The Einstein-Podolsky-Rosen (EPR) problem

  • The EPR reasoning
    • figure 8
      The EPR problem must be seen as an ultimate attempt from the part of Einstein to prove `incompleteness of quantum mechanics', while circumventing Bohr's quantum postulate by (allegedly) `measuring a physical quantity without interaction with a measuring instrument'. According to the EPR proposal this is achieved by making use of a system of two correlated particles (1 and 2) that are far apart after having interacted during preparation by a source23. Their initial state is an entangled state. A measurement procedure is considered in which only particle 1 is interacting with a measuring instrument (see figure 8). It is assumed that the state vector of the two-particle system is such that the positions of particles 1 and 2 are strictly correlated. Then, from a measurement of particle 1's position (Q1) the value of particle 2's position (Q2) can be inferred.
    • Since the position of particle 2 can be predicted with certainty, and can be determined without interaction with particle 2, it is assumed to be an element of physical reality, that is, a property the object objectively possesses, independent of any measurement. Such an `element of physical reality' is yielding an explanation by determinism of the `experimentally obtained value of a quantum mechanical observable' (in the sense that the object is supposed to already possess that value before the measurement, compare the possessed values principle).
    • The state of the two-particle system is chosen such that not only the positions of the particles but also their momenta are strictly correlated. By measuring the momentum (P1) of particle 1 it is therefore possible to determine the momentum (P2) of particle 2 without interacting with the latter particle. Therefore also momentum is assumed to be an `element of physical reality' of particle 2.
    • Since particle 2 cannot "know" which observable of particle 1 has been chosen to be measured, position and momentum must simultaneously be objective properties of particle 2.
    • EPR conclude that, since within the quantum mechanical formalism there does not exist any mathematical entity that can represent a state having sharp values of the two incompatible observables position and momentum (of particle 2), quantum mechanics must be incomplete.
    • Remarks on the EPR reasoning:
      • Note that the notion of an `element of physical reality' as used in the EPR paper (viz. as an `objectively possessed value of a quantum mechanical observable') is potentially applicable only in the objectivistic-realist interpretation of quantum mechanics, since in the contextualistic-realist and empiricist interpretations measurement results are thought to be possibly (co-)determined by the interaction with the measuring instrument; in the empiricist interpretation a measurement result for particle 2 does not even exist because there is no measuring instrument corresponding to that observable (compare the distinction between EPR and EPR-Bell experiments). Bohr's attempt at a refutation of the EPR challenge, as contained in Bohr's answer to EPR, hinges on the possibility of a contextualistic-realist interpretation of quantum mechanical observables, in which the possessed values principle is circumvented as a consequence of the `contextual meaning of a quantum mechanical measurement result'.
      • Since position and momentum have continuous spectra they pose mathematical problems that are not essential but are didactically undesirable (compare), the problem is usually discussed in terms of Bohm's formulation using spin observables5, for both particles the observables Q and P being replaced by Sz and Sx, respectively. As the entangled state then is taken the singlet state
        (S = 0)> = |S1z = + >|S2z = − > − |S1z = −> |S2z = + > =
        = |S1x = + >|S2x = − > − |S1x = − >|S2x = + >,
        being entangled both with respect to the Sz and the Sx observables of the two particles.
        More generally, the gist of the EPR problem is represented by a two-particle state described by a state vector that is entangled in two different representations:
        |Ψ> = ∑m cm|a1m>|a2m> = ∑n dn|b1n>|b2n>,
        in which |a1m> and |a2m> are eigenvectors of observables A1 and A2, respectively (and analogously for |b1n> and |b2n> with respect to observables B1 and B2). The essential point is the strict correlations between on one hand values of A1 and A2, and on the other hand between B1 and B2, Ai and Bi being incompatible for i = 1,2.

  • Bohr's answer to EPR
    • Bohr's answer to EPR is based on the `Copenhagen interpretation', viz. the strong correspondence principle (stating that physical quantities are only defined within the context of the measurement serving to measure it), the quantum postulate (emphasizing a certain `wholeness' of a quantum phenomenon), and complementarity. According to Bohr in the EPR experiment the phenomenon must include the measurement arrangement for particle 1, even if the measurement is only regarding particle 2. Since measurement arrangements for position and momentum of particle 1 are mutually exclusive (due to the incompatibility of these observables) they cannot be simultaneously measured. Hence, the strictly correlated observables (position and momentum of particle 2) cannot be `simultaneously defined'. As Bohr puts it: there is an essential ambiguity in the definition of the EPR elements of physical reality (of position and momentum of particle 2), because these are defined within different experimental contexts.
