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arXiv:quant-ph/0201058 · Physical Review Letters

Bell-Type Inequalities to Detect True n -Body Nonseparability

Daniel Collins, Nicolas Gisin, Sandu Popescu, David Roberts, Valerio Scarani

Published Apr 12, 2002• 298 citations
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Abstract

We analyze the structure of correlations among more than two quantum systems. We introduce a classification of correlations based on the concept of nonseparability, which is different a priori from the concept of entanglement. Generalizing a result of Svetlichny [Phys. Rev. D 35, 3066 (1987)] on three-particle correlations, we find an inequality for n-particle correlations that holds under the most general separability condition and that is violated by some quantum-mechanical states.

DOI
10.1103/physrevlett.88.170405

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We analyze the structure of correlations among more than two quantum systems. We introduce a classification of correlations based on the concept of nonseparability, which is different a priori from the concept of entanglement. Generalizing a result of Svetlichny [Phys. Rev. D 35, 3066 (1987)] on three-particle correlations, we find an inequality for n-particle correlations that holds under the most general separability condition and that is violated by some quantum-mechanical states.
An information-theoretic inequality analogous to the well-known result of Bell [Physics 1, 195 (1964)] is formulated using the concept of information distance. This inequality, like Bell's, is true for all local-hidden-variable theories, but not for quantum mechanics. The metric space structure of this new inequality suggests a reformulation of familiar Bell inequalities in terms of a ``covariance distance.'' Quantum nonseparability can be demonstrated through violations of these inequalities even in cases where the correlation between two systems is extremely weak. The connection between nonseparability and complementarity is also briefly discussed in this paper.
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.
The notions of three-particle entanglement and three-particle nonlocality are discussed in the light of Svetlichny's inequality [Phys. Rev. D 35, 3066 (1987)]. It is shown that there exist sets of measurements, which can be used to prove three-particle entanglement, but which are nevertheless useless at proving three-particle nonlocality. In particular, it is shown that the quantum predictions giving a maximal violation of Mermin's three-particle Bell inequality [Phys. Rev. Lett. 65, 1838 (1990)] can be reproduced by a hybrid hidden variables model in which nonlocal correlations are present only between two of the particles. It should be possible, however, to test the existence of both three-particle entanglement and three-particle nonlocality for any given quantum state via Svetlichny's inequality.
The correlations between local measurement results are characteristic for multi-partite quantum systems. Under the condition of local realism, which assumes that properties of physical systems exist independent of being measured, and that space-like separated actions on the system cannot influence each other, these correlations are constrained by Bell-inequalities. Quantum mechanics predicts correlations which are not explainable under these two assumptions. The resulting violation of Bell inequalities was originally studied for pairs of entangled particles (Einstein et al.,1935). Meanwhile these ideas were extended to three-and more partite quantum states in particular with respect to the different kinds of multi-particle entanglement and their implications on the obtainable correlations (Mermin, 1990). Here we present a method to construct characteristic Bell inequalities for particular four-qubit states. These inequalities are a versatile tool, not only to provide evidence for non-locality, but also for the characterization of the states, with respect to detection of genuine multipartite entanglement and to state discrimination. Exemplarily we apply our approach to the symmetric four qubit Dicke-state (Dicke, 1954) with two excitations. (A symmetric N-qubit Dicke state with M excitations is the equal superposition of N-qubit pure product states having M logical 1 's and (N-M) logical 0's.) Experimentally the state was observed with photons as carrier of the qubits, using polarization encoding.