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The Einstein Podolsky Rosen paradox and its solution

The numbers with the publications refer to the numbers that these publications have in the other lists (chronological and year by year) and also the numbers that are used to refer to publications in the research webpage.

While I elaborated the mathematical framework for the general description of the joint entity consisting of separated quantum entities, focusing on the result that I had obtained already then: namely that standard quantum mechanics cannot describe the joint entity consisting of separated quantum entities because of two of its axioms (see section "The separated product and failing quantum axioms" for details), I also started to read some papers on Bell inequalities. There was indeed quite some activity in this field at that time within the foundations of quantum mechanics community. First of all I have to mention that it occurred almost immediately to me that it should be possible to violate Bell inequalities by situations other than the typical situation of two entangled quantum entities. Perhaps it was my experience with the elaboration of the operational description for separated quantum entities that made this clear to me, because indeed I have had to reflect deeply on how experiments performed on two separated entities behave to be able to elaborate this description, and of course, also situations of nonseparated entities had passed through my mind many times in this respect. And indeed, I managed quickly to invent a situation of two macroscopic nonseparated entities and experiments performed on the entities that violate Bell inequalities (see section "The violation of Bell inequalities" for details). That this had not been seen by the many scientists working in this field at the time intrigued me very much. It was even openly claimed in some of the articles written on the subject at that time that the violation of Bell inequalities was as much a characteristic of 'quantum nature' as was the requirement of Heisenberg's inequalities, and that Bell inequalities were never violated by classical systems.

Digging further into the literature on Bell inequalities brought me into contact with the source of it all: the paper written by Albert Einstein, Boris Podolsky and Nathan Rosen in 1935: "Can quantum mechanical reality considered to be complete?", Phys. Rev., 47, 777, and the problem announced there, now commonly referred to as 'the Einstein Podolsky Rosen paradox'. I remember very well how I held my breath and how my heart was pounding while I read again and again the original EPR paper. because I could see it from an angle that had not been seen before, taking into account all the subsequent papers, on Bell inequalities etc..., that I meanwhile also had read. I could see it from another angle than other scientist had been seeing it, and this was obviously due to my research on the description of separated quantum entities, as I will make clear now. It is a subtle matter, and that is way I have to expose it in some length to make it clear, and that is also the reason why nobody else had been able to see this angle.

What Einstein, Podolsky and Rosen do in the article, and what had been identified by most of the scientists having studied the paper, is the following. First of all they introduce the idea of 'element of reality'. "If without in any way disturbing the state of a physical entity the outcome of a certain observable can be predicted with certainty, there exists an element of reality corresponding to this outcome and this observable". This is a first subtle matter of the whole EPR reasoning and it contains a very deep insight of Einstein in the nature of reality. It means "something real is there if one can predict it". Related with this prediction is an experiment that one can eventually execute, but it must be possible to make the prediction without disturbing the state of the thing that is there. The second step of the EPR paper is to consider the situation of two quantum entities that have interacted but now are flying apart and are so far apart already that they do not interact any longer. In my mind this corresponded to the situation of two separated quantum entities. They have once interacted, but are separated now. The third step in the paper is to consider the quantum mechanical description of this situation. This quantum mechanical description is calculated explicitly in the paper, and it is seen that from the description it follows that position as well as momentum are correlated: in the sense that if one of the entities has position x, the other must have position -x (taking 0 as the place where they both interacted before flying apart), and if one of the entities has momentum p, the other one must have momentum -p. Let us remark that the appearance of the correlation of position and momentum from the quantum mechanical calculation of the situation is a priori not mysterious. A similar effect happens for the situation of classical entities that fly apart having been united earlier on: consider for example the situation of a rock that explodes into two equal pieces that fly apart. The positions and momenta of the two pieces of rock will be correlated in exactly the same way. Now the most subtle part of the whole EPR reasoning appears.

