Abstract
Quantum theory predicts and experiments confirm that nature can produce correlations between distant events that are nonlocal in the sense of violating a Bell inequality^{1}. Nevertheless, Bell’s strong sentence ‘Correlations cry out for explanations’ (ref. 2) remains relevant. The maturing of quantum information science and the discovery of the power of nonlocal correlations, for example for cryptographic key distribution beyond the standard quantum key distribution schemes^{3,4,5}, strengthen Bell’s wish and make it even more timely. In 2003, Leggett proposed an alternative model for nonlocal correlations^{6} that he proved to be incompatible with quantum predictions. We present here a new approach to this model, along with new inequalities for testing it. These inequalities can be derived in a very simple way, assuming only the nonnegativity of probability distributions; they are also stronger than previously published and experimentally tested Leggetttype inequalities^{6,7,8,9}. The simplest of the new inequalities is experimentally violated. Then we go beyond Leggett’s model, and show that we cannot ascribe even partially defined individual properties to the components of a maximally entangled pair.
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Main
Formally, a correlation is a conditional probability distribution P(α,βa,b), where α,β are the outcomes observed by two partners, Alice and Bob, when they make measurements labelled by a and b, respectively. On the abstract level, a and b are merely inputs, freely and independently chosen by Alice and Bob. On a more physical level, Alice and Bob hold two subsystems of a quantum state; in the simple case of qubits, the inputs are naturally characterized by vectors on the Poincaré sphere, hence the notation a,b.
How should we understand nonlocal correlations, in particular those corresponding to entangled quantum states? A natural approach consists in decomposing P(α,βa,b) into a statistical mixture of hopefully simpler correlations:
Bell’s locality assumption is P_{λ}(α,βa,b)=P_{λ}^{A}(αa)P_{λ}^{B}(βb), admittedly the simplest choice, but an inadequate one as it turns out: quantum correlations violate Bell’s locality^{1}. Setting out to explore other choices, it is natural to require first that the P_{λ} fulfil the socalled nosignalling condition, that is, that none of the correlations P_{λ} results from a communication between Alice and Bob. This can be guaranteed by ensuring spacelike separation between Alice and Bob. Nonsignalling correlations happen without any time ordering: there is not a first event, let us say on Alice’s side, that causes the second event via some spooky action at a distance. We may phrase it differently: nonsignalling correlations happen from outside spacetime, in the sense that there is no story in spacetime that tells us how they happen. This is the case in orthodox quantum physics, or in some illuminating toy models such as the nonlocal box of Popescu and Rohrlich (PR box)^{10}. Mathematically, the nosignalling condition reads P_{λ}(αa,b)=P_{λ}(αa) and P_{λ}(βa,b)=P_{λ}(βb): Alice’s local statistics are not influenced by Bob’s choice of measurement, and reciprocally.
In 2003, Leggett proposed another model of the form (1), which can also be experimentally tested against quantum predictions^{6}. This model was recently brought into focus by the work of Gröblacher et al. ^{7}. The basic assumption of Leggett’s model is that locally everything happens as if each single quantum system would always be in a pure state. We shall be concerned here with the case of binary outcomes α,β=±1, though generalizations are possible. In this case, the supplementary variables λ in Leggett’s model describe pure product states of two qubits, denoted by normalized vectors u,v on the Poincaré sphere,
and the local expectation values have the usual form as predicted by quantum physics:
If the qubits are encoded in the polarization of photons, as in Leggett’s initial idea, then the assumption is that each photon should locally behave as if it were perfectly polarized (in the directions u and v), and the local observations, conditioned on each λ, should fulfil Malus’s law. It is worth emphasizing that Leggett’s assumption concerns exclusively the local marginals 〈α〉_{λ} and 〈β〉_{λ} of the probability distributions P_{λ}, whereas nothing is specified about the correlation coefficients 〈α β〉_{λ}. Leggett’s model can thus still be nonlocal, and can in general violate a Bell inequality.
