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Non-locality and quantum theory: new experimental evidence

Starting from the late 60's many experiments have been performed to verify the violation Bell's inequality by Einstein-Podolsky-Rosen (EPR) type correlations. The idea of these experiments being that: (i) Bell's inequality is a consequence of locality, hence its experimental violation is an indication of non locality; (ii) this violation is a typical quantum phenomenon because any classical system making local choices (either deterministic or random) will produce correlations satisfying this inequality. Both statements (i) and (ii) have been criticized by quantum probability on theoretical grounds (not discussed in the present paper) and the experiment discussed below has been devised to support these theoretical arguments. We emphasize that the goal of our experiment is not to reproduce classically the EPR correlations but to prove that there exist perfectly local classical dynamical systems violating Bell's inequality.

(1) Introduction The experiment described in the present paper has been performed in the attempt to clarify a question that has accompanied quantum theory since its origins: is it true that the theory of relativity, quantum mechanics and a realistic interpretation of natural phenomena are mutually contradictory?
In the past 30 years the answer to this question, accepted by the majority of physicists has been: yes they are and this can be proved by theory and confirmed by experiment. Furthermore, since the experimentally confirmed EPR type correlations are . . . necessarily nonlocal in character . . . [GrHoZe93], the experiments also solve the contradiction in favor of quantum theory by showing that the basic pillar of relativity theory is violated in nature. Usually this statement is tempered by the clause that quantum nonlocality cannot be used to send superluminal signals. This amounts to downgrade the locality principle from a law of nature (no action at distance exists) to a principle of telecommunications theory (the existing actions at distance cannot be used to build up a superluminal television). Of course, if this downgrade is the only reasonable way to interpret the experimental data, we have to accept it. However, given the relevance of the issue, the question whether it effectively is the only reasonable way to interpret the experimental data, naturally arises. In several papers starting from [Ac81a], (cf. [Ac97] for a general presentation, [Ac99], [AcRe99a] for more recent results) the arguments, relating Bell's inequality to locality, have been criticised on a theoretical ground. In the present paper we will substantiate these theoretical arguments with an experiment.
Before describing our experiment let us briefly review the usual proof of the contradiction between relativity, quantum mechanics and realism, which is based on a combination of Einstein, Podolsky, Rosen (EPR) type arguments with Bell's inequality and goes as follows. In the EPR type experiments a source emits pairs of systems with the following properties: i) the members of each pair are distinguishable after separation and we denote them 1 and 2. After emission the two become spatially separated, say: 1 goes to the left, 2 goes to the right. ii) for each member j = 1, 2 of each pair we can measure a family of observables (spin) j = 1, 2 parametrized by an index set T (a ∈ T ). To fix the ideas let us say that T is the unit circle so that each a is a unit vector in the plane. iii) each observableŜ  ; j = 1, 2 cannot be simultaneously measured on the same system iv) The singlet condition is satisfied, i.e. if the same observable is measured on both particles, then the results are opposite.
According to EPR the values of the spins must be pre-determined otherwise, if one assumes that these values are created by a choice of nature at the act of measurement, the only way to explain the strict validity of the singlet condition would be to postulate a mechanism of instantaneous action at distance through which a particle instantaneously knows which choice nature is going to do on its distant partner so that the two choices can be matched into the singlet law. The pre-determination of the result is called a realism condition. Thus the negation of the realism condition in the sense of EPR, would imply a nonlocality effect. We anticipate that, in our experiment, a crucial role will be played by a distinction between the EPR realism , or ballot box realism, on which the whole of classical statistics is based, from what might be called the chameleon realism which seems to be more appropriate when dealing with quantum systems.
Before explaining this distinction let us shortly outline the role of Bell's inequality: if the values of the spins must be pre-determined, then there must be a hidden parameter λ which determines these values. In other words the spin variables must be functions S (j) a of this hidden parameter λ:Ŝ and the statistical fluctuations of the spins reflect a statistical distribution of this unknown parameter. The nature of the hidden parameter and of its statistical distribution are both unknown to us, but Bell proves that, whatever this nature may be, the correlations must satisfy a certain inequality that now brings his name. The quantity of experimental data accumulated in EPR type experiments is now quite remarkable and the common consensus on the interpretation of these data is that they exhibit a violation of Bell's inequality. A survey of the situation up to 1988 was published in Nature by J. Maddox [Mdx88] and a multiplicity of results in the past 12 years have further confirmed the experimental violation of this inequality by a variety of quantum systems.