Einstein, Podolsky and Rosen consider the situation where one could eventually measure the position of one of the quantum entities, let us say entity 2, that flies to the right (while entity 1 flies to the left). Suppose that such a measurement of position was carried out, and the position of entity 2 would be registered, for example x, then from the quantum description of the whole situation follows that the position of entity 1 can be predicted, namely -x. Overall this means that a measurement can be performed (the measurement of the position of entity 2) that does not disturb the state of entity 1 (since it is separated from entity 2), and predicts the position of entity 1 (namely position -x, if position x is registered for the measurement of the position of entity 2). A similar reasoning can be made for the momentum. Suppose that the momentum of entity 2 is measured and it found to be p. The quantum description of the situation predicts then that the momentum of entity 1 must be -p. This means that again a measurement can be made (measuring the momentum of entity 2), that does not disturb the state of entity 1 (since entity 1 and 2 are separated), and makes it possible to predict with certainty the outcome -p for entity 1, if p is registered for the momentum of entity 2. This means however that as well the value of the position (-x) as the value of the momentum (-p) can be predicted for entity 1, by means of measurements that do not disturb the state of entity 1. As a consequence both position and momentum must have definite values (-x and -p in the situation considered) at once, because without disturbing the state of entity 1 in any way, they can both be predicted. This is in contradiction with the Heisenberg uncertainty relations, that indeed forbid position and momentum of a quantum entity to have definite values at once. Hence the fundamental contradiction in the Einstein Podolsky Rosen paper is reached. Einstein, Podolsky and Rosen conclude in their paper that this contradiction proves that quantum mechanics is an incomplete theory, in the sense that it cannot represent all elements of reality of a physical entity.

Bohr's reaction to the EPR argument was obscure but determinate following the lines of the common Copenhagen style of reasoning: "it is not allowed in quantum mechanics to make the type of reasoning proposed by Einstein, Podolsky and Rosen, and more specifically, the notion of element of reality does not make sense for quantum mechanical entities". The authority of Bohr and the general influence of the Copenhagen interpretation made that the majority of leading quantum physicists -- with the notable exception of Erwin Schrodinger -- believed that there was not really a deep problem involved in this EPR paradox. Later, perhaps under influence of David Bohm, who certainly took for serious the EPR argument, and invented the entangled spin example as a new and more transparent EPR-like situation, a small group of physicists did believe that Einstein, Podolsky and Rosen had touched on a fundamental problem in quantum mechanics. John Bell was one of them. In the introduction of his 1964 paper: "One the Einstein Podolsky Rosen paradox", 1964, Physics, 1, 195-200, where also the Bell inequalities are formulated, he writes literally the following: "The paradox of Einstein, Podolsky and Rosen was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables." This sentence expresses probably well what most physicists in the little group of the ones that believed that something deep was touched by the EPR reasoning taught, and also what Einstein himself taught to be the problem. We are ready now to come back to my own contribution to the problem and to explain why I started this section by mentioning that my insight in the problems of standard quantum mechanics with the description of separated quantum entities made it possible to look at the EPR reasoning from a completely new angle. This new angle proves that both groups were wrong, more specifically, that quantum mechanics is indeed an incomplete theory -- so this part of the EPR conclusion is correct -- but that the incompleteness is of a totally different nature than the one advocated by Einstein, Bohm, Bell and the others, the lacking of additional variables. The incompleteness is exactly what we encountered: the impossibility for quantum mechanics to deliver a model for the description of the joint entity consisting of two separated quantum entities. At the end of his life Einstein claimed that in his opinion nobody had given a satisfactory answer to the problem put forward in his 1935 paper, and we definitely agree with him on this. We hope, and believe with a certain degree of probability, that Einstein would have liked the solution of the paradox that we put forward. Lets proceed explaining our solution.