Both in the original paper^{6} and in ref. 7, the model was presented by implicitly assuming a time ordering of the events (indeed, Bob’s output is assumed to depend on Alice’s input: her input must therefore have been defined before Bob’s output materializes). Any model based on such an assumption had already been falsified by the socalled before–before experiment^{11,12}, as Suarez emphatically stressed^{13}. However, assumptions (2)–(3) clearly define nonsignalling correlations, and Leggett’s model can be defined without any reference to time ordering. As a consequence, its study does add something to our understanding of quantum nonlocality. But what exactly? In what are such P_{λ} ‘simpler’ than the usual quantum correlations? To answer these questions, we recall that, in quantum theory, the singlet state is such that the properties of the pair are sharply defined (the state is pure), but the properties of the individual particles are not. In this perspective, Leggett’s model is an attempt at keeping the correlations and reintroducing sharp properties at the individual level as well.
Leggett’s model cannot reproduce the correlations of the singlet state. Experimental falsifications have already been reported, first under additional assumptions^{7}, then more directly^{8,9}. These works relied on the violation of socalled Leggetttype inequalities. Analogous to Bell’s inequalities, these criteria say that, under Leggett’s assumptions (2) and (3), a measurable quantity L should satisfy L≤L_{max}, whereas quantum theory predicts that L>L_{max} can be observed for suitable measurements. An important feature of Leggetttype inequalities is that, in contrast to Bell’s inequalities, the bound L_{max} is not a fixed number: instead, like the model itself, it depends on the quantum measurements that are made. Consequently all experimental data aiming at disproving Leggett’s model should present evidence that the settings used in the experiment have been properly adjusted.
All previously available derivations of Leggetttype inequalities were quite lengthy and failed to suggest possible improvements or generalizations. We have found a much more straightforward derivation (see the Methods section), simply based on the fact that each P_{λ} must be a valid probability distribution, so in particular P_{λ}(α,βa,b)≥0. Remarkably, this constraint of nonnegativity of probabilities, weak as it may seem, is enough to induce an observable incompatibility between Leggett’s model and quantum predictions. In our derivation, it also seems that the previously derived Leggetttype inequalities are suboptimal; among the improved inequalities that our approach suggests, the simplest one reads
where is the usual correlation coefficient. This inequality holds provided the three measurements on Alice’s side and six on Bob’s fulfil some relations; a possible set of measurements is given in Fig. 1. For the singlet state, quantum mechanics predicts C_{Ψ}^{−}(a,b)=−a · b. Thus, for the settings defined in Fig. 1, L_{3}(ϕ) is
This expression violates inequality (4) for a large range of values ϕ.
To test the Leggetttype inequality (4) in an experiment, we prepared polarizationentangled photon pairs in singlemode optical fibres in a close approximation to a singlet state, similarly as in ref. 9. In our setup (Fig. 2), we choose the settings of polarization measurements a_{i},b_{i} and b_{i}′ for the individual photons by dialling in appropriate orientation angles γ_{A,B} for two quarterwave plates (λ/4) and angles θ_{A,B} for two absorptive polarization filters. Details of the experimental implementation can be found in Supplementary Information, section 1. Through four consecutive coincidence measurements between photodetectors D_{A,B} for all combinations of settings a,−a and b,−b, we establish an experimental value for a correlation coefficient C(a,b).
The correlation coefficients necessary to compose values for L_{3}(ϕ=±30^{∘}) were obtained with an integration time of T=60 s per point, leading to values of 1.9068±0.0009 for ϕ=−30^{∘} and 1.9005±0.0010 for ϕ=30^{∘}. This corresponds to a violation of the bound for L_{3}(ϕ=−30^{∘}) and L_{3}(ϕ=+30^{∘}) in Leggett’s model by 83.7 and 74.5 standard deviations, respectively.
The asymmetry in the measured values of L_{3}(ϕ) is an indication for experimental imperfections in the accuracy of the settings, such as a possible misalignment of one of the quarterwave plates with respect to the polarizing filters. To test this alignment, we collected values for L_{3} over a larger range of ϕ with an integration time of T=15 s per setting (Fig. 3). The variation of L_{3} with ϕ is compatible with the quantummechanical prediction for a singlet state with residual coloured noise and an orientation uncertainty of the quarterwave plate of 0.2^{∘}.
The falsification of Leggett’s model proves that it is impossible to reconstruct quantum correlations from hypothetical, more elementary correlations in which individual properties would be sharply defined. Let us argue that a much stronger statement holds, namely, that individual properties cannot be even partially defined.
