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arXiv:quant-ph/9905095 · Physical Review A

Two-particle entanglement as a property of three-particle entangled states

Jose Luis Cereceda

Published Sep 1, 1997• 2 citations
arXivHTMLPDF

Abstract

In a recent article [Phys. Rev. A 54, 1793 (1996)] Krenn and Zeilinger investigated the conditioned two-particle correlations for the subensemble of data obtained by selecting the results of the spin measurements by two observers 1 and 2 with respect to the result found in the corresponding measurement by a third observer. In this paper we write out explicitly the condition required in order for the selected results of observers 1 and 2 to violate Bell's inequality for general measurement directions $e{\ensuremath{\rightarrow}}_{1},$ $e{\ensuremath{\rightarrow}}_{1}^{\ensuremath{'}},$ $e{\ensuremath{\rightarrow}}_{2},$ $e{\ensuremath{\rightarrow}}_{2}^{\ensuremath{'}},$ and $e{\ensuremath{\rightarrow}}_{3}.$ It is shown that there are infinitely many sets of directions giving the maximum level of violation. Further, we extend the analysis by the authors to the class of triorthogonal states $|\ensuremath{\Psi}〉{=c}_{1}|{z}_{1}〉|{z}_{2}〉|{z}_{3}〉{+c}_{2}|\ensuremath{-}{z}_{1}〉|\ensuremath{-}{z}_{2}〉|\ensuremath{-}{z}_{3}〉.$ It is found that a maximal violation of Bell's inequality occurs provided the corresponding three-particle state yields a direct (``all or nothing'') nonlocality contradiction.

DOI
10.1103/physreva.56.1733

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In a recent article [Phys. Rev. A 54, 1793 (1996)] Krenn and Zeilinger investigated the conditioned two-particle correlations for the subensemble of data obtained by selecting the results of the spin measurements by two observers 1 and 2 with respect to the result found in the corresponding measurement by a third observer. In this paper we write out explicitly the condition required in order for the selected results of observers 1 and 2 to violate Bell's inequality for general measurement directions e\ensuremath→1,e{\ensuremath{\rightarrow}}_{1},e\ensuremath→1​, e\ensuremath→1\ensuremath′,e{\ensuremath{\rightarrow}}_{1}^{\ensuremath{'}},e\ensuremath→1\ensuremath′​, e\ensuremath→2,e{\ensuremath{\rightarrow}}_{2},e\ensuremath→2​, e\ensuremath→2\ensuremath′,e{\ensuremath{\rightarrow}}_{2}^{\ensuremath{'}},e\ensuremath→2\ensuremath′​, and e\ensuremath→3.e{\ensuremath{\rightarrow}}_{3}.e\ensuremath→3​. It is shown that there are infinitely many sets of directions giving the maximum level of violation. Further, we extend the analysis by the authors to the class of triorthogonal states ∣\ensuremathΨ〉=c1∣z1〉∣z2〉∣z3〉+c2∣\ensuremath−z1〉∣\ensuremath−z2〉∣\ensuremath−z3〉.|\ensuremath{\Psi}〉{=c}_{1}|{z}_{1}〉|{z}_{2}〉|{z}_{3}〉{+c}_{2}|\ensuremath{-}{z}_{1}〉|\ensuremath{-}{z}_{2}〉|\ensuremath{-}{z}_{3}〉.∣\ensuremathΨ〉=c1​∣z1​〉∣z2​〉∣z3​〉+c2​∣\ensuremath−z1​〉∣\ensuremath−z2​〉∣\ensuremath−z3​〉. It is found that a maximal violation of Bell's inequality occurs provided the corresponding three-particle state yields a direct (``all or nothing'') nonlocality contradiction.
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 original Bell inequality was obtained in a statistical derivation assuming three mutually cross-correlated random variables (four in the later version). Given that observations destroy the particles, the physical realization of three variables from an experiment producing two particles per trial requires two separate trial runs. One assumed variable value (for particle 1) occurs at a fixed instrument setting in both trial runs while a second variable (for particle 2) occurs at alternative instrument settings in the two trial runs. Given that measurements on the two particles occurring in each trial are themselves correlated, measurements from independent realizations at mutually exclusive settings on particle 2 are conditionally independent, i.e., conditionally dependent on particle 1, through probability. This situation is realized from variables defined by Bell using entangled particle pairs. Two correlations have the form that Bell computed from entanglement, but a third correlation from conditionally independent measurements has a different form. When the correlations are computed using quantum probabilities, the Bell inequality is satisfied without recourse to assumptions of non-locality, or non-reality.
In a recent article [Phys. Rev. A 53, R1927 (1996)] Wu et al. give a proof of nonlocality for the maximally entangled state of two particles without using inequalities. Here we point out that in order for their argument to go through it is necessary to consider a minimum total of six dimensions in Hilbert space. Indeed, the above argument is found to fail whenever the dimensionality of the product Hilbert space is reduced to four. The explanation for this failure is provided by the fact according to which no direct contradiction between the quantum-mechanical and locally realistic predictions can arise for any fraction of an ensemble of 2\ifmmode\times\else\texttimes\fi{}2 systems configured in the maximally entangled state.
Bell's inequalities are important to our understanding of quantum foundations and critical to several quantum technologies. A recent work [E. Wolfe and S. F. Yelin, Phys. Rev. A 86, 012123 (2012)] derived three parametrized families of two-particle, two-setting Bell inequalities. These inequalities are important as they theoretically explore a larger volume of allowed quantum correlations over local hidden-variable models than previous results [A. Cabello, Phys. Rev. A 72, 012113 (2005)] by exploiting marginal, or single particle measurements. In this work we subject those predictions to experimental test using nonmaximally entangled photon pairs to optimize the expected violation. We find excellent agreement with the upper bounds predicted by quantum mechanics with violations of the limits imposed by local hidden-variable models as large as almost 30$\ensuremath{\sigma}$ for the optimal parameters and a significant violation over a wide range of parameters.
The original formula of Bell inequality (BI) in terms of two-spin singlet has to be modified for the entangled-state with parallel spin polarization. Based on classical statistics of the particle-number correlation, we prove in this paper an extended BI, which is valid for two-spin entangled states with both parallel and antiparallel polarizations. The BI and its violation can be formulated in a unified formalism based on the spin coherent-state quantum probability statistics with the state-density operator, which is separated to the local and non-local parts. The local part gives rise to the BI, while the violation is a direct result of the non-local quantum interference between two components of entangled state. The Bell measuring outcome correlation denoted by PBP_{B}PB​ is always less than or at most equal to one for the local realistic model (PBlc≤1P_{B}^{lc}\leq1PBlc​≤1) regardless of the specific superposition coefficients of entangled state. Including the non-local quantum interference the maximum violation of BI is found as PBmax⁡P_{B}^{\max}PBmax​ =2=2=2, which, however depends on state parameters and three measuring directions as well. Our result is suitable for entangled photon pairs.
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.