    • Critique of Bohr's answer to EPR
      According to Bohr66 in analyzing the EPR experiment no essential difference is met from the usual way the notion of `complementarity' is applied to quantum measurements like the double-slit experiment. However, since this judgment is based on the `(semi-)objective existence of the strict correlations between measurement results of the two particles of the EPR pair' criticized here, this judgment does not seem warranted. Indeed, the circumstance of `strict correlation' is alien to the "simple examples" previously discussed by Bohr. It plays a crucial role in the EPR reasoning, and it is overlooked by Bohr that it can only be consistently applied to EPR-Bell experiments (in which for each EPR pair both measurement results are available). Probably Bohr's oversight in this matter is caused by his `classical way of thinking about quantum mechanical observables', inducing him to extend `classical objectivity' to quantum mechanical quantities.
      According to Jammer the EPR problem has forced Bohr to change his interpretation from an interactional to a relational one. Not only the interaction between the microscopic object and the measuring instrument (for particle 1) is thought to be instrumental in defining the `element of physical reality' of particle 2, but also the correlation of the quantities of the two particles. Thus,
      relation = interaction + correlation,
      once again illustrating the difference with Bohr's earlier emphasis on `interaction of microscopic object and measuring instrument'.
    • The nonlocality conundrum
      It was realized by Einstein that this relationalism was introducing a feature of nonlocality into the Copenhagen interpretation: the measuring instrument for particle 1 is contributing to the definition of the `element of physical reality' of particle 2 evidently in a nonlocal way. This so-called EPR nonlocality was unacceptable to Einstein, who referred to it as "spooky action-at-a-distance." Its ontological status has been much debated, giving rise to paradoxical ideas awarding it the status of a `conundrum'.
    • An alternative to Bohr's correspondence reasoning, based on a consideration of states rather than observables, but also yielding `EPR nonlocality', makes use of the `individual-particle aspect of the Copenhagen interpretation'. An individual-particle interpretation of the EPR experiment requires that, since the position observable Q1 of particle 1 is well-defined as soon as Q1 has been measured (compare figure 8), the state vector of particle 2 must change discontinuously to the corresponding eigenvector (this, of course, can only be approximately so in case of a continuous spectrum). This discontinuous change (often interpreted as strong von Neumann projection, cf. Publ. 16 and Publ. 57) must have been brought about in a nonlocal way by the measurement of particle 1, since it would happen simultaneously with a measurement event taking place far away.
    • It was Einstein's conviction that the EPR experiment can be understood in a local way if the state vector is not considered a description of an individual object but of an ensemble. Then the discontinuous change of the particle 2 state vector can be understood as a selection of a subensemble, which does not seem to imply any real influence on particle 2 by the measurement of particle 1 (however, compare).
    • `Bohr's answer to EPR' is not consistent with his `correspondence principle (strong form)'
      It has seldom been noticed that Bohr's answer, being based on the strong form of his correspondence principle, is actually inconsistent (see Publ. 16). It assumes that the correlation between the positions of particles 1 and 2 (as well as the correlation between their momenta) is well-defined independent of whether this correlation is measured or not. However, these correlations are described by quantum mechanical observables (Q1Q2 and P1P2, respectively), and, according to the `strong correspondence principle', should be defined only within the contexts of measurements of these observables. Hence, Bohr's answer can only be consistently applied within an experimental context in which a measurement is performed on both particles. A consistent rebuttal of the EPR challenge by Bohr would have denied the definability, within the proposed EPR experiment, of the `correlations employed by EPR to prove the incompleteness of quantum mechanics'. That Bohr did not do so presumably is a consequence of his realist interpretation of quantum mechanical observables, the only difference with Einstein being that Bohr's `realist interpretation' was not an objectivistic but a contextualistic one.
    • Bohr actually accepted the EPR measurement arrangement as a joint measurement of an observable of particle 1 and the correlated observable of particle 2. This acceptance probably is a consequence of the Copenhagen confusion of preparation and measurement.
