On some EPR (Einstein, Podolsky, Rosen) issues

On some EPR (Einstein, Podolsky, Rosen) issues Giuseppe Giuliani Formerly at: Dipartimento di Fisica, Università degli Studi di Pavia. Retired. email: giuseppe.giuliani@unipv.it arXiv:2001.00553v1 [quant-ph] 2 Jan 2020 website: www.fisica.unipv.it/percorsi/ Abstract. A critical reconsideration of the EPR (Einstein - Podolsky Rosen) paper shows that the EPR argument can be developed without using the concept of ‘element of physical reality’, thus eliminating any philosophical element in the logical chains of the paper. Deprived of its philosophical ornament, the EPR argument plainly reduces to require what quantum mechanics can not do: to assign definite values to two incompatible physical quantities. Hidden variables theories built up according to Bell - type theorems are formulated on the basis of the assumption that the locality condition implies the statistical independence between two measurements space - like separated. This assumption is valid only with the additional one that statistical dependence between two measurements requires a causal connection between them. This additional assumption rules out the possibility that statistical dependence may due to an intrinsic property of the physical system under study. Therefore, hidden variables theories are built up with a restriction which leads them to be disproved by experiment. They appear to be ‘straw man’ theories whose main role is that of putting under fire philosophical realism. However, a philosophy can not be disproved by an experiment unless it is shown that this experiment disproves a theory whose postulates are logical consequence of the philosophy. Quantum mechanical non - locality, invoked for describing EPR - type experiments, is strictly connected to the hypothesis (NDV hypothesis) according to which the twin photons of entangled pairs do not have a definite polarization before measurements. Both hypotheses are used only for describing EPR experiments and not for making predictions. Therefore, they can be dropped without reducing the predictive power of quantum mechanics concerning entangled photons pairs. Furthermore, both hypotheses can be experimentally tested by a modification of a standard experimental apparatus designed for studying entangled photons pairs. PACS 03.65.Ud Entanglement and quantum nonlocality PACS 03.67.Mn Entanglement measures, witnesses, and other characterizations Contents 1 Introduction 2 2 Philosophical and epistemological background of this paper 2.1 Theoretical entities and physical quantities . . . . . . . . . . . 2.2 Measurements and ontological statements . . . . . . . . . . . . 3 4 4 3 The EPR paper 7 4 Bohm’s version of the EPR’s thought experiment 11 5 The role of locality in Bell - type theorems 13 6 The Orsay experiment 15 7 How to experimentally test the NDV and the non - locality hypotheses 19 8 Discussion 20 9 Conclusions 23 1 Introduction In 1935, Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) published a paper in which the completeness of quantum mechanics was called into question [1]. In a historical perspective, the origin of the paper must be viewed in the contest of the epistemological and philosophical debate that accompanied the foundation of quantum mechanics. Broadly speaking, the philosophical inspiration of the EPR paper is a realist one and, apparently, the EPR argument hinges upon the concept of ‘physical reality’ (section 3). This choice has mixed up epistemological considerations about the completeness of a theory with more general philosophical stands, thus entangling in an intricate way, physics, epistemology and philosophy. The literature about EPR issues is huge: for instance, the review paper on ‘Bell non - locality’ [2], essentially dedicated to the physical offsprings of the EPR paper, counts over five hundred references; while the site philpapers.org, returns 170 philosophical papers at the query ‘Einstein Podolsky Rosen’ and other 664 papers on related issues. In this situation, it is almost a foolhardy idea to enter the debate. Nonetheless, I dare to submit 2 some considerations on three specific EPR issues: the EPR paper, the role of locality in Bell - type theorems and the concept of non - locality. In section 2 the philosophical background of the present paper will be expounded. It will be used for analyzing the philosophical content of the EPR paper and for bringing to light its role in the EPR argument. This background will further be used for disentangling philosophy from physics in much more recent papers. The discussion of the EPR paper (section 3) will show that the concept of ‘element of physical reality’, apparently so substantial to the paper, can be dropped without weakening the EPR argument. Section 4 is dedicated to the Bohm’s version of the thought experiment devised by EPR and to a summary of the ensuing developments. The role of locality condition in Bell’s - type theorems is discussed in section 5. In the following section 6, a typical experiment with entangled photon pairs (Orsay, experiment, 1982) is presented and its standard interpretation discussed. In section 7, it is shown how to experimentally test the two hypotheses used in this standard interpretation: the hypothesis that the photons of an entangled pair do not have a definite value of polarization before measurement (NDV hypothesis, Not Definite Value) and the subsequent necessary hypothesis of non - locality. Finally, the last section 8 is dedicated to some recent entanglements of physics and philosophy in EPR issues. 2 Philosophical and epistemological background of this paper In this section – for sake of clarity – I outline the philosophical and epistemological positions that inspire this paper. These positions play a heuristics role, in the sense that they suggest how to deal with some philosophical EPR issues. But – of course – the theses held in the present paper, can not be logically derived from these philosophical and epistemological positions 1 . The historical development of Science, broadly suggests that experimental disciplines have developed on the basis of three main philosophical assumptions: 1. There is a World whom the observer belongs to. 2. Causality principle. 1 A more detailed exposition of the epistemological stands of the present paper can be found in [3, pp. 1 - 25]. Furthermore, in this reference, these epistemological positions are tested by applying them to fundamental turning points in the history of physics. 