According to Bell, given the realism condition (as defined above), the vital assumption in the deduction of his inequality is a locality condition: ... The vital assumption [2] [i.e. the EPR paper [EPR35]] is that the result B for particle 2 does not depend on the setting a of the magnet for particle 1, nor A on b . . . [Be64] A theory which satisfies both assumptions is called a local realistic theory. With this terminology Bell's result is often reformulated in the form: a theory that violates Bell's inequality must be either non-local or non-realistic.
Since, by the original EPR argument, a non-realistic theory is necessarily non-local, it follows that in any case the violation of Bell's inequality implies non-locality. This is Bell's original thesis and, even if balanced by a multiplicity of subtle differences on minor points, from the huge literature now available on this topic (cf. for example [Red87], [Be87], [CuMcM89], [Maud94], ...), a substantial consensus emerges with its main points which one may summarize in the scheme: -locality implies Bell's inequality -quantum mechanical (EPR type) experimental correlations violate Bell's inequality -therefore quantum mechanics is non-local -furthermore, no classical system can violate Bell's inequality by local choices.
Therefore if we can construct a classical system (even deteministic and macroscopic) which, by means of purely local choices produces a set of statistical correlations violating Bell's inequality, this will be an experimental proof of the fact that (i) locality does not imply Bell's inequality (ii) the experimental violation of Bell's inequality achieved by the quantum mechanical (EPR type) correlations is not an indication of non-locality. (iii) the experimental violation of Bell's inequality by purely local experiments is not a typically quantum phenomenon b , . . . are functions defined on the same probability space then the inequality (4) below must be fulfilled even if we postulate a strong non locality condition (e.g. thatŜ (1) a depends on the values assumed byŜ Therefore, if we want to violate (4) we have to find a local mechanism of generation of the answers so that each pair (Ŝ (1) a ,Ŝ (2) b ), . . . have a joint probability distribution, but this is not true for triples. Our goal in this paper is to construct such a system.
The goal of the present experiment is not to build a hidden variable model for the quantum mechanical singlet correlations (although a variant of it may play this role), but to construct a classical system showing that the three conditions of pre-determination, locality and singlet condition are not incompatible with a violation of Bell's inequality. Consequently the violation of these inequalities in the EPR-type experiment cannot be interpreted as an incompatibility of the three above mentioned conditions with quantum theory.
(2) Description of the experiment Our experiment describes a classical situation analogous, but not identical, to the one which takes place in the EPR type experiments. The classical system considered is made of three computers: (2.1) one, playing the role of the source of the singlet pairs, will produce pseudo-random points in the unit disk p 1 , p 2 , . . . , p N (1) Each point p j plays the role of a singlet pair, therefore in the following we shall speak indifferently of the point p j or of the singlet pair p j . The choice of the pseudo-random generator is not relevant for the results, provided the algorithm has a reasonably good performance. The algorithm we use has been taken from [PrTe93]. An order of N = 50.000 points is already sufficient to obtain acceptable results. Increasing the number of points gives a higher precision in the computation of the areas, but does not change the order of magnitude of the violation of the inequality. This also shows that the violation we obtain cannot be attributed to round-off errors in the measurements of the areas. (2.2) the other two computers play the role of the two measurement apparata. The three computers are separated (e.g. they may be in different rooms or different countries, 3) The role of the experimentalists will be played by two persons, one for each of the "measuring computers". Given a point p j (a singlet pair), operator 1 makes a local independent choice of a unit vector a in the plane (analogue of the direction of the magnetic field) and activates a programme in computer 1 which computes the value of a function S  (answers) can only be +1 or −1.
-A priori, for any given point (singlet pair) p j , experimenter 1 (resp. 2) can choose among infinitely many questions to be asked to system 1 (resp. 2). However only one question at a time can be asked to any single system.
is purely local, however the programme has been devised so that if, by chance, the same question is asked to both systems then they will give opposite answers (singlet condition).
(2.6) Finally we introduce, in our classical model, a strong form of the disturbance effect of a measurement of an observable on the other observables of the same system, incompatible with the given one, by requiring that, if the observableŜ x is measured, then both types of particles instantaneously change the value of all the other observables fromŜ y (y = x) to −Ŝ y (if I measure the weight of a chameleon in a closed box, its color need not to be the same I would have found if I would have measured it on a leaf). This additional prescription plays a role in the deduction of the Bell inequality (section (5)), but is not necessary for the Zeilinger-Harne-Greenberger type contradiction of section (7).