A solution of the Einstein Podolsky Rosen paradox

First of all we point out that there is a subtle point in the EPR reasoning that, as far as we know, has been overlooked by everybody: "The Einstein Podolsky Rosen reasoning is a reasoning ex absurdum". What Einstein, Podolsky and Rosen prove is that 'if quantum mechanics is a correct and complete theory' then 'it is an incomplete theory'. This is so because in their reasoning they use quantum mechanics to describe the situation of the two separated quantum entities, entity 1 and entity 2, flying apart after having interacted. In using this quantum description as part of their reasoning, they explicitly use the fact that quantum mechanics is correct and complete in the sense that it describes well this situation. Now of course, even if the reasoning is a reasoning ex absurdum, it is possible to draw a conclusion when a contradiction is reached. The conclusion is that one of the premises of the reasoning is false. So from the proof of the implication ('quantum mechanics is correct and complete' implies 'quantum mechanics is incomplete') can be concluded that (quantum mechanics is incorrect 'or' quantum mechanics is incomplete). Suppose for a moment that we do not want to consider one of these alternatives, the one 'quantum mechanics is incorrect' (we come back to this point later), because this was not at stake at the time of the EPR paradox, then remains as the only possible conclusion 'quantum mechanics is incomplete'. That is the reason why we claim that the EPR reasoning proves, under the hypothesis that quantum mechanics is a correct theory, that quantum mechanics is incomplete. But the status of this proof is the one of a proof ex absurdum. That makes a big difference. We know for example very well that the steps encountered during such a proof ex absurdum have no truth value. This means that one of the intermediate steps in the EPR reasoning, namely that quantum mechanics has to be supplemented by additional variables, has no truth value at all. So it is incorrect to say that quantum mechanics is incomplete and has to be supplemented with additional variables to solve this incompleteness. The EPR reasoning does not give us any hint at all to what would be the nature of the incompleteness that it reveals. This is a point that has been overlooked by the scientist studying the EPR paper, and also Einstein, Podolsky and Rosen were probably not aware of the 'ex absurdum' status of their proof of incompleteness of quantum mechanics.

The forgoing reasoning makes it also easy to understand why I could read the EPR paper from a completely new angle. Indeed, in my work on the description of the joint entity consisting of two separated quantum entities I had proven, in a constructive way, hence not by a reasoning ex absurdum, that standard quantum mechanics could not describe this situation. This can be interpreted as an incompleteness of quantum mechanics (or incorrectness) and hence, this makes it clear what is the origin of the contradiction identified by Einstein, Podolsky and Rosen in their reasoning. So the situation is the following: Quantum mechanics is an incomplete theory in the sense that it cannot describe the situation of separated quantum entities. Since in the EPR reasoning quantum mechanics is used to describe a situation of quantum entities that have interacted at a certain moment and then fly apart and then behave as separated quantum entities, this leads to a contradiction. At first sight -- but this is the tricky part - the contradiction identified in the EPR reasoning is that additional variables have to be introduced that allow position and momentum to have definite values at once contrary to what the Heisenberg uncertainty relations prove. But a more careful analysis of the reasoning, as the one we present here, shows that the real contradiction is that the hypothesis of a correct an complete description of quantum mechanics of the situation of separated quantum entities leads to these additional variables being necessary and hence to a contradiction with this hypothesis of correctness and completeness of quantum mechanics, from which can only be concluded that quantum mechanics is incomplete, but cannot be concluded that additional variables are the remedy for this incompleteness. Although it is years ago now that I made this analysis and published it, I also have to take a breath now, and the thought again comes in my mind 'that it is a subtle matter'. So reader, do not worry, if what I tried to explain in a very clear way now, is not yet clear, just read it over again, and again ...