3 3. The World behaves constantly in the same way (phenomena are reproducible). The epistemic status of these assumptions is different. The first is a reasonable hypothesis and, though it has been challenged with various argumentations, often paradoxical, it constitutes the cornerstone of experimental inquiry. The second has proved to be of great heuristic value because it asks for a methodological search of the causes of phenomena. The third is a necessary prerequisite for any knowledge of the World and has been sustained by thousands of years of observations and experimental inquires. These same assumptions constitute the foundations of a rationally oriented common sense and guide us in our daily life. Assumption (P1) asks for an answer to a basic question: which are the relationships between scientific descriptions of the World (or part of it) and the World? Answering this question amounts to sketch a ‘theory of scientific knowledge’: whatever it would be, this theory should be independent from the various disciplines, their theories and experimental results. We shall try to answer in some degree this question with reference to Physics: to see if and how the following considerations apply also to other experimental disciplines goes beyond the scope of this paper and of my capacities. 2.1 Theoretical entities and physical quantities The descriptions of Physics in their mature stage come in as theories which use - among others - two basic types of concepts: theoretical entities and physical quantities. Examples of theoretical entities are the concepts of reference frame, particle, wave, electron, proton. . . Physical quantities describe properties of theoretical entities or relations between them. For instance, mass, charge, spin and magnetic moment describe properties of the theoretical entity ‘electron’. The physical quantity ‘velocity’ is attributed to, for instance, an electron with respect to a reference frame. The physical quantity ‘force’ describes an interaction between two theoretical entities, for instance, between two masses. Physical quantities enter equations of Physics and can be measured. 2.2 Measurements and ontological statements A measurement can be defined as a set of experimental procedures that allow to attribute to a physical quantity a definite value (within a range of experimental inaccuracy). The result of a measurement of the quantity G that 4 describes a property of the theoretical entity E depends on the interaction between the theoretical entity E and the apparatus A (also considered as a theoretical entity): the result of the measurement depends on the property of the theoretical entity E described by the quantity G. As an example, let us consider the measurement of the mass of an ion with a mass spectrometer: the outcome of the measurement depends on a property (which we call ‘mass’) of the ions we are testing. In this case, our acquired knowledge suggests that the apparatus does not influence the result of the measurement. However, this is not, in general, the case. For instance, the insertion of an ammeter in an electrical circuit changes its electrical resistance and, therefore, the measured value of the current is different from that of the circuit without the ammeter. On the basis of assumption (P1), we state that the result of the measurement reflects a property PQE of a quid QE that, in the World, corresponds to the theoretical entity E. We can only establish a correspondence between theoretical entities and quid and properties of theoretical entities and properties of quid. For instance, we can say that in the World there is a quid that corresponds to our theoretical entity ‘electron’: this means that in the World there is a quid which has properties that correspond to the properties attributed by our theory to the ‘electron’ and that this quid behaves in accordance with the laws of our theory and with properties that are described by the measured values of the quantities Gi that our theory attributes to the ‘electron’. We can convene that the statement ‘the electron exists in the World’ is simply and only a shorthand of the previous one. Ontological statements, like the previous one about the existence of the electron, can be made only a posteriori by looking at our entire acquired knowledge. They can not be deduced by logical chains from the acquired knowledge, but they must be compatible with it: ontological statements can be only plausible. While nowadays the statement ‘the electron exists’ (as a shorthand of the longer one given above) is plausible, the statement ‘the Ether exists’ cannot be reasonably considered as compatible with our present acquired knowledge. However, the existence of the Ether was plausible in Maxwell’s times. If the plausibility of an ontological statement holds up for a long time, it turns into likelihood. For instance, around 1900, the electron was endowed with a mass and an electric charge, but its role in the constitution of matter was uncertain if not obscure: its existence was loosely plausible. Later, it acquired an intrinsic angular momentum (spin) and an intrinsic magnetic moment. Also a wavelength has been associated to it: the de Broglie wavelength λdB = h/p. However, the physicists, who increasingly learned how to manipulate the ‘electrons’ in spite of their changing properties, wisely kept on referring to them as to the same ‘particle’. Meanwhile, the 5 theory of the electron has changed and we can not exclude that it will change again. But the statement of the existence of the electron has become more and more plausible: it has become probable. The statement of existence of a theoretical entity may become more stable than the theory which originally used it. We can measure physical quantities which describe properties of theoretical entities that do not exist. For instance, Maxwell, in the item Ether, included in the IX edition of the Enciclopedia Britannica, showed how to measure the rigidity of the Ether starting from the measurement of the intensity of light coming from the Sun [4]. This is possible because the theories themselves indicate or suggest what to measure. Another, more intriguing, example is given by the ‘hole’, a concept developed in solid state physics in the late Twenties of last century. Theoretically, a ‘hole’ is constituted by a vacant site in the valence band of a semiconductor: the behavior of the remaining electrons under the action of electric or magnetic fields can be described as due to the motion of a ‘particle’, the hole, endowed with electric charge +e, (effective) mass and electric mobility. Solid state physicists measure these quantities, which are properties of the theoretical entity ‘hole’. But we can hardly say that the ‘hole’ exists, as we can say that the ‘electron’ exists [5]. We shall define ‘realism’ a philosophical position who accepts the three postulates above. It is a loose definition that allows many versions. We shall define ‘tempered realism’ a realism which incorporates the analysis given above of the process of measurement with its implications; in particular, a ‘tempered realism’ agrees with the above definition and use of ontological statements. A ‘tempered realist’ rejects naive assumptions. Let us consider a theory that, like classical electromagnetism, is in good agreement