Abstract
Bell's theorem^{1} states that certain statistical correlations predicted by quantum physics for measurements on twoparticle systems cannot be understood within a realistic picture based on local properties of each individual particle—even if the two particles are separated by large distances. Einstein, Podolsky and Rosen first recognized^{2} the fundamental significance of these quantum correlations (termed ‘entanglement’ by Schrödinger^{3}) and the twoparticle quantum predictions have found everincreasing experimental support^{4}. A more striking conflict between quantum mechanical and local realistic predictions (for perfect correlations) has been discovered^{5,6}; but experimental verification has been difficult, as it requires entanglement between at least three particles. Here we report experimental confirmation of this conflict, using our recently developed method^{7} to observe threephoton entanglement, or ‘Greenberger–Horne–Zeilinger’ (GHZ) states. The results of three specific experiments, involving measurements of polarization correlations between three photons, lead to predictions for a fourth experiment; quantum physical predictions are mutually contradictory with expectations based on local realism. We find the results of the fourth experiment to be in agreement with the quantum prediction and in striking conflict with local realism.
Main
We first analyse certain quantum predictions for the entangled threephoton GHZ state:
where H and V denote horizontal and vertical linear polarizations respectively. This state indicates that the three photons are in a quantum superposition of the state H〉_{1}H〉_{2}H〉_{3} (all three are horizontally polarized) and the state V〉_{1}V〉_{2}V〉_{3} (all three are vertically polarized) with none of the photons having a welldefined state on its own.
We consider now measurements of linear polarization along directions H′/V′ rotated by 45° with respect to the original H/V directions, or of circular polarization L/R (lefthanded, righthanded). These new polarizations can be expressed in terms of the original ones as:
For convenience we will refer to a measurement of H′/V′ linear polarization as an x measurement and one of R/L circular polarization as a y measurement.
Representing the GHZ state (equation (1)) in the new states by using equations (2) and (3), one obtains the quantum predictions for measurements of these new polarizations. For example, for the case of measurement of circular polarization on, say, both photon 1 and 2, and linear polarization H′/V′ on photon 3, denoted as a yyx experiment, the state may be expressed as:
This expression implies, first, that any specific result obtained in any individual or in any twophoton joint measurement is maximally random. For example, photon 1 will exhibit polarization R or L with the same probability of 50%, or photons 1 and 2 will exhibit polarizations RL, LR, RR or LL with the same probability of 25%. Second, given any two results of measurements on any two photons, we can predict with certainty the result of the corresponding measurement performed on the third photon. For example, suppose photons 1 and 2 both exhibit righthanded R circular polarization. Then by the third term in equation (4), photon 3 will definitely be V′ polarized.
By cyclic permutation, we can obtain analogous expressions for any experiment measuring circular polarization on two photons and H′/V′ linear polarization on the remaining one. Thus, in every one of the three yyx, yxy, and xyy experiments, any individual measurement result—both for circular polarization and for linear H′/V′ polarization—can be predicted with certainty for every photon given the corresponding measurement results of the other two.
Now we will analyse the implications for local realism. As these predictions are independent both of the spatial separation and of the relative time order of the three measurements, we consider them performed simultaneously in a given reference frame—say, for conceptual simplicity, in the reference frame of the source. Then, as Einstein locality implies that no information can travel faster than the speed of light, this requires any specific measurement result obtained for any photon never to depend on which specific measurements are performed simultaneously on the other two nor on their outcome. The only way then for local realism to explain the perfect correlations predicted by equation (4) is to assume that each photon carries elements of reality for both x and y measurements that determine the specific individual measurement result^{5,6,8}.
For photon i we call these elements of reality X_{i} with values +1(1) for H′(V′) polarizations and Y_{i} with values +1(1) for R(L); we thus obtain the relations^{8} Y_{1}Y_{2}X_{3} = 1, Y_{1}X_{2}Y_{3} =  1 and X_{1}Y_{2}Y_{3} = 1 , in order to be able to reproduce the quantum predictions of equation (4) and its permutations.
We now consider a fourth experiment measuring linear H′/V′ polarization on all three photons, that is, an xxx experiment. We investigate the possible outcomes that will be predicted by local realism based on the elements of reality introduced to explain the earlier yyx, yxy and xyy experiments.
Because of Einstein locality any specific measurement for x must be independent of whether an x or y measurement is performed on the other photon. As Y_{i}Y_{i} = +1, we can write X_{1}X_{2}X_{3} = (X_{1}Y_{2}Y_{3})(Y_{1}X_{2}Y_{3})(Y_{1}Y_{2}X_{3}) and obtain X_{1}X_{2}X_{3} =  1. Thus from a local realist point of view the only possible results for an xxx experiment are V′V′V′, H′H′V′, H′V′H′, and V′H′H′.
How do these predictions of local realism for an xxx experiment compare with those of quantum mechanics? If we express the state given in equation (1) in terms of H′/V′ polarization using equation (2) we obtain:
Thus we conclude that the local realistic model predicts none of the terms occurring in the quantum prediction and vice versa. This means that whenever local realism predicts that a specific result will definitely occur for a measurement on one of the photons based on the results for the other two, quantum physics definitely predicts the opposite result. For example, if two photons are both found to be H′ polarized, local realism predicts the third photon to carry polarization V′ while quantum physics predicts H′. This is the GHZ contradiction between local realism and quantum physics.
In the case of Bell's inequalities for two photons, the conflict between local realism and quantum physics arises for statistical predictions of the theory; but for three entangled particles the conflict arises even for the definite predictions. Statistics now only results from the inevitable experimental limitations occurring in any and every experiment, even in classical physics.
