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arXiv:0712.3268 · Physical Review Letters

Necessary and Sufficient Detection Efficiency for the Mermin Inequalities

Adán Cabello, David Rodríguez, Ignacio Villanueva

Published Sep 16, 2008• 49 citations
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Abstract

We prove that the threshold detection efficiency for a loophole-free Bell experiment using an n-qubit Greenberger-Horne-Zeilinger state and the correlations appearing in the n-partite Mermin inequality is n/(2n-2). If the detection efficiency is equal to or lower than this value, there are local hidden variable models that can simulate all the quantum predictions. If the detection efficiency is above this value, there is no local hidden variable model that can simulate all the quantum predictions.

DOI
10.1103/physrevlett.101.120402

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We prove that the threshold detection efficiency for a loophole-free Bell experiment using an n-qubit Greenberger-Horne-Zeilinger state and the correlations appearing in the n-partite Mermin inequality is n/(2n-2). If the detection efficiency is equal to or lower than this value, there are local hidden variable models that can simulate all the quantum predictions. If the detection efficiency is above this value, there is no local hidden variable model that can simulate all the quantum predictions.
In Bell experiments, one problem is to achieve high enough photodetection to ensure that there is no possibility of describing the results via a local hidden-variable model. Using the Clauser-Horne inequality and a two-photon nonmaximally entangled state, a photodetection efficiency higher than 0.67 is necessary. Here we discuss atom-photon Bell experiments. We show that, assuming perfect detection efficiency of the atom, it is possible to perform a loophole-free atom-photon Bell experiment whenever the photodetection efficiency exceeds 0.50.
The Greenberger-Horne-Zeilinger (GHZ) paradox is subject to the detector-efficiency ``loophole'' in a similar manner as the Bell inequality. In a paper by J.-\AA{}. Larsson [Phys. Rev. A 57, R3145 (1998)], the issue is investigated for very general assumptions. Here, the assumptions of constant efficiency and independent errors will be imposed, and it will be shown that the necessary and sufficient efficiency bound is not lowered, but remains at 7575%75. An explicit local-variable model is constructed in this paper to show the necessity of this bound. In other words, it is not possible to use the independence of experimental nondetection errors to rule out local realism in the GHZ paradox below 7575%75 efficiency.
Local hidden-variable models can reproduce Bell violations if settings depend on hidden variables. Using total-variation distance, we determine the minimum dependence needed to reproduce all outcome probabilities for Pauli X and Y measurements on n-qubit Greenberger-Horne-Zeilinger states, for every n ≥ 3, visibility and coherence phase. For odd n, the larger magnitude of the real and imaginary coherence components sets the cost; for even n, the smaller can raise it. States with identical, Bell-violating Mermin data can thus have different costs.
The nonclassical properties of the noisy three-qubit Greenberger-Horne-Zeilinger (GHZ) states, ${\ensuremath{\rho}}_{v}=v|\text{GHZ}\ensuremath{\rangle}\ensuremath{\langle}\text{GHZ}|+(1\ensuremath{-}v)\mathbb{1}/8$ parameterized by the visibility $0\ensuremath{\le}v\ensuremath{\le}1$ are investigated. Based on the violation of the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$-setting Mermin inequality, ${\ensuremath{\rho}}_{v}$ is nonclassical for the parameter range $1/2<v\ensuremath{\le}1$. It has been posed whether additional settings would allow to lower the threshold visibility. Here we report on Bell inequalities giving a threshold value smaller than v=1/2v=1/2v=1/2. This rules out the possibility of a local hidden variable model in the limit of v=1/2v=1/2v=1/2. In particular, the lowest threshold visibility we found is v=0.496057v=0.496\phantom{\rule{0.16em}{0ex}}057v=0.496057, attainable with $5\ifmmode\times\else\texttimes\fi{}5\ifmmode\times\else\texttimes\fi{}5$ settings, whereas the most economical one in number of settings corresponds to $3\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}4$ settings. The method which enabled us to obtain these results, and in particular the about 10 000 tight Bell inequalities giving v<1/2v<1/2v<1/2, are also discussed in detail.
We consider Bell experiments with NNN spatially separated qubits where loss is present and restrict ourselves to two measurement settings per site. We note the Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell inequalities do not present a tight bound for the predictions of local hidden variable (LHV) theories. The Holder-type Bell inequality derived by Cavalcanti, Foster, Reid, and Drummond [E. G. Cavalcanti, C. J. Foster, M. D. Reid, and P. D. Drummond, Phys. Rev. Lett. 99, 210405 (2007)] provides a tighter bound, for high losses. We analyze the actual tight bound for the MABK inequalities, given the measure $W={\ensuremath{\prod}}_{k=1}^{N}{\ensuremath{\eta}}_{k}$ of overall detection efficiency, where ${\ensuremath{\eta}}_{k}$ is the efficiency at site kkk. Using these inequalities, we confirm that the maximally entangled Greenberger-Horne-Zeilinger state enables loophole-free falsification of LHV theories provided ${\ensuremath{\prod}}_{k=1}^{N}{\ensuremath{\eta}}_{k}>{2}^{(2\ensuremath{-}N)}$, which implies a symmetric threshold efficiency of $\ensuremath{\eta}\ensuremath{\rightarrow}50%$, as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Furthermore, loophole-free violations remain possible, even when the efficiency at some sites is reduced well below 0.50.50.5, provided N>3N>3N>3.
The Greenberger–Horne–Zeilinger (GHZ) puzzle has been used to study quantum nonlocality and provide an all-or-nothing, no-go theorem for local hidden-variable models. Recent experiments using coincident-detected entangled photons prepared in a three-particle GHZ state have been used to test quantum nonlocality, but fail to rule out local realism due to a reliance on the fair-sampling hypothesis and insufficient detection efficiency. This paper describes a physically motivated local hidden-variable model based on amplitude-threshold detection that is capable of producing similar results. Detection efficiencies for the model are within the bounds permitted for local realism and, interestingly, exhibit statistical correlations between detectors, even when the detection events are spacelike separated. Increasing the detection threshold improves agreement with the ideal quantum predictions at the cost of decreased detection efficiency. A curious emergent feature of the model is that detection efficiencies may depend upon which observables are chosen for measurement.