with experiments (in its domain of application). We conclude that the phenomena described by classical electromagnetism happen in the World exactly as described by the theory. This conclusion implies that all the theoretical entities used by the theory exist in the World with the properties describes by the associated physical quantities. But we have seen that ontological statements are only plausible or verisimilar; furthermore, we should know, independently from our theory, that in the World things happen exactly as described by our theory. This kind of naive realism, that can be denoted as ‘realism of theories’, appears as untenable. In dealing with the EPR paper and some EPR issues, we shall encounter many versions of realism. We shall also find out that, somewhere, it is held that experiments can falsify a philosophy. This can be true only in one case: when the postulates of a theory are logical consequences of a philosophy and the theory is disproved by an experiment. To my knowledge, in the history 6 of physics there is not an example of this kind and, reasonably, it never will there be. 3 The EPR paper The premise of the paper is a philosophical one: Any serious consideration of a physical theory must take into account the distinction between the objective reality, which is independent of any theory, and the physical concepts with which the theory operates. These concepts are intended to correspond with the objective reality, and by means of these concepts we picture this reality to ourselves (my italics) [1, p. 777]. Between the objective reality and its physical descriptions there is a correspondence, but EPR do not clarify the properties of this correspondence. In the words of the philosophical background of section 2, EPR hold that there is a World (the objective reality), i.e. they accept the supposition 1 of page 3. Without defining what is meant by ‘physical reality’, EPR formulated a sufficient condition (SC): If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity [1, p. 777]. At a first sight, it is not clear if the ‘physical reality’ coincides with the ‘objective reality’ or if the ‘physical reality’ belongs to a ‘physical picture of the World’ that must be distinguished from the ‘objective reality’. In the first case, the sufficient condition (SC) states that an algorithm will allow to establish the existence in the World of a physical quantity, and, therefore, also of the theoretical entity of which the physical quantity describes a property. This interpretation is incompatible with the concept of objective reality (the World with its properties as a datum, distinguished from the physical descriptions of it), because some basic properties of the World will be established directly by an algorithm. Therefore, we are left with the latter case: the concept of physical reality belongs to a physical description of the World. The sufficient condition – as it stands – leads to paradoxical implications. For example. If a photon has passed through a linear polarizer whose optical axis is aligned along the z axis, the photon – after the polarizer – is 7 linearly polarized along the z axis. Then, we can predict with certainty, i.e. with probability equal to one, that the photon will pass through a second polarizer with it axis aligned along the z axis. Therefore – according to the sufficient condition (SC) – the linear polarization of the photon is an element of physical reality. Instead, if the axis of the second polarizer makes an angle θ with the z axis, we can predict only that the photon will pass through the second polarizer with probability cos2 θ (Malus’ law). In this case, the sufficient condition is not satisfied: therefore, the linear polarization is not an element of physical reality. Then, being an element of physical reality depends on the type of measurement we are planning to do: this result is hardly acceptable. Rephrasing this conclusion with the words of EPR: “No reasonable definition of reality could be expected to permit this [1, p. 780]. In order to overcome this embarrassing conclusion, the sufficient condition should be reformulated with the specification that it must be satisfied in at least one physical situation. In this way, the linear polarization of the photon will be an element of the physical reality in both cases. Again, without defining what is meant by ‘complete’ when speaking of a theory, EPR formulated a necessary condition (NC): Every element of the physical reality must have a counterpart in the physical theory [1, p. 777]. This formulation is ambiguous. It could be interpreted by saying that every element of physical reality must be quantitatively described in a complete theory. But this interpretation does not allows to demonstrate the EPR thesis. For doing so, it must be interpreted by stating that a complete theory must attribute – in every physical situation – a definite value to a physical quantity endowed with physical reality by the sufficient condition. There is a tension between the sufficient and the necessary condition. While the sufficient condition must be satisfied at least in one physical situation, the necessary condition must be satisfied in every physical situation. Then, it is easier to find out physical quantities endowed with physical reality than to build up a complete theory that describes these physical quantities. Finally, though devised for quantum mechanics, these conditions should be applicable also to classical physics; otherwise, it should be explained why these two descriptions of the objective reality (or part of it) must obey different epistemological criteria. After these definitions, EPR considered a system composed of two parts. The two parts are permitted to interact from t = 0 to t = T ; for t > T no further interaction between the two parts is possible. If A and B are two physical quantities, when the two parts no longer interact, a measurement of the quantity A on part 1 will leave part 2 in a definite state, say ψ2 (reduction 8 of the wave function). If, instead, the quantity B is measured on part 1, part 2 will be left in a different state, say ϕ2 . It may happen that ψ2 and ϕ2 are eigenfunctions of two non - commuting operators corresponding to some physical quantities P and Q. Therefore, without in any way perturbing 2, we can predict with certainty the values of the non - commuting quantities P and Q of part 2. Then, the quantities P and Q of part 2 are ‘elements of physical reality’: as such, they should be described by a complete theory. But quantum mechanics says that two incompatible quantities P and Q can not be described by a common eigenfunction; therefore, quantum mechanics can not attribute definite values to both P and Q. In this precise, technical sense, P and Q can not have simultaneous definite values. Therefore, quantum mechanics is incomplete. As an example, EPR applied this argument to the