A diagram of our experimental setup is given in Fig. 1. The method to produce GHZ entanglement for three spatially separated photons is a further development of the techniques that have been used in our previous experiments on quantum teleportation^{9} and entanglement swapping^{10}. GHZ entanglement has also been inferred for nuclear spins within single molecules from NMR measurements^{11}, though there a test of nonlocality is impossible.
In the experiment GHZ entanglement is observed under the condition that the trigger detector T and the three GHZ detectors D_{1}, D_{2} and D_{3} all actually register a photon. As there are other detection events where fewer detectors fire, this condition might raise doubts about whether such a source can be used to test local realism. The same question arose earlier for certain experiments involving photon pairs^{12,13} where a violation of Bell's inequality was only achieved under the condition that both detectors used register a photon. It was often believed^{14,15} that such experiments could never, not even in their idealized versions, be genuine tests of local realism. However, this has been disproved^{16}. Following the same line of reasoning, it has recently been shown^{17} that our procedure permits a valid GHZ test of local realism. In essence, both the Bell and the GHZ arguments exhibit a conflict between detection events and the ideas of local realism.
As explained above, demonstration of the conflict between local realism and quantum mechanics for GHZ entanglement consists of four experiments, each with three spatially separated polarization measurements. First, we perform yyx, yxy and xyy experiments. If the results obtained are in agreement with the predictions for a GHZ state, then for an xxx experiment, our consequent expectations using a localrealist theory are exactly the opposite of our expectations using quantum physics.
For each experiment we have eight possible outcomes of which ideally four should never occur. Obviously, no experiment either in classical physics or in quantum mechanics can ever be perfect; therefore, even the outcomes which should not occur will occur with some small probability in any real experiment. The question is how to deal with such spurious events in view of the fact that the original GHZ argument is based on perfect correlations.
We follow two independent possible strategies. In the first strategy we simply compare our experimental results with the predictions both of quantum mechanics and of a local realist theory for GHZ correlations, and assume that the spurious events are attributable to experimental imperfection that is not correlated to the elements of reality a photon carries. A local realist might argue against that approach and suggest that the nonperfect detection events indicate that the GHZ argument is inapplicable. In our second strategy we therefore accommodate localrealist theories, by assuming that the nonperfect events in the first three experiments indicate a set of elements of reality which are in conflict with quantum mechanics. We then compare the local realist prediction for the xxx experiment obtained under that assumption with the experimental results.
The observed results for two possible outcomes in a yyx experiment are shown in Fig. 2. The six remaining possible outcomes of a yyx experiment have also been measured in the same way and likewise in the yxy and xyy experiments. For all three experiments this results in 24 possible outcomes whose individual fractions thus obtained are shown in Fig. 3.
Adopting our first strategy, we assume that the spurious events are attributable to unavoidable experimental errors; within the experimental accuracy, we conclude that the desired correlations in these experiments confirm the quantum predictions for GHZ entanglement. Thus we compare the predictions of quantum mechanics and local realism with the results of an xxx experiment (Fig. 4) and we observe that, again within experimental error, the triple coincidences predicted by quantum mechanics occur and not those predicted by local realism. In this sense, we believe that we have experimentally realized the first threeparticle test of local realism following the GHZ argument.
We then investigated whether local realism could reproduce the xxx experimental results shown in Fig. 4c, if we assume that the spurious nonGHZ events in the other three experiments (Fig. 3) actually indicate a deviation from quantum physics. To answer this we adopt our second strategy and consider the best prediction a local realistic theory could obtain using these spurious terms. How, for example, could a local realist obtain the quantum prediction H′H′H′? One possibility is to assume that triple events producing H′H′H′ would be described by a specific set of local hidden variables such that they would give events that are in agreement with quantum theory in both an xyy and a yxy experiment (for example, the results H′LR and LH′R), but give a spurious event for a yyx experiment (in this case, LLH′). In this way any local realistic prediction for an event predicted by quantum theory in our xxx experiment will use at least one spurious event in the earlier measurements together with two correct ones. Therefore, the fraction of correct events in the xxx experiment can at most be equal to the sum of the fractions of all spurious events in the yyx, yxy, and xyy experiments, that is, 0.45 ± 0.03. However, we experimentally observed such terms with a fraction of 0.87 ± 0.04 (Fig. 4c), which violates the local realistic expectation by more than eight standard deviations.
Our latter argument is equivalent to adopting an inequality of the kind first proposed by Mermin^{18}. This second analysis succeeds because our average visibility of 71% ± 4% clearly surpasses the minimum of 50% necessary for a violation of local realism^{18,19,20}. However, we realise that, as for all existing twoparticle tests of local realism, our experiment has rather low detection efficiencies. Therefore we had to invoke the fair sampling hypothesis^{21,22}, where it is assumed that the registered events are a faithful representative of the whole.
Possible future experiments could include: further study GHZ correlations over large distances with spacelike separated randomly switched measurements^{23}; extending the techniques used here to the observation of multiphoton entanglement^{24}; observation of GHZcorrelations in massive objects like atoms^{25}; and investigation of possible applications in quantum computation and quantum communication protocols^{26,27}.
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Acknowledgements
We thank D. M. Greenberger, M. A. Horne and M. Zukowski for useful discussions. This work was supported by the Austrian Fonds zur Förderung der Wissenschaftlichen Forschung, the Austrian Academy of Sciences and the Training and Mobility of Researchers programme of the European Union.
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Pan, JW., Bouwmeester, D., Daniell, M. et al. Experimental test of quantum nonlocality in threephoton Greenberger–Horne–Zeilinger entanglement. Nature 403, 515–519 (2000). https://doi.org/10.1038/35000514
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DOI: https://doi.org/10.1038/35000514
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