbitrary number of settings and any spectral type of outcomes at each site. Specifying this general representation for correlation functions, we prove that the form of any correlation Bell-type inequality does not depend on spectral types of outcomes, in particular, on their numbers at different sites, and is determined only by extremal values of outcomes at each site. We also specify the general form of bounds in Bell-type inequalities on joint probabilities. Our approach to the derivation of Bell-type inequalities is universal, concise and can be applied to a multipartite correlation experiment with outcomes of any spectral type, discrete or continuous. We, in particular, prove that, for an N -partite quantum state, possibly, infinite dimensional, admitting the -setting LHV description, the Mermin–Klyshko inequality holds for any two bounded quantum observables per site, not necessarily dichotomic.
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.