case of two particles described by a particularly chosen wavefunction in one dimension: the difference x1 − x2 and the sum p1x + p2x have definite values, x0 and 0, respectively. This is permitted by the formalism because the quantities x1 − x2 and p1x + p2x are compatible, i.e. their operators commute. For this system, if a measurement of x1 yields x1 = xm , then we know with certainty that a measurement of x2 will yield x2 = xm + x0 . On the other hand, if a measurement of p1x yields p, we know with certainty that a measurement of p2x will yield −p. Hence, both x2 and p2x are elements of physical reality. But quantum mechanics says that x2 and p2x can not have simultaneous definite values. Consequently, quantum mechanics is incomplete because it does not account for all elements of physical reality. However, EPR stress that: Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted (original italics). On this point of view, since either one or the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this [1, p. 780]. As we have seen, the EPR argument is developed starting from the definition of the ‘element of physical reality’. However, this concept is superfluous and obscures the intimate nature of the EPR argument. In fact, it is possible to restate both EPR conditions in the following way: 9 If we can predict with certainty (i.e., with probability equal to one) the value of a physical quantity in at least one physical situation, then a complete theory must allow this quantity to have, in every physical circumstance, a definite value. Clearly, this conditions can not be satisfied by quantum mechanics, when a couple of incompatible quantities like position and momentum are considered. Then, quantum mechanics is incomplete. This necessary reformulation of the EPR argument reduces to plainly requiring what quantum mechanics can not do: to assign definite values to two incompatible quantities. The above analysis implies that the value of the EPR paper must be reconsidered. It is not a definite proof that quantum mechanics is incomplete, because its incompleteness is founded on a questionable definition of completeness. In fact, it can be held that other criteria are important in the evaluation of a theory: is the theory internally coherent? its predictions are corroborated by experiment? it describes quantitatively all phenomena that reasonably fall in its application domain? Historically, these criteria have decided the fate of physical theories. Quantum mechanics can not be an exception. Of course, a theory can be challenged also on the basis of philosophical considerations. However, in this case, the thing to do, is to develop an equivalent or better theory, i.e. a theory with an application domain equal to or larger than that of quantum mechanics. In the EPR paper, the concept of locality never appears explicitly. EPR speak only of two parts of a system that no longer interact: On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. This is, of course, merely a statement of what is meant by the absence of an interaction between the two systems [1, p. 779]. This statement must be compared with the one concerned with the locality condition: If the measurement on part II of the system is made at the instant tII while that on part I was performed at the previous instant tI and l > c(tII − tI ) – where l is the distance between the two points in which the two measurements are performed – then the result obtained on part II can not be influenced by a physical interaction coming from part I. The locality condition as stated above, is implied by special relativity in which no physical interaction can propagate at a speed higher than that of light in vacuum. 10 The aim of the EPR paper was that of proving that quantum mechanics is an incomplete theory on the basis of a quite questionable definition of completeness. Moreover, the concept of ‘element of physical reality’ used in the paper appears as ambiguous and, above all, superfluous. What is worst, this concept has entangled in a spurious way philosophical considerations with technical ones concerning the completeness of a theory. This original sin of the EPR paper has never been redeemed: indeed, the spurious intertwining of philosophy and physics has been accentuated by some subsequent developments to the detriment of the necessary and vital separation between physical theories and philosophical reflections on them. However, in spite of the flaws of the EPR paper, the thought experiment contained in it – opportunely reformulated – has become the starting point of new physics and new applications. Some of these developments, will be discussed in the remaining sections. 4 Bohm’s version of the EPR’s thought experiment In the Fifties, David Bohm tried to clarify some aspects of the EPR argument from two sides. Firstly, he reformulated the EPR thought experiment in the case of a system of two particles with spin h̄/2 in the singlet state [6]. The two particles, are initially permitted to interact (for instance, they are somehow produced by the same source); then, they fly apart in opposite directions 2 . When the two particles do not interact further, a measurement of, say, the spin component of particle 1 along the x direction is made. If, for example, the result is h̄/2, then we know with certainty that a measurement of the spin component along the x direction of particle 2 will yield −h̄/2. Therefore, this spin component of particle 2 is an element of physical reality. However, since we could have measured, instead of the x component of particle 1, the y component, we would have found that also the y component of the spin of particle 2 is an element of physical reality. Since, quantum mechanics can not attribute definite values to both spin components of particle 2, because they are incompatible quantities, quantum mechanics is incomplete. Secondly, in order to reintroduce a deterministic behavior of an individual system, Bohm tried to reformulate the formalism of non relativistic quantum mechanics by introducing ‘hidden’ variables [7, 8]. While the first step transformed the EPR ‘thought experiment’ in an experiment in principle 2 The example discussed by Bohm was that of an excited Hydrogen molecule that splits into two hydrogen atoms. 