It is clear from our rules that the answers are: i) pre-determined , i.e. each system of the pair knows a priori what his/her answer will be if any question, determined by an input point p j and a local choice a or b, will be asked. ii) local, i.e. if, for a given point p j , operator 1 makes the local choice a, the answer of system 1 depends only on p j and a. In particular it does not depend on which question b has been asked to system 2. The same is true exchanging, in the above statement, the roles of 1 and 2. iii) entangled , i.e. the singlet condition is satisfied.
We want to study the statistics of these answers. In order to do so we have to ask the same pair of questions to many, say N , pairs: p 1 , . . . , p N . In particular, denoting the answer given by system ν(= 1, 2) of the j-th pair to the local questions S (ν) a , we can define the empirical correlation for any pair (S The question we want to answer with our experiment is the following: can the members of each pair p j make an agreement on their answers so that the inequality is violated for some choices of the indices a, b, c? According to Bell's analysis, if the inequality (4) is violated, then at least one of the conditions (i), (ii), (iii) must be violated (cf. section 5 for further discussion of this point). Our experiment contradicts this conclusion.

(3) Intuitive interpretation of the experiment
The present experiment describes an ensemble of pairs of classical particles which can interact with a vector observable B, that we call magnetic field . We assume that there are two types of particles, type I (left handed) and type II (right handed) and that each pair contains exactly one particle of type I and one particle of type II.
The pairs decay, i.e. split, sequentially, one after the other, and we suppose that left handed particles always go on the left and right handed particles always go on the right, so that after a short time they become spatially separated. On the path of each type of particles there is an experimentalist and the two have coordinated their space-time reference frames so that it makes sense to say that they make simultaneous measurements and that they orient a physical vector quantity in a common direction. Without loss of generality we can assume that: -all the experiments take place in the same plane -in this plane we have fixed an oriented reference frame e = (e 1 , e 2 ) -this reference frame can be parallel transported without holonomy to different points of the (affine) plane (Figure (1)).
In particular, with respect to this frame, we can speak of upper and lower half-plane and of counterclockwise or clockwise measurement of the angles (Figures (2) and (3)). Moreover for any point c in the disk,we denote Rc its reflection with respect to the origin (Figure (3)) Denote D the unit disk in the plane: ∂D the unit circle D + = (resp. ∂D + ) the upper semi-disk (resp. semi-circle) D − = (resp. ∂D − ) the lower semi-disk (resp. semi-circle) (cf. Figure (2)). To both types of particles we associate a family of observables (that we call spin) parametrized by the points on the unit circle in the plane {Ŝ a : a ∈ ∂D} and with values ±1 in appropriate units. The two types of particles differ because a magnetic field of unit strength in direction a ∈ ∂D splits a beam of particles of type I according to the values of the observableŜ a , and a beam of particles of type II according to the values of the observableŜ Ra . More precisely: if a is any unit vector in the upper half plane (which includes the vector e 1 , but not the vector Re 1 ) then, under the action of a magnetic field B(a) in direction a, a particle of type I will deviate in the upward direction ifŜ a = +1, in the downward direction ifŜ a = −1 where the first arrow means input and the second one denotes the effect of the interaction with the magnetic field. Under the action of the same magnetic field B(a), a particle of type II will deviate in the upward direction ifŜ Ra = +1, in the downward direction if For a as above, a magnetic field in direction Ra will have the symmetric effect: i.e. particles of type I will go up ifŜ Ra = −1, down ifŜ Ra = +1 and particles of type II will go up ifŜ a = −1, down ifŜ a = +1 In this sense the observableŜ a can be understood as measuring the response of a particle (either of Type or of I Type II) to a magnetic field in direction a, being understood that no other interaction has to take place with the same particle in the same time. In fact what we measure is only this response and the values ±1 are simply a conventional way to code these two mutually exclusive physical alternatives.
From this definition it is clear that for any two different vectors a, b ∈ ∂D the corresponding observablesŜ a ,Ŝ b are incompatible because it is impossible to place two different magnetic fields B(a), B(b) in the same point and, even if it were possible, a single particle cannot interact only with B(a) and, at the same time, only with B(b). A biological analogue of this situation would be whenŜ a denotes the color of a chameleon on a leaf andŜ b its color on a log: it is self-contradictory to say that the color is simultaneously measured only on a leaf and only on a log. This expression of logical impossibility has to be distinguished from the usual statement of the Heisenberg principle, expressing the physical (but by no means logical) impossibility of simultaneously measuring, with arbitrary precision and on the same system, position and momentum.
Remark. In standard quantum mechanics it is well known that the spin in direction a is a pseudo-scalar quantity, i.e. under proper rotations it transforms as a scalar but, under parity, according to the rule ( [Sak85], chap. 4) S Ra = −S a and obviously the two hermitean matrices S a and −S a commute. However the above considerations on the impossibility, for a single particle, to interact simultaneously with two magnetic fields oriented in opposite directions and placed in the same point also apply to spin variables and show that some care is needed when identifying the simultaneous physical measurability of two observables with the commutativity of the associated operators, at least when pseudo-scalar quantities are involved.
