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arXiv:quant-ph/0505100 · Physical Review A

Addendum to “Sufficient conditions for three-particle entanglement and their tests in recent experiments”

Géza Tóth, Otfried Gühne, Michael Seevinck, Jos Uffink

Published Jul 8, 2005• 22 citations
arXivHTMLPDF

Abstract

A recent paper [] presented a bound for the three-qubit Mermin inequality such that the violation of this bound indicates genuine three-qubit entanglement. We show that this bound can be improved for a specific choice of observables. In particular, if spin observables corresponding to orthogonal directions are measured at the qubits (e.g., $X$ and $Y$ spin coordinates), then the bound is the same as the bound for states with a local hidden variable model. As a consequence, it can straightforwardly be shown that in the experiment described by ], genuine three-qubit entanglement was detected.

DOI
10.1103/physreva.72.014101

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•Jan 1, 2019•0 citations
A recent paper [] presented a bound for the three-qubit Mermin inequality such that the violation of this bound indicates genuine three-qubit entanglement. We show that this bound can be improved for a specific choice of observables. In particular, if spin observables corresponding to orthogonal directions are measured at the qubits (e.g., XXX and YYY spin coordinates), then the bound is the same as the bound for states with a local hidden variable model. As a consequence, it can straightforwardly be shown that in the experiment described by ], genuine three-qubit entanglement was detected.
It was shown by Dur [Phys. Rev. Lett. 87,87,87, 230402 (2001)] that N (N> 4)(N>~4)(N> 4) qubits described by a certain one-parameter family F\mathcal{F}F of bound entangled states violate the Mermin-Klyshko inequality for N> 8.N>~8.N> 8. In this paper we prove that the states from the family F\mathcal{F}F violate Bell inequalities derived by \ifmmode \dot{Z}\else \.{Z}\fi{}ukowski and Kaszlikowski [Phys. Rev. A 56,56,56, R1682 (1997)], in which each observer measures three noncommuting sets of orthogonal projectors, for N> 7.N>~7.N> 7. We also derive a simple one-parameter family of entanglement witnesses that detect entanglement for all the states belonging to F.\mathcal{F}.F. It is possible that these entanglement witnesses could be generated by some Bell inequalities.
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.
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.
It is already known that one can always find a set of measurements on any two-qubit entangled state that will lead to a violation of the CHSH inequality. We provide an explicit state in terms of the angle between Alice's choice of measurements and the angle between Bob's choice of measurements, such that the CHSH inequality is always violated provided Alice's or Bob's choice of inputs are not collinear. We prove that inequalities with a corresponding Bell operator written as a linear combination of tensor products of Pauli matrices, excluding the identity, will generate the most nonlocal correlations using maximally entangled states in our experiment. From this result and a proposition from Horodecki et. al., we are able to construct the state that generates these optimal correlations. To achieve this state in a lab, one party must rotate their qubit using the orthogonal operation we provide and also rotate their Bloch sphere such that all their measurements lie in the same plane. We provide a comprehensive study of how Bell inequalities change when experiments introduce error via imperfect detection efficiency. The original cases of perfect efficiency are covered first and then a more realistic approach, when inefficient detectors are used, will follow. It is shown that less entanglement is needed to demonstrate more nonlocality in some Clasure-Horne-Shimony-Holt (CHSH) experiments when detector inefficiency is introduced. An example of this is shown for any given specific set of measurements in the CHSH Bell experiment. This occurs when one party has a detector of efficiency for each choice of input and the other party makes projective measurements. The efficiency can be pushed down to fifty percent while still violating the CHSH inequality, and for the experimental set-up illustrated, there is more nonlocality with less entanglement. Furthermore, it is shown that if the first party has an imperfect detector for only one choice of inputs rather than two, the efficiency can be brought down arbitrarily close to zero percent while still violating the CHSH inequality. Historically, nonlocality and entanglement were viewed as two equivalent resources, but recently this equality has come under question; these results further support this fundamental difference. Further more, we introduce Mermin's game in the case of relaxed conditions. The original constraints were that when the detectors in separate labs of a two-qubit experiment are in the same setting, then the results should be the same. We require that the outcomes are the same at least part of the time, given by some epsilon variable. Initially, one could find a maximum violation of one-fourth by allowing to parties to share the singlet state and have measurement settings one-hundred and twenty degrees apart from one another. By allowing some epsilon error in the perfect correlations regime, one can find a maximum violation of minus one plus the square root of two using the singlet state and measurement inputs that achieve Tsirelson's bound for the CHSH experiment. The reason is that we show Mermin's inequality is technically the CHSH inequality "in disguise", but with using constraints the CHSH experiment does not use. We derive Mermin's inequality under new conditions and give the projective measurements needed to violate maximally.