11 realizable, the introduction of hidden variables gave new fuel to the debate. Therefore, the appearance of a paper by John Bell was welcomed as a turning point: nothing would have been the same as before [9]. Bell translated into a mathematical, general, form the idea of hidden variables applied to the Bohm’s version of the EPR thought experiment. He showed that an appropriately defined correlation coefficient of a hidden variables theory must satisfy an inequality (Bell’s inequality), susceptible of an experimental check. Soon after, Clauser, Horn, Shimony and Holt (CHSH) generalized Bell’s theorem and proposed an experimental test based on the polarization correlations of photon pairs emitted by Calcium atoms in a cascade process 3 [10]. Aspect, Grangier and Roger realized the experiment proposed by CHSH using linear polarizers [12] and, soon after, two channels polarizers [13]. Both experiments strongly violated CHSH inequalities, showing that, in some circumstances, any hidden variables theory is disproved by experiment which, instead, corroborates the predictions of quantum mechanics. From then on, increasingly sophisticated experiments have been carried out in order to close several loopholes: among them, the communication loophole [14], due to the necessity of correlating polarization measurements space - like separated; the detector loophole, due to the limited efficiency of the photon detectors [15]; and the coincidence loophole connected to the necessity of determining which local events form a pair [16]. For a review of the loopholes issue, see, for instance, [17]. In spite of the considerable amount of unequivocal experimental data, the debate about some EPR issues is not over. In particular, I believe that two questions require further critical inquiry: the role of the locality condition in hidden variables theories and the interpretation of EPR - type experiments. In hidden variables theories, built up according to Bell - type theorems, the locality condition implies the statistical independence between the results of two measurements space - like separated: this implication leads hidden variables theories to be fatally disproved by experiment (section 5). On the other end, in standard interpretations of EPR - type experiments, a key role is played by the hypothesis that the twin photons of an entangled pair do not have a definite polarization before measurement (NDV hypothesis) and by the correlated hypothesis of non - locality (section 6). These two hypotheses are used only for describing experiments and never for making predictions. In section 7, it will be shown that the two hypotheses can be experimentally tested. 3 This proposed experiment was an extension of a previous one, performed by Kocher and Commins [11]. 12 5 The role of locality in Bell - type theorems John Bell put it very clearly in the Introduction of his paper 4 : It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty [9, p. 195]. In order to clarify this point, it is necessary to revisit the deduction chain of a Bell - type theorem. Consider a system constituted by two parts which having interacted in the past – because, for example, somehow produced by the same source – are now separated by the distance l. Let us further suppose that l > c∆t where ∆t is the time interval that separates the two measurements carried out on the two parts of the system. It is not necessary to specify which measurements we are talking about. That stated, a Bell type theorem can be formulated in the following way. Bell - type theorem A hidden variables theory formulated on the basis of the following assumptions: P1 The system constituted by the two parts has properties not described by the formalism of quantum mechanics. P2 These properties, symbolized by the parameter λ, may vary from one measurement cycle to another. Hence the opportunity of introducing a normalized probability distribution ρ(λ) of the parameter λ. P3 The locality condition (p. 10) implies that the results of the measurements space - like separated performed on the two parts of the system are statistically independent. yields predictions that, in some circumstances, are different from those of quantum mechanics. The assumptions P1 - P3 lead to the equation 5 : p(ab|xy) = Z ρ(λ)p(a|x, λ)p(b|y, λ)dλ 4 (1) It must be stressed that in Bell’s paper no philosophical element perturbs the logical chain of the argumentation. 5 See, for instance, [2, p. 421]. 13 Where: the p’s are probabilities; x and y represent the types of measurement (not specified) made on part I and II of the pair; a and b are the results of these measurements. The statistical independence between the results a and b is represented in equation (1) by the product of the probabilities p(a|x, λ)p(b|y, λ). From equation (1) Bell - type inequalities can be straightforwardly derived; among them, Bell and CHSH inequalities. These inequalities must be satisfied by hidden variables theories built up according to Bell - type theorems. As discussed in detail in section 6, these inequalities are violated by EPR experiments which, on the other hand, corroborate the predictions of quantum mechanics. The vital role played by equation (1) in Bell - type theorems has been stressed by many. For instance, [2] write: It is relatively frequent to see a paper claiming to “disprove” Bell’s theorem or that a mistake in the derivation of Bell inequalities has been found. However, once one accepts the definition (1), it is a quite trivial mathematical theorem that this definition is incompatible with certain quantum predictions [2, p. 421] (my italics). Equation (1) relies on assumptions P1, P2 and P3. EPR experiments show that the results of measurements performed on the separated parts of the systems are statistical dependent. Then, at least one of the assumptions is false. The most suspected is P3, because it establishes explicitly the statistical independence of the two measurements on behalf of the locality condition. Let us consider the following logical chain (the symbol ⇓ represents logical implication): A = Special relativity ⇓ B = Locality condition ⇓ C = Statistical independence of the measurements space like separated [l > c(tII − tI )] performed on the two parts of the pair EPR experiments show that C is false. If we believe that special relativity is true, we must conclude that the implication B ⇒ C is false, i.e. that the locality condition does not implies the statistical independence of the two space like separated measurements. As a matter of fact, the assumption (P3) that locality implies statistical independence is based on the additional, implicit assumption that a statistical dependence between two events must be due to a physical interaction between them. This additional hypothesis rules out the possibility that the correlation between these two events might 14 be due to an intrinsic property of the physical system under study. If P3 is false, the relevance of Bell - type theorems will be highly reduced 6 . 6 The Orsay experiment In 1982, Aspect, Grangier and Roger carried out at Orsay a renowned EPR experiment [13]. The source is a beam of Calcium atoms. Two photon transitions produced by two laser beams, perpendicular to the atoms’ beam, excite the electrons from the fundamental state 4s2 (J = 0) to the excited state 4p2 (J=0). From this state, the electrons fall back into the fundamental state by passing through the intermediate state 4s4p1 (J = 1) and by emitting two photons corresponding to λ1 = 551.3 nm and λ2 = 422.7 nm, respectively (fig. 1). 