On two different particles any pair of observables can be simltaneously measured. Accordingly an ordered pair (B(a), B(b)) of magnetic fields in directions a, b ∈ ∂D + in the upper half plane will describe a simultaneous measurement of observableŜ a , for particle I, and of observableŜ Rb , for particle II, of the pair. We will also use the notation (S Given that the two types of particles react differently to the same apparatus, we have to distinguish between the two statements: (i) the same measurement operation is performed on both particle 1 and particle 2 (ii) the same observable is measured on both particle 1 and particle 2 For example, case (i) corresponds to use the same magnetic field B(a) on both particles of the pair. In this case we know that we are measuring observableŜ a on particle 1 and observableŜ Ra on particle 2.
Case (ii) corresponds to use the magnetic field B(a) for particle 1 and the magnetic field B(Ra) for particle 2. In this case we know that we are measuring the same observablê S a on both particles.
With these premises we will discuss the statistics, over an esemble of pairs, of measurements of pairs of observables of the form (S corresponding to the measurements of the pairs of observables which, in terms of the originalŜ-observables can be written Notice that we have three experiments, corresponding to the three magnetic fields but the observables involved cannot be reduced to three in fact, as made explicit by the notation (4) they are four , i.e.Ŝ a ,Ŝ Rb ,Ŝ c ,Ŝ Rc (9) due to the different dynamical reaction of particle I and particle II to the same magnetic field B(c). This is an example, in a particularly simple and idealized classical situation, of the chameleon effect (the interaction Hamiltonian of a system with an apparatus, hence the dynamical evolution of the system , depends on the observable one wants to measure) described, for example, in [Ac99], [AcRe99a].

(4) Prescriptions for the local answers
The state space of our classical particles is the unit disk D and, for c in the unit circle, we denote S (j) c the response of particle j to the magnetic field B(c) as described in section (3). We will describe these responses by means of functions S x : p ∈ D → S x (p) = ±1, parametrized by points p in the unit circle. Throughout this section, the symbol S x shall denote such a function and should not be confused with the observablesŜ x of Section (3): for example S (2) x measures the observabalesŜ Rx , but is calculated using the function −S Rx (p) = S x (p). The values of the observables S (j) c are distributed according to the rule described in Figures (4) and (5): Notice that the two rules give exactly the same function for c = b, i.e.
As explained in the previous section, this corresponds to the measurement of S c on particle 1 and of S Rc on particle 2 while, to measure S c on both particles we have to place a magnetic field oriented in direction c for particle 1 and in direction Rc for particle 2. This gives, according to (1): S (2) which is the singlet law.
The programme to calculate the hidden functions S a is the following. For a ∈ ∂D + , making a angle α ∈ [0, π) (measured counterclockwise) with the x-axis (cf. Figure (6) Figure (7).

(5) The chameleon effect
In this section we show that, if one pretends to apply to our experiment the same type of arguments which are applied to the EPR type experiments, then one arrives to the conclusion that Bell's inequality should be satisfied and this leads to a contradiction with the experimental data.
Recall that the chameleon effect means that the dynamical evolution of a system depends on the observable A one is going to measure. In particular, if we measure A , the evolution of another observable B, hence its value at the time of measurement, might be quite different from what it would have been if B, and not A, had been measured. As explained in item (2.6) of section (2), in our classical model, this disturbance effect is not only deterministic but also known, so that we can use it in our calculations.
For a given pair (point in the unit disk) p ∈ D, we do not know a priori the values assumed by S (j) x (p) (because p is a hidden parameter for the experimentalists). However whatever these values are they must satisfy the inequality Since, in the second experiment, S (2) b has been measured and b = c, we now, from item (2.6) of section (2), that particle 2 of the pair p, has changed the value of S From which Bell's inequality c is easily deduced by taking averages.
In the following section we show that the experimental results contradict this argument.

(6) Violation of the Bell's inequality
We will now prove that there exist three directions a, b, c in the plane such that the correlations violate the Bell inequality. To this goal we fix a to be the x-axis and consider a generic choice of both vectors c, b in the upper semi-circle. According to our rules the corresponding statistics is described by Figure ( Then, according to equation (2), the two sides of the Bell inequality are respectively: and we are reduced to compare cb − ab = ca and ac But, because of our choice of the axes (cf. Figure 9), one has ca > ac Thus we conclude that c which violates the Bell inequality.