We first consider the following straightforward generalization of Leggett’s model: we allow the ‘local states’ λ to be mixed states, for example photons with a degree of polarization η. Therefore, we replace (2) and (3) by
The derivation of inequalities for this model follows exactly the same pattern as for usual Leggetttype inequalities. In particular, the generalized version of (4) is
which, for angles ϕ small enough, is violated by L_{Ψ}^{−}(ϕ) for any value of η>0. Thus, as soon as the degree of purity of the ‘local states’ is nonzero, this generalization of Leggett’s model also fails to reproduce quantummechanical predictions (see Supplementary Information, section 3 for a more complete analysis of this generalization).
Experimentally, we cannot expect to conduct a meaningful comparison between these two predictions down to η=0, owing to imperfections in the state preparation. From the measurement of Fig. 3, however, we can claim experimental evidence of a violation for all η≥0.56, with a statistical significance of at least 3.65 standard deviations, thus putting a lower bound for this class of models (Fig. 4).
It is then natural to conjecture that no model of form (1), with nonsignalling correlations P_{λ}, can perfectly reproduce the correlations of the singlet state C_{Ψ}^{−}(a,b)=−a · b, unless
for all measurements a and b (except perhaps for a few λ in a zeromeasure set). In the Methods section we prove this conjecture for models with discrete supplementary variables λ; continuity arguments should enable us to extend the result to more general models. We thus have a necessary condition for a nonsignalling model to be compatible with quantum mechanics. However, given a model with nontrivial marginals, finding an explicit inequality that can be tested experimentally against quantum predictions is another problem; for this, we need the specific details of the model.
In summary, with the general goal of improving our understanding of quantum correlations, we reformulated Leggett’s model. No time ordering of the events was assumed, and all assumptions were made on the local part of the correlations. We derived new Leggetttype inequalities, simpler and stronger than previously known. The simplest version of our inequalities has been experimentally violated. Finally we investigated more general models à la Leggett, for which we imposed only the nosignalling condition. We argued that any such model with biased marginals is incompatible with quantum predictions for the singlet state. This shows that quantum correlations cannot be reconstructed from ‘simpler’ correlations in which the individual properties would be even partially defined. Nature is really such that, in some cases, individual properties are completely lost whereas global properties are sharply defined.
Our result is in good agreement with the recent work in of Colbeck and Renner^{14}, who have derived general inequalities to falsify such models with nontrivial marginals. An example of a nonsignalling model that successfully reproduces the singlet correlations can be found in ref. 15; indeed, this model has unbiased marginals. It is also worth mentioning the nonlocal model of Toner and Bacon^{16}, which reproduces analytically the singlet correlations with one bit of communication. In this model, the probability distributions P_{λ} have nonvanishing marginals; however, the P_{λ} are signalling. The remarkable property of the Toner–Bacon model is that the communication is cleverly hidden, such that the final probability distribution P is nonsignalling.
This work is part of the general research programme that looks for nonlocal models compatible or incompatible with quantum predictions. The goal is to find out what is essential in quantum correlations. Here we have found that to simulate or to decompose the singlet correlations we cannot use nonsignalling correlations P_{λ} with nontrivial marginals. This nicely complements the result of ref. 17, that the correlations corresponding to very partially entangled states, hence large marginals, cannot be simulated by a single PR box, which has trivial marginals.
Methods
Simple derivation of inequalities that test Leggett’s model
Convenient notations
As mentioned, in this paper we focus on the case of binary outcomes α,β=±1. In this case, the correlations can conveniently be written as
This expression enables us to clearly distinguish the marginals on Alice’s side and on Bob’s, and the correlation coefficient Throughout the Methods section, we shall use these notations; in the main text, we have used more standard and simplified notations, the correspondence being 〈α〉_{λ}=M_{λ}^{A}(a,b), 〈β〉_{λ}=M_{λ}^{B}(a,b). The nosignalling condition is M_{λ}^{A}(a,b)=M_{λ}^{A}(a) and M_{λ}^{B}(a,b)=M_{λ}^{B}(b).
In order for the decomposition (1) to be a valid mixture of correlations, all distributions P_{λ} should be nonnegative. As we said, this constraint is enough to derive Leggetttype inequalities. From equation (6), we can see that the nonnegativity implies the general constraints
Constraints on the marginals M_{λ}^{A} or M_{λ}^{B} thus imply constraints on the correlation coefficients C_{λ}, and vice versa.