2 1 4p S0 581 nm 551.3 nm n1 4s4p1 P1 406 nm n2 422.7 nm 2 1 4s S0 Figure 1: Energy levels of Calcium atoms used in the Orsay experiment. Let us consider the twin photons of a pair which propagate in opposite directions ±z. Two filters allow the propagation of photons ν1 along +z and of photons ν2 along −z. A two channels analyzer A, followed by two photomultipliers, is situated along +z; an identical analyzer B is situated along −z. The experiment consists in measuring the polarization correlations of the twin photons as a function of the angle θ between the directions a and b that identify the orientations of the two polarizers. The results are confronted with the predictions of quantum mechanics and those of Bell - type hidden variables theories. For the conservation of angular momentum in the emission process, the photons pairs are described by the state vector: 1 |ψ(ν1 , ν2 ) > = √ (|R1 , R2> + |L1 , L2>) 2 6 (2) Of course, also P1 may be false: this amounts to say that it is not possible to introduce hidden variables in the formalism of quantum mechanics. 15 (+1) (+1) n2 D (-1) S n1 D A B (-1) z Figure 2: Experimental setup of the Orsay experiment. S is the source; A and B are two channels polarizers; D are photon detectors. Two filters, not shown in the figure, allow the passage of only photons ν1 along +z and of photons ν2 along −z. See the text. The state vector (2) is an example of what are called entangled states. Since a circularly polarized photon can always be expressed in terms of two orthogonal linear polarizations, the photon pairs can be described by the state vector: 1 (3) |ψ(ν1 , ν2 ) > = √ (|x1 , x2> + |y1, y2>) 2 Let us suppose that the measurement on photon ν1 is made before that on photon ν2 and that the space - time interval separating the two measurements is space - like 7 . The probability that the photon ν1 enters one of the two channels of the analyzer A is 1/2. If a is the polarization direction of photon ν1 after the measurement, then the photon pair, after the measurement made by A, is described by the state vector (reduction of the state vector due to the measurement): |ψ ′ (ν1 , ν2 )>= |a, a> (4) After the measurement by A, we know that photon ν1 is polarized along a (because it has come out from a polarizer oriented along a), and we learn, from the state vector reduction, that the polarization of photon ν2 is the same as that measured for ν1 . Therefore, if the analyzer B is oriented as A (i.e. if the direction a and b are parallel), the photon ν2 will pass through, with certainty, the same channel passed through by ν1 . If, instead, θ is the angle between a and b, photon ν2 will pass through the same channel passed through by photon ν1 with probability cos2 θ (Malus law). Hence, the probability that ν1 and ν2 pass through the same channel and, then, will have parallel polarizations after the measurements, is given by: Pkk (a, b) = 7 1 cos2 θ 2 (5) Here, I am following the description given by Aspect [18]. In the Orsay experiment, the measurements on the twin photons were not space - like separated. The first experiment of this kind, has been carried out by Weihs et al. in 1998 [14]. 16 This equation can, of course, be derived directly from (3) without considering the details of the two steps description. Analogously, we can derive the probability that the two photons pass through different channels: Pk⊥ (a, b) = 1 2 sin θ 2 (6) These equations, together with similar ones for P⊥⊥ and P⊥k , are used to calculate the quantum mechanical value of the quantity S appearing in the CHSH inequality. The quantity S is defined as: S = E(a, b) − E(a, b′ ) + E(a′ , b) + E(a′ , b′ ) (7) where the E’s are the correlation coefficients corresponding to the pairs of orientations of the two analyzers A and B. We recall that, for example: E(a, b) = Pkk (a, b) + P⊥⊥ (a, b) − Pk⊥ (a, b) − P⊥k (a, b) = cos 2(a, b) (8) The Orsay experiment shows that the maximum violation of the CHSH inequality occurs in correspondence of a value of SQM = √±2.697 ± 0.015. This value must be compared with the theoretical one ±2 2 = ±2.8284 and the fact that, according to hidden variables theories −2 ≤ S ≤ 2. Therefore, the Orsay experiment shows that hidden variables theories can not fully reproduce the predictions of quantum mechanics and that, on the other hand, these predictions are in satisfactory agreement with experiment. In the above two steps description of the Orsay experiment two points are crucial: I No supposition is made about the value of the polarization of photon ν1 before its measurement. The reason is that, for using the state vector reduction due to the measurement, only the value of the polarization after the measurement is of interest. II The reduction of the state vector due to the measurement on photon ν1 , shows that the polarizations of the twin photons are strongly correlated: see equation (4). This description – which we shall denote as the ‘formal description’ – uses only the formalism of quantum mechanics without any further assumption. It asserts only that the measurement implies – at the theoretical level – a reduction of the state vector of the photon pair and that this reduction uncovers the polarization correlation between the twin photons. Then, on the basis of the new state vector, it predicts which will be the result of the polarization measurement on photon ν2 . This description follows strictly the 17 prescriptions of quantum mechanics: i) write the state vector for the photon pairs on the basis of the production (preparation) process of the twin photons; ii) make predictions by using the formalism; iii) compare the predictions with the experimental results. Instead, the standard description uses two additional hypothesis 8 : A The twin photons do not have a definite polarization before measurement (hypothesis NDV (Not Definite Value)). B When the measurement on photon ν1 is made, photon ν2 assumes instantaneously the same polarization of photon ν1 . If we want to describe the outcomes of the experiments on the basis of hypothesis A , we must necessarily adopt also hypothesis B . As we have seen in point I , the polarization of photon ν1 before the measurement is not used in the calculation of the quantum mechanical predictions. Hence, the NDV hypothesis is used only for describing, a posteriori, the predictions obtained by I and II . Anyway, in next section, it is shown how the A (NDV) and B hypotheses can be tested by experiment. Statement B has the same logical structure of the statement: when I push the button, the lamp turns on. The statement on the lamp implies a causal connection between the pushing of the button and the turning on of the lamp. This causal connection is suggested by the detailed knowledge of the physical system to which the button and the lamp belong. Also the statement about photon ν2 should contain a similar causal