We have to mention another train of happenings that lead to even more obscurification of the real meaning of the EPR reasoning. Somebody could say: "All right, I follow your argument, coming to the conclusion that the nature of the incompleteness of quantum mechanics already touched by a reasoning ex absurdum in the original EPR paper is the impossibility for quantum mechanics to describe separated quantum entities. But, did the EPR experiments, culminating the experiment of Alain Aspect in 1982, not show, that exactly the situation considered by Einstein, Podolsky and Rosen in their paper, and later reformulated by David Bohm for entangled spins, prove that quantum mechanics 'does' deliver a good description of the two quantum entities flying apart after having interacted? Was this not all what is was about? Namely that, since Bell inequalities still got violated by the far apart coupled spin entities, it is true that quantum mechanics delivers a good model for this situation?" The answer is 'yes' and 'no' and hence needs some explanation. At the time that Einstein proposed the example of quantum entities having interacted and then flying apart, one would have guessed that being sufficiently apart, the entities would start to behave as separated quantum entities. This was still the case when Bohm proposed the spin model and Bell derived his inequalities. The experiments have shown that in this typical situation of quantum entities having interacted and then flying apart it is 'possible', making a big effort, to have the quantum entities flying apart while still remaining non separated entities. This quantum effect has meanwhile been called nonlocality. So the experiments prove that quantum nonlocality (or entanglement) can be retained for quantum entities being distances apart that are of a macroscopic nature, and were 'at first sight' one would not expect this to happen. This is a very interesting and intriguing possibility offered by quantum entities, and we analyse it in great length in the section "Nonlocality, entanglement and the role of space". But it has very little to do with the EPR reasoning. Einstein, Podolsky and Rosen, to be able to make their reasoning, need the situation of quantum entities that fly apart, and do get separated after a while. Only on such separated entities the reasoning can be made. The reasoning can not be made on entities that fly apart and remain nonseparated or entangled. This does not mean that I underestimate the findings of the experiments (as can be seen in the space I give to them in section "Nonlocality, entanglement and the role of space"), the nonlocality effect is one of the most important ones identified for quantum entities in the last decades. But of course, also experiments could be made where the quantum entities that fly apart do get separated and the entanglement gets broken. They have never been made consciously, because the experimentators involved did not understand that these would be the situations leading to the Einstein Podolsky Rosen paradox. But obviously most of the badly performed EPR experiments will separate the flying apart quantum entities, as Alain Aspect once told me when I informed about this, of course being amazed why this information even would interest me. Hence, the EPR reasoning needs these experimental situation where entanglement gets broken, and the flying apart entities get separated, and that is the physical situation that reveals the incompleteness of quantum mechanics. The incompleteness of quantum mechanics is not revealed in the physical situation of quantum entities flying apart and remaining nonseparated, which means that if these situations are well described by quantum mechanics, as the violation of Bell inequalities proves, these is no contradiction.

Some reading over these paragraphs may remark that the situation of separated quantum entities can be described by taking product states of the Hilbert space tensor product that describes the joint entity. Indeed, the incompleteness of standard quantum mechanics of not being able to deliver a model for the joint entity consisting of two separated quantum entities is not revealed on the level of the states, but on the level of the properties (represented by orthogonal projections in standard quantum mechanics), and the dynamics. We explain this in detail in the section "The description of separated entities and failing quantum axioms". The results that we have outlined here are contained in my doctoral thesis, hence publication [10], and also in publications [12] and [14]. In publications [18] and [20] we construct explicitly the missing elements of reality in the standard quantum mechanical description of the joint entity of two separated quantum entities. We do this by defining in a operational way tests and corresponding properties of this joint entity and prove that these properties cannot be represented by orthogonal projection operators. In publications [21] and [22] we analyse in detail what we have put forward here in broad lines. Publication [25] contains an humorist opinion pole that Duch and myself made during a conference on the foundations of quantum mechanics in Urbino in 1995. It is here that for the first time that notion of 'passion at a distance' was introduced for the situation of quantum nonlocality. I remember that we invented the notion as opposed to 'action at a distance', such that we could ask the question "what would happen, for the believers of passion at a distance, if one would fall in love with ones own grandmother, as opposed to killing ones grandfather in the case of belief in action at a distance. The notion of passion at a distance was later also used by Abner Shimony and has been credited to him now. Publications [28,64,65,74] are less directly about the Einstein Podolsky Rosen paradox. The solution of the paradox that we propose here and in the publications that we mentioned already however plays a role in the overal analysis put forward in these publications. In publication [97] we put forward a new and unsolved paradox in relation with the description of the joint entity of two quantum entities.

Aerts, D. (1981). The One and the Many: Towards a Unification of the Quantum and Classical Description of One and Many Physical Entities. Doctoral dissertation, Brussels Free University.

Aerts, D. (1982) Description of many physical entities without the paradoxes encountered in quantum mechanics. Foundations of Physics, 12, pp. 1131-1170.

Abstract: We show that it is impossible in quantum mechanics to describe two separated systems. This is due to the mathematical structure of quantum mechanics. It is possible to give a description of two separated systems in a theory which is a generalization of quantum mechanics and of classical mechanics, in the sense that this theory contains both theories as special cases. We identify the axioms of quantum mechanics that make it impossible to describe separated systems. One of these axioms is equivalent to the superposition principle. We show how these findings throw a different light on the paradox of Einstein, Podolsky, and Rosen.