Let us now consider one measurement setting a for Alice and two measurement settings b,b′ for Bob, and let us combine the previous inequalities (7) that we get for (a,b) and (a,b′). Using the triangle inequality, we get
These constraints must hold for all probability distributions P_{λ}. After integration over the λ, we get, for the averaged correlation coefficients ,
This inequality is general for all models with ‘local marginals’, that is, that fulfil the nosignalling condition.
Derivation of a simple Leggetttype inequality
Now we derive an inequality satisfied by Leggett’s specific model, that can be experimentally tested. Inequality (8) implies, for the particular form of equation (3) for Bob’s marginals,
Let us consider three triplets of settings (a_{i},b_{i},b′_{i}), with the same angle ϕ between all pairs (b_{i},b′_{i}), and such that b_{i}−b′_{i}=2sin(ϕ/2)e_{i}, where {e_{1},e_{2},e_{3}} form an orthogonal basis (Fig. 1). After combining the three corresponding inequalities (9), using the fact that and the normalization , we finally get the Leggetttype inequality (4).
For a pure singlet state, inequality (4) is violated when ϕ<4arctan(1/3)≃73.7^{∘}, and the maximal violation is obtained for ϕ=2arctan(1/3)≃36.9^{∘}. In the case of imperfect interference visibility V (), a violation can still be observed as long as .
Note that other Leggetttype inequalities can be derived, as we mention in Supplementary Information, section 2.
Any nonsignalling model must have vanishing marginals
We prove here that all marginals in a nonsignalling model must necessarily satisfy the constraints (11) and (12) below; and we argue that this in turn implies the claim made in the main text, namely, all the marginals must vanish (apart perhaps on a zeromeasure subset of λ).
In order for a general nonsignalling model to reproduce all quantum correlations C_{Ψ}^{−}(a,b)=−a · b of the singlet state, we must have, according to equation (8), for all a,b,b′ on the Poincaré sphere
and therefore, for all b,b′,
where · is the Euclidian norm.
In the case where b′=−b, the first constraint of equation (10) implies that for all b
In the case b′→b, the two vectors being normalized, we have and therefore the second constraint of (10) implies that for all b
Let us now prove, in the case of discrete λ (λ∈{λ_{i}}), that (11) and (12) in turn necessarily imply that the marginals M_{λ}^{B}(b) must vanish. In this case, the integral should be changed to a discrete sum .
Constraint (11) indeed implies that for all λ_{i} (such that p_{λ}_{i}>0) and for all b, M_{λ}_{i}^{B}(−b)=−M_{λ}_{i}^{B}(b), that is, M_{λ}_{i}^{B} must be an odd function of b. This is a very natural property that we would impose on such a model; in particular, Leggett’s model indeed has odd marginals.
In addition, constraint (12) implies that for all λ_{i} and for all b, lim_{b′→b}(M_{λ}_{i}^{B}(b)−M_{λ}_{i}^{B}(b′)/b−b′)=0, that is, that all M_{λ}_{i}^{B} are differentiable, and their derivative is zero for all b: therefore, they are constant. As they have to be odd functions, then necessarily they are equal to zero.
In conclusion, for discrete λ, Bob’s marginals M_{λ}_{i}^{B}(b) must all vanish; of course, the same reasoning holds for Alice’s marginals M_{λ}_{i}^{A}(a). This result should also be valid for any distribution ρ(λ), at least those physically reasonable (for example piecewise continuous): we conjecture that for any reasonable model to reproduce the quantum correlations of the singlet state, the marginals must necessarily vanish, in the sense that
that is, for all a,b, M_{λ}^{A}(a)=M_{λ}^{B}(b)=0 for ‘almost all’ λ (except for a zeromeasure subset of the λ, that could possibly depend on a,b).
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Acknowledgements
We thank A. Ekert and C. Simon for comments. C.B., N.B. and N.G. acknowledge financial support from the EU project QAP (ISTFET FP6015848) and Swiss NCCR Quantum Photonics. A.L.L. and C.K. acknowledge ASTAR for financial support under SERC grant No 052 101 0043. This work is partly supported by the National Research Foundation and Ministry of Education, Singapore.
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Branciard, C., Brunner, N., Gisin, N. et al. Testing quantum correlations versus singleparticle properties within Leggett’s model and beyond. Nature Phys 4, 681–685 (2008). https://doi.org/10.1038/nphys1020
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DOI: https://doi.org/10.1038/nphys1020
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