We first analyse certain quantum predictions for the entangled three-photon 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 |H1|H2|H3 (all three are horizontally polarized) and the state |V1|V2|V3 (all three are vertically polarized) with none of the photons having a well-defined 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 (left-handed, right-handed). 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 two-photon 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 right-handed 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 result5,6,8.

For photon i we call these elements of reality Xi with values +1(-1) for H′(V′) polarizations and Yi with values +1(-1) for R(L); we thus obtain the relations8 Y1Y2X3 = -1, Y1X2Y3 = - 1 and X1Y2Y3 = -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 YiYi = +1, we can write X1X2X3 = (X1Y2Y3)(Y1X2Y3)(Y1Y2X3) and obtain X1X2X3 = - 1. Thus from a local realist point of view the only possible results for an xxx experiment are VVV′, HHV′, HVH′, and VHH′.

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 set-up 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 teleportation9 and entanglement swapping10. GHZ entanglement has also been inferred for nuclear spins within single molecules from NMR measurements11, though there a test of nonlocality is impossible.

Figure 1: Experimental set-up for Greenberger–Horne–Zeilinger (GHZ) tests of quantum nonlocality.
figure 1

Pairs of polarization-entangled photons28 (one photon H polarized and the other V) are generated by a short pulse of ultraviolet light ( 200 fs, λ = 394 nm). Observation of the desired GHZ correlations requires fourfold coincidence and therefore two pairs29. The photon registered at T is always H and thus its partner in b must be V. The photon reflected at the polarizing beam-splitter (PBS) in arm a is always V, being turned into equal superposition of V and H by the λ/2 plate, and its partner in arm b must be H. Thus if all four detectors register at the same time, the two photons in D1 and D2 must either both have been VV and reflected by the last PBS or HH and transmitted. The photon at D3 was therefore H or V, respectively. Both possibilities are made indistinguishable by having equal path lengths via a and b to D1 (D2) and by using narrow bandwidth filters (F ≈ 4 nm) to stretch the coherence time to about 500 fs, substantially larger than the pulse length30. This effectively erases the prior correlation information and, owing to indistinguishability, the three photons registered at D1, D2 and D3 exhibit the desired GHZ correlations predicted by the state of equation (1), where for simplicity we assume the polarizations at D3 to be defined at right angles relative to the others. Polarizers oriented at 45° and λ/4 plates in front of the detectors allow measurement of linear H′/V′ (circular R/L) polarization.

In the experiment GHZ entanglement is observed under the condition that the trigger detector T and the three GHZ detectors D1, D2 and D3 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 pairs12,13 where a violation of Bell's inequality was only achieved under the condition that both detectors used register a photon. It was often believed14,15 that such experiments could never, not even in their idealized versions, be genuine tests of local realism. However, this has been disproved16. Following the same line of reasoning, it has recently been shown17 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 local-realist 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 non-perfect detection events indicate that the GHZ argument is inapplicable. In our second strategy we therefore accommodate local-realist theories, by assuming that the non-perfect 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.

Figure 2: A typical experimental result used in the GHZ argument.
figure 2

This is the yyx experiment measuring circular polarization on photons 1 and 2 and linear polarization on the third. a, Fourfold coincidences between the trigger detector T, detectors D1 and D2 (both set to measure a right-handed polarized photon), and detector D3 (set to measure a linearly polarized H′ (lower curve) and V′ (upper curve) photon as a function of the delay between photon 1 and 2 at the final polarizing beam-splitter). We could adjust the time delay between paths a and b in Fig. 1 by translating the final polarizing beam-splitter (PBS) and using additional mirrors (not shown in Fig. 1) to ensure overlap of both beams, independent of mirror displacement. At large delay, that is, outside the region of coherent superposition, the two possibilities HHH and VVV are distinguishable and no entanglement results. In agreement with this explanation, it was observed within the experimental accuracy that for large delay the eight possible outcomes in the yyx experiment (and also the other experiments) have the same coincidence rate, whose mean value was chosen as a normalization standard. b, At zero delay maximum GHZ entanglement results; the experimentally determined fractions of RRV′ and RRH′ triples (out of the eight possible outcomes in the yyx experiment) are deduced from the measurements at zero delay. The fractions were obtained by dividing the normalized fourfold coincidences of a specific outcome by the sum of all possible outcomes in each experiment—here, the yyx experiment.

Figure 3: All outcomes observed in the yyx, yxy and xyy experiments, obtained as in Fig. 2.
figure 3

a, yyx; b, yxy; c, xyy. The experimental data show that we observe the GHZ terms predicted by quantum physics (tall bars) in a fraction of 0.85 ± 0.04 of all cases and 0.15 ± 0.02 of the spurious events (short bars) in a fraction of all cases. Within our experimental error we thus confirm the GHZ predictions for the experiments.

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 three-particle test of local realism following the GHZ argument.

Figure 4: Predictions of quantum mechanics and of local realism, and observed results for the xxx experiment.
figure 4

a, b,The maximum possible conflict arises between the predictions for quantum mechanics (a) and local realism (b) because the predicted correlations are exactly opposite. c, The experimental results clearly confirm the quantum predictions within experimental error and are in conflict with local realism.

We then investigated whether local realism could reproduce the xxx experimental results shown in Fig. 4c, if we assume that the spurious non-GHZ 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 HHH′? One possibility is to assume that triple events producing HHH′ 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 HLR and LHR), 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 Mermin18. This second analysis succeeds because our average visibility of 71% ± 4% clearly surpasses the minimum of 50% necessary for a violation of local realism18,19,20. However, we realise that, as for all existing two-particle tests of local realism, our experiment has rather low detection efficiencies. Therefore we had to invoke the fair sampling hypothesis21,22, 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 space-like separated randomly switched measurements23; extending the techniques used here to the observation of multi-photon entanglement24; observation of GHZ-correlations in massive objects like atoms25; and investigation of possible applications in quantum computation and quantum communication protocols26,27.