Aerts, D. (1983). The description of one and many physical systems. In C. Gruber (Ed.), Foundations of Quantum Mechanics (pp. 63-148). Lausanne: AVCP.

Aerts, D. (1984). The missing elements of reality in the description of quantum mechanics of the EPR paradox situation. Helvetica Physica Acta, 57, pp. 421-428.

Abstract:We show that quantum mechanics is not a complete theory. We do not as in the case of Einstein Podolsky and Rosen derive this incompleteness by a logical reasoning ex absurdum, but indicate explicitly which are the missing elements of reality in the description by quantum mechanics of separated physical systems.

Aerts, D. (1984). The missing elements of reality in the description of quantum mechanics of the EPR paradox situation. Annales de la Fondation Louis de Broglie, 2, pp. 163-175.

Aerts, D. (1985). The physical origin of the Einstein Podolsky Rosen paradox. In G. Tarozzi and A. van der Merwe (Eds.), Open Questions in Quantum Physics: Invited Papers on the Foundations of Microphysics (pp. 33-50). Dordrecht: Kluwer Academic.

Aerts, D. (1985). The physical origin of the EPR paradox and how to violate Bell inequalities by macroscopical systems. In P. Lathi and P. Mittelstaedt (Eds.), Symposium on the Foundations of Modern Physics: 50 years of the Einstein-Podolsky-Rosen Gedankenexperiment (pp. 305-320). Singapore: World Scientific.

Duch, W. and Aerts, D. (1986). Microphysical reality. Physics Today, 39, pp. 13-14.

Aerts, D. (1990). An attempt to imagine parts of the reality of the micro-world. In J. Mizerski, A. Posiewnik, J. Pykacz and M. Zukowski (Eds.), Problems in Quantum Physics (pp. 3-25). Singapore: World Scientific.

Abstract: Quantum mechanics is the theory used to 'describe' the processes that take place in the micro-world. From the start quantum mechanics has been a 'strange' theory, in the sense that it seemed to contradict in various ways the image of a micro-world consisting of 'objects' moving around in a three dimensional space, and interacting with each other in this three dimensional space. So from the advent of the theory a lot of disagreement existed as to the 'physical meaning' of this quantum theory, and a lot of discussions of a philosophical nature have taken place among the founding fathers. Only however during the last years experiments have been performed that, independently of the strangeness of the quantum theory, confront us directly with the strangeness of the reality of the micro-world. We have in mind the experiments on the EPR problem. In our opinion to be able to 'understand' the reality of this micro-world, it will be necessary to introduce new concepts, and become aware of old 'classical' prejudices. Certainly in not such a radical way as proposed by what is sometimes called the 'California interpretation' of quantum mechanics, but also in not such a vague way as is proposed by what is called the 'Copenhagen interpretation' of quantum mechanics. Since we nowadays have very 'specific' results, on very refined experiments, we should start 'imagining' how this 'micro-reality' is. The aim of this paper is to try something in this direction, and to propose what could be called a new discipline in theoretical physics. This discipline should investigate whether different kinds of realities (world-models) can correspond with the results of the experiments that we now have, and with the theoretical descriptions given by the quantum theory. And so although we agree that the quantum-world is a very strange one, our aim will be to show that it is not so strange as it looks at the first place. Just because 'a reality' can be much more complicated than one would imagine.

Aerts, D. (1998). The hidden measurement formalism: what can be explained and where paradoxes remain. International Journal of Theoretical Physics, 37, pp. 291-304. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0105126.