      Bohr's acceptance of the EPR proposal as a measurement of a particle 2 observable (for instance, Q2, if Q1 is measured) also brings him in conflict with complementarity: it would enable a simultaneous measurement of Q2 and P2; indeed, since Q1 and P2 are compatible, it is possible to measure these observables without any mutual disturbance (see figure 9).
      figure 9

      Then, if the Q1 measurement would be interpreted as a Q2 measurement, it would be possible to obtain in this way sharp values for both position and momentum of one and the same individual particle 2.
    • Fundamental difference between EPR and EPR-Bell experiments
      The fundamental difference between the experiments of figures 8 and 9 marks the difference between the original EPR experiment and `EPR-Bell experiments (Alice measuring Q1 while Bob is measuring P2) intended to test the Bell inequality'. The fundamental difference of the two experimental arrangements (in EPR-Bell experiments measurements are performed on both particles) makes the habit of referring to EPR-Bell experiments as `EPR experiments' very undesirable. In particular, application of von Neumann's projection postulate in EPR-Bell experiments (to the effect that the state change of particle 2 would satisfy this postulate on the basis of a measurement performed on particle 1) is impossible because particle 2 would also be subjected to the `interaction with its own measuring instrument'.
    • On the other hand, it is easy to see that in both EPR and EPR-Bell experiments `von Neumann projection' is applicable to EPR experiments as a preparation principle of particle 2, conditional on the measurement result obtained for particle 1. Thus, applying the theory of conditional preparation to an EPR experiment in which observable A1 is measured in the entangled state |ψ> = ∑m cm|a1m>|a2m>, we obtain as the final state of the pre-measurement the state
      f> = ∑m cm1m>|a2m> |θm>,
      m> pointer states. The states of particle 2 remain unchanged because this particle is not influenced by the measuring instrument.
      By considering the possibility of joint measurements of the pointer observable and arbitrary observables of particles 1 and 2 it follows that the conditionally prepared state (conditional on measurement result m) is the state |ψ1m>|a2m>, reducing to |a2m> if particle 1 is ignored. Evidently, this result coincides with the one obtained by an intuitive application of von Neumann projection to particle 2. Note, however, that this obtains only because the latter particle is not interacting with a measuring instrument (if it were, the state of particle 2 would be influenced just like particle 1's state was changed).
      Note also that for more general initial states |ψ> = ∑mn cmn|a1m>|a2n> we in the same way find as the conditionally prepared state ∑n cmn|a2n> (up to normalization), which generalizes von Neumann projection, the latter states for different values of m not even needing to be orthogonal.
    • The general `lack of distinction between EPR and EPR-Bell experiments' is a consequence of a `realist interpretation', in which no distinction is made between a measurement result as a `reading of a pointer position of a measuring instrument' and a `property of the microscopic object'. In an `empiricist interpretation' an EPR experiment simply does not yield a measurement result for particle 2. Indeed, the EPR "paradox" is an important reason to prefer an `empiricist interpretation' over a realist one (compare), even though, maybe, a consistent application of Bohr's strong correspondence principle (in which a correlation is well-defined only within a measurement of that correlation) would also allow a local contextualistic-realist interpretation in which an observable of particle i (i= 1,2) is (co-)determined by the local context of the measuring instrument of that same particle i (compare).

  • Einstein's reaction to Bohr's answer
    • According to Einstein the `nonlocality' inherent in Bohr's relationalism is a consequence of the latter's assumption of completeness. This would imply a trade-off of `completeness' and `locality', in the sense that, since
      it would logically follow that
      Both choices (either `completeness' or `locality') are possible to the exclusion of the other one. Whereas Bohr's choice was `completeness', was Einstein's choice `locality'. It seems that, whereas Einstein was aware of the possibility that either choice could be made (although he preferred 'locality'), it seems that for Bohr only the choice of `completeness' was open (compare).
    • Critique of Einstein's reaction
      It should be noted that here `completeness' means completeness in the restricted sense, associated with `contextuality due to the interaction of object and measuring instrument'. Hence, in the EPR discussion `incompleteness' is associated in the first place with objectivity (i.e. noncontextuality) rather than with an ensemble interpretation of the state vector (although for Einstein the `ensemble interpretation' was crucial). Actually, the `Bohr-Einstein discussion on the interpretation of quantum mechanics' was not about whether a realist interpretation of quantum mechanics is possible (both having in mind a classically-realist notion of `physical quantity'), but rather whether a `realist interpretation of quantum mechanical observables' (assumed by both) can be objectivistic (i.e. independent of the `observer including his measuring instrument', Einstein) or should be contextualistic (