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arXiv:quant-ph/0510019 · Physical Review A

Detection of N -particle entanglement with generalized Bell inequalities

Wiesław Laskowski, Marek Żukowski

Published Dec 30, 2005• 24 citations
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Abstract

We show that the generalized Bell-type inequality, explicitly involving rotational symmetry of physical laws, is very efficient in distinguishing between true $N$-particle quantum correlations and correlations involving less particles. This applies to various types of generalized partial separabilities. We also give a rigorous proof that the Bell inequalities are maximally violated by the GHZ states, and find a very handy description of the $N$-qubit correlation function.

DOI
10.1103/physreva.72.062112

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We show that the generalized Bell-type inequality, explicitly involving rotational symmetry of physical laws, is very efficient in distinguishing between true NNN-particle quantum correlations and correlations involving less particles. This applies to various types of generalized partial separabilities. We also give a rigorous proof that the Bell inequalities are maximally violated by the GHZ states, and find a very handy description of the NNN-qubit correlation function.
We derive N-particle Bell-type inequalities under the assumption of partial separability, i.e., that the N-particle system is composed of subsystems which may be correlated in any way (e.g., entangled) but which are uncorrelated with respect to each other. These inequalities provide, upon violation, experi-mentally accessible sufficient conditions for full N-particle entanglement, i.e., for N-particle entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. The inequalities are shown to be maximally violated by the N-particle Greenberger-Horne-Zeilinger states.
Most of known multipartite Bell inequalities involve correlation functions for all subsystems. They are useless for entangled states without such correlations. We give a method of derivation of families of Bell inequalities for NNN parties, which involve, e.g., only $(N\ensuremath{-}1)$-partite correlations, but still are able to detect proper NNN-partite entanglement. We present an inequality which reveals five-partite entanglement despite using only four-partite correlations. Classes of inequalities introduced here can be put into a handy form of a single nonlinear inequality. An example is given of an NNN-qubit state, which strongly violates such an inequality, despite having no NNN-qubit correlations. This surprising property might be of potential value for quantum information tasks.
We derive a set of Bell-type inequalities for arbitrarily-high-dimensional systems based on the assumption of partial separability in the hybrid local-nonlocal hidden-variable model. Partially entangled states would not violate the inequalities, and thus, upon violation, these Bell-type inequalities are sufficient conditions to detect the full NNN-particle entanglement and invalidity of the hybrid local-nonlocal hidden-variable description.
We develop a necessary condition of quantum correlation. It is utilized to construct a ddd-level bipartite Bell-type inequality which is strongly resistant to noise and requires only analyses of O(d)O(d)O(d) measurement outcomes compared to the previous result O(d2)O({d}^{2})O(d2). Remarkably, a connection between the arbitrary high-dimensional bipartite Bell-type inequality and entanglement witnesses is found. Through the necessary condition of quantum correlation, we propose that the witness operators to detect truly multipartite entanglement for a generalized Greenberger-Horne-Zeilinger (GHZ) state with two local measurement settings and a four-qubit singlet state with three settings. Moreover, we also propose a robust entanglement witness to detect a four-level tripartite GHZ state with only two local measurement settings.