connection. However, a causal connection is forbidden by the fact that the two measurements on the twin photons are space - like separated. Therefore, it is necessary to invoke a kind of instantaneous action at a distance between the two measurements. This action at a distance is labeled as non - locality. However, non - locality is used only for describing the experiment and not for making predictions: these are obtained only from the formalism of quantum mechanics. Therefore, the concept of non - locality can be dropped without reducing the predictive power of quantum mechanics concerning entangled pairs. It is worth stressing that the description of the experiment based on the NDV and the non - locality hypotheses presupposes that the things go in the World, step by step, exactly as described by the theory. Therefore, this description is a paradigmatic example of what is called realism of theories, a type of realism that appears as untenable (section 2). Finally, the standard description of the Orsay experiment faces what may be called the simultaneity paradox. Let us suppose that the two measurements are made at the same instant. If ν1 enters channel (+1), then ν2 enters 8 See, for instance, the paper by Aspect [18]. 18 simultaneously the same channel. However, since the two measurements are simultaneous, ν2 might enter channel (−1): in this case, ν1 should enter the same channel, in contradiction with the starting hypothesis. 7 How to experimentally test the NDV and the non - locality hypotheses The NDV hypothesis is strictly connected to that of non - locality: both are used in the standard description of the Orsay experiment or, more generally, of EPR - type experiments. The NDV and the non - locality hypothesis can be experimentally tested by a modification of the apparatus of the Orsay experiment. During the emission of each photon of a pair, the total angular momentum (atom + emitted photon) must be conserved. Therefore, the twin photons emitted in a single cascade process should be both right (R) or left (L) circularly polarized. This implies that the photons of each pairs have definite polarization when emitted by the Calcium atoms, i.e. before any polarization measurement. This assertion is the negation of the NDV (Not Definite Value) hypothesis (the twin photons do not have a definite polarization before measurement) that, instead, is used in the standard description of the Orsay experiment. Luckily, there is an experimental procedure that allows to verify which is the polarization of a light beam before the measurement. The procedure requires the use of a quarter wavelength plate and a linear polarizer and it can be applied to the twin photons emitted by Calcium atoms 9 . In an Orsay - type apparatus, the double channel analyzers must be replaced, in both arms, by a quarter wavelength plate and a linear polarizer. We shall suppose that the two measurements on photon ν1 and photon ν2 are made at two instants such that the space - time interval separating the two measurements is space - like. The quarter wavelength plate transforms the incoming photon supposed to be R (L) polarized in a photon linearly polarized along a direction which makes an angle of π/4 (−π/4) with the optical axis of the plate 10 . Therefore, if after the plate, we insert a linear polarizer whose optical axis is parallel to the direction π/4 (−π/4), the photomultiplier situated after the polarizer will click if the incoming photon is R (L) as emitted by the source. If in the two arms of the apparatus the linear polarizers are 9 A similar experiment has been already proposed in a previous work of mine [19, p. 273]. 10 Here, we are adopting the convention according to which a photon is R (L) polarized if its intrinsic angular momentum vector is directed along (opposite to) its propagation direction. 19 both oriented along a direction which makes an angle of π/4 (−π/4) with the optical axis of the associated plate, both photomultipliers placed behind the linear polarizers will click only if the twin photons are produced as R (L) circularly polarized by the source, i.e. before the polarization measurements. Now, let us try to describe this proposed experiment on the basis of hypothesis A (NDV hypothesis) and of hypothesis B (non - locality hypothesis). The NDV hypothesis affirms that photon ν1 does not have a definite polarization before measurement. However, if photon ν1 is detected, this photon, after the measurement of its polarization (i.e. after the linear polarizer) is linearly polarized along the direction of the optical axis of the polarizer. Let us denote this direction with the symbol k. According to the non - locality hypothesis, photon ν2 assumes instantaneously the same linear polarization k. Then, photon ν2 , going through the plate, will be transformed into an R circularly polarized photon and will have a probability 1/2 of going through the subsequent linear polarizer. Therefore: while the hypothesis that the twin photons are both produced as R or L polarized yields a probability of being detected for photon ν2 equal to one, the NDV and non - locality hypotheses yield for the same probability the value of 1/2. Then, the proposed experiment allows to experimentally test the two theoretical predictions and shows that the question of the polarization of the twin photons as produced by the source in a physical question, answerable with an experiment. 8 Discussion Initially, EPR experiments have been devised in order to disprove hidden variables theories (built up as indicated by Bell - type theorems). EPR experiments have disproved these theories and ascertained that the polarizations of twin photons described by entangled states obey quantum mechanics. But this is only a piece of the story. In fact, these experiments have taught us how to cleverly manipulate entangled pairs of photons and basically contributed to new fields of applied research like that of quantum information and quantum cryptography [2]. These applications depend only on the correlation properties of entangled states. The validity of Bell - type theorems or the way in which we describe EPR experiments are irrelevant for these applications. These questions are instead of interest from an epistemological viewpoint. The following considerations are dedicated to these issues. Typical consequences usually drawn from EPR experiments are: 1. Hidden variables theories formulated according to Bell - type theorems are disproved by experiments 20 2. Quantum mechanics is non - local 3. Realism and locality are incompatible The epistemological status of these statements is different. Statements 1 and 2 are physical statements, while statement 3 mixes up physical and philosophical issues. As we have seen, statement 1 is true. However, hidden variables theories considered by Bell - type theorems are built up with an intrinsic lethal flaw, constituted by equation (1): this equation is a direct consequence of the implication ‘locality condition → statistical independence of the results of two space like separated measurements’. This implication is valid only if it is additionally assumed that statistical dependence between two events requires a physical interaction between them. This additional assumption forbid hidden variables theories to describe statistical dependencies due to intrinsic properties of the physical system. Therefore, hidden variables theories are built up with rules that fatally lead them to be disproved by experiments: they are ‘straw man’ theories. Statement 2 follows from statement 1 if the implication ‘locality ⇒ statistical independence of the results of two measurements space like separated’ is true. However, the logical chain considered in section 5 shows that this implication is false (if we believe that special relativity is true) and, therefore statement 2 appears as questionable. However, the experiment proposed in the previous section allows a direct experimental test of non - locality. Statement 3 entangles physical elements (locality) with philosophical ones (realism), following the temper of the EPR paper. It has been phrased in many ways. Here are some representative examples: The violation of a Bell inequality is an experimental observation that forces the abandonment of a local realistic viewpoint – namely, one in which physical properties are (probabilistically) defined before and independently of measurement, and in which no physical influence can propagate faster than the speed of light [15, p. 227]. Bell’s theorem does not prove the validity of quantum mechanics, but it does allow us to test the hypothesis that nature is governed by local realism. The principle of realism says that any system has preexisting values for all possible measurements of the system. In local realistic theories, these preexisting values depend only on events in the past light cone of the system [16, p. 250402 - 2]. 21 Most working scientists hold fast to the concept of ‘realism’ – a viewpoint according to which an external reality exists independent of observation. But quantum physics has shattered some of our cornerstone beliefs. According to Bell’s theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenable. Maintaining realism as a fundamental concept would therefore necessitate the introduction of ‘spooky’ actions that defy locality [21, p. 871]. The meaning of the word ‘realism’ is not the same in these quotations. In the first two, by ‘realism’ it is meant that a physical quantity has a definite value before measurement. It is true that this assumption is suggested by a philosophical realistic stand. However, as shown in section 7, the assertion that the photons of an entangled pair have or not have a definite value of polarization before measurement is a physical question, answerable by an experiment. The experiment tests a physical assertion and not the philosophy that might have inspired it. In the third quotation, by realism it is meant ‘a viewpoint according to which an external reality exists independent of observation.’ This philosophical viewpoint has been at the basis of the scientific endeavor and of what we may call the rationally oriented common sense. This philosophical stand (like any other) can not be falsified by any experiment. As already pointed out in section 2, in order to experimentally disprove a philosophy, one should build up a theory whose entire set of assumptions are logically derived from the philosophy and shows that this theory is falsified by experiment. However, the philosophical stand of a scientist in not a neutral one with respect to his attitude in elaborating theories, devising or explaining experiments. Different philosophies produce different heuristics. For instance, a realistic stand will suggest to search for the causes of a phenomenon, or, more specifically, to investigate further the polarization properties of a photon in an EPR experiment in order to better understand its behavior. Instead, other philosophies will hold that the probabilistic nature of quantum mechanics reflects the indeterministic behavior of the microscopic world. Consequently, it is useless to search for causes or, more specifically, to investigate further the polarization properties of the photons of an entangled pair. The difference between these two heuristics showed up in the discussion of the role of locality in Bell - type theorems (section 5) and of the Orsay experiment 22 (section 6) or in the proposal of experiments for testing the NDV and the non - locality hypothesis (section 7). 9 Conclusions A critical reconsideration the EPR (Einstein - Podolsky - Rosen) paper shows that the EPR argument can be developed without using the concept of ‘element of physical reality’, thus eliminating any philosophical element in the logical chains of the paper. Deprived of its philosophical ornament, the EPR argument plainly reduces to require what quantum mechanics can not do: to assign definite values to two incompatible physical quantities. However, the thought experiment devised by EPR, opportunely revised, has become a laboratory experiment and has given birth to new fields of fundamental ad applied research. Hidden variables theories built up according to Bell - type theorems are formulated on the basis of the assumption that the locality condition implies the statistical independence between two measurements space - like separated. This assumption is valid only with the additional one that statistical dependence between two measurements requires a physical interaction between them. This additional assumption rules out the possibility that statistical dependence may be due to an intrinsic property of the physical system under study. Therefore, hidden variables theories are built up with a restriction which leads them to be disproved by experiment. They appear to be ‘straw man’ theories whose main role seems that of allowing to call into question philosophical realism. However, a philosophy can not be disproved by an experiment unless it is shown that this experiment disproves a theory whose postulates are logical consequences of the philosophy. Quantum mechanical non - locality, invoked for describing EPR - type experiments, is strictly connected to the hypothesis (NDV hypothesis) according to which the twin photons of entangled pairs do not have a definite polarization before measurements. Both hypotheses are used only for describing EPR experiments and not for making predictions. Therefore, they can be dropped without reducing the predictive power of quantum mechanics concerning entangled photons pairs. Furthermore, both hypotheses can be experimentally tested by a modification of a standard experimental apparatus designed for studying entangled photons pairs. Acknowledgements. 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