(7) A Greenberger-Horne-Zeilinger type contradiction Greenberger, Horne and Zeilinger [GrHoZe93] have constructed an example showing that the attempt to attribute simultaneous values to 2-valued observables represented by non commuting operators may lead to a contradiction independently of any statistical consideration. In this section we show that our experiment also provides a classical analogue of the construction of these authors.
From (3.1) and (3.3) we know that the answers of a particle of type I to the two (mutually exclusive) measurements B(a) and B(Ra) will be opposite: Exactly in the same sense we know that Moreover we know that Notice that, if we interpret S a (resp. S Ra ) as the response of a type I particle to the only action of the magnetic field B(a) (resp. B(Ra)) the joint event [S a = +] and [S Ra = −] makes no sense as a simultaneous statement on the same particle. Now let us show that, if we pretend (as done in the original Bell's argument and in all discussions of the EPR type experiments) that all these relations hold simultaneously, in the sense of joint events, then we arrive to a contradiction. Interpreting the diagrams below as explained in section (3) (both columns in each diagram are referred to the same particle), we see that relations (1) and (3.1) imply that, for any vector a in the upper half plane: and relations (2) and (3.2) imply: Relations (1) and (3.3) imply: and relations (2) and (3.4) imply: But, if we pretend to attribute to S (2) a the values given by the second column of (6) then, because of (5) this would contradict (3) because it gives Similarly also (7) and (8)  Summing up: the functions of the "hidden parameter" p ∈ D correctly describe the behavior of pairs of observables referred to different particles. In fact this behavior can be simulated on the computer with arbitrary precision. However, if we pretend to extend this descriptions to triples (or quadruples) of observables by including pairs of observables referred to the same particle, then we arrive to a logical contradiction, independent of any statistics. This kind of contradictions should not be considered as pathological but rather as natural manifestation of the paradoxes that might arise if one attempts to apply the conceptual schemes, elaborated having in mind a passive (ballot-box like) reality, to a physical situation involving an adaptive (chameleon like) reality. In the latter case the value of an observable may express the dynamical reaction to an interaction. Two incompatible interactions (e.g. only magnetic field B(a) or only magnetic field B(Ra)), might give rise to opposite physical reactions (e.g. going up or going down). If we codify this reactions with numbers, e.g. +1 or −1 and then we pretend to interpret these numbers as values of observables, in the same sense of passive reality (e.g. color or weight of a ball in a ballot box) then it is clear that joint events of the form [S a = +1] and [S Ra = −1], when referred to the same particle at the same time, are in trinsically contradictory. In fact the same particle cannot simultaneously go up and go down.

(8) Conclusions and acknowledgments
Our experiments show that there is no contradiction between lcality, realism and quantum theory, thus confirming the results of the theoretical analysis of [Ac81],..., [AcRe99b]. The three directions in Figure (9) parametrize three experiments I (a, b), II (c, b), III (a, c), in which the experimentalists 1 and 2 locally choose a unit vector according to the scheme illustrated in Table 1. I II III 1 a c a 2 b b c Table 1 Having fixed a to be the x-axis, as in Figure (9) this is equivalent to choose the angles ab and ac of Figure 9.
For the choice of the angles illustrated in Table 2 (angles are given in radiants, so the local choices consist in numbers between 0 and 3, 14) I II III 1 0 1.989675 0 2 0.3141593 0.3141593 1.989675 Table 2 we verify that the difference between the left and the right hand side of Bell's inequality is of order 0.521 > 0 leading to a violation of the inequality. Since our scheme is obviously stable under small perturbations, we have violation of the inequality for infinitely many directions.
The present experiment has been first performed on july 10, 1999 at the Centro Volterra and first publicly performed on september 13, 1999 at the Centre for Philosophy of Natural and Social Science of the London School of Economics. Then, in the same year: on September 29 at the Steklov Institute (Moscow), in occasion of the conference dedicated to the 90-th birthday of N.N.Bogoliubov; on November 9 at the 3-rd Tohwa University International Meeting on Statistical Physics (Tohwa University, Fukuoka, Japan); in January 10, 2000 at the IV International Workshop on Stochastic Analysis and Mathematical Physics, in Santiago; on February 24, 2000 at the Department of Physics of the University of Pavia; on March 7, 2000 at the 3rd Meijo Conference on Quantum Information. It was described without performance of the experiment (because of problems with the computer) on December 17, 1999 at the conference on Quantum Probability and Infinite Dimensional Analysis, Jointly organised by Indian Statistical Institute and the Jawaharlal