Abstract: In the hidden measurement formalism that we develop in Brussels we explain the quantum structure as due to the presence of two effects, (a) a real change of state of the system under influence of the measurement and, (b) a lack of knowledge about a deeper deterministic reality of the measurement process. We show that the presence of these two effects leads to the major part of the quantum mechanical structure of a theory describing a physical system where the measurements to test the properties of this physical system contain the two mentioned effects. We present a quantum machine, where we can illustrate in a simple way how the quantum structure arises as a consequence of the two effects. We introduce a parameter epsilon that measures the amount of the lack of knowledge on the measurement process, and by varying this parameter, we describe a continuous evolution from a quantum structure (maximal lack of knowledge) to a classical structure (zero lack of knowledge). We show that for intermediate values of epsilon we find a new type of structure that is neither quantum nor classical. We analyze the quantum paradoxes in the light of these findings and show that they can be divided into two groups: (1) The group (measurement problem and Schrodingers cat paradox) where the paradoxical aspects arise mainly from the application of standard quantum theory as a general theory (e.g. also describing the measurement apparatus). This type of paradox disappears in the hidden measurement formalism. (2) A second group collecting the paradoxes connected to the effect of non-locality (the Einstein-Podolsky-Rosen paradox and the violation of Bell inequalities). We show that these paradoxes are internally resolved because the effect of non-locality turns out to be a fundamental property of the hidden measurement formalism itself.

Aerts, D. (1998). The entity and modern physics: the creation-discovery view of reality. In E. Castellani (Ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics (pp.223-257). Princeton: Princeton University Press.

Abstract: The classical concept of 'physical entity', be it particle, wave, field or system, has become a problematic concept since the advent of relativity theory and quantum mechanics. The recent developments in modern quantum mechanics, with the performance of delicate and precise experiments involving single quantum entities, manifesting explicit non-local behavior for these entities, brings essential new information about the nature of the concept of entity. Such fundamental categories as space and time are put into question, and only a recourse to more axiomatic descriptions seems possible. In this contribution we want to put forward a 'picture' of what an 'entity' might be, taking into account these recent experimental and theoretical results, and using fundamental results of the axiomatic physical theories (describing classical as well as quantum entities) such as they have been developed during the last decade. We call our approach the 'creation-discovery view' because it considers measurements as physical interactions that in general entail two aspects: (1) a discovery of an already existing reality and (2) a creation of new aspects of reality during the act of measurement. We analyze the paradoxes of orthodox quantum mechanics in this creation-discovery view and point out the pre-scientifc preconceptions that are contained in the well-known orthodox interpretations of quantum mechanics. Finally we identify orthodox quantum mechanics as a first order non classical theory, and explain in this way why it is so successful in its numerical predictions.

Aerts, D. (1999). The stuff the world is made of: physics and reality. In D. Aerts, J. Broekaert and E. Mathijs (Eds.), Einstein meets Magritte: An Interdisciplinary Reflection (pp. 129-183). Dordrecht: Kluwer Academic. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0107044.

Abstract: Taking into account the results that we have been obtained during the last decade in the foundations of quantum mechanic we put forward a view on reality that we call the 'creation discovery view'. In this view it is made explicit that a measurement is an act of a macroscopic physical entity on a microphysical entity that entails the creation of new elements of reality as well as the detection of existing elements of reality. Within this view most of the quantum mechanical paradoxes are due to structural shortcomings of the standard quantum theory, which means that our analysis agrees with the claim made in the Einstein Podolsky Rosen paper, namely that standards quantum mechanics is an incomplete theory. This incompleteness is however not due to the absence of hidden variables but to the impossibility for standard quantum mechanics to describe separated quantum entities. Nonlocality appears as a genuine property of nature in our view and makes it necessary to reconsider the role of space in reality. Our proposal for a new interpretation for space makes it possible to put forward an new hypothesis for why it has not been possible to unify quantum mechanics and relativity theory.

Aerts, D. (2000). The description of joint quantum entities and the formulation of a paradox. International Journal of Theoretical Physics, 39, pp. 485-496. Archive reference and link: http://uk.arxiv.org/abs/quant-ph/0105106.

Abstract: We formulate a paradox in relation to the description of a joint entity consisting of two subentities by standard quantum mechanics. We put forward a proposal for a possible solution, entailing the interpretation of 'density states' as 'pure states'. We explain where the inspiration for this proposal comes from and how its validity can be tested experimentally. We discuss the consequences on quantum axiomatics of the proposal.

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Last modified December 15, 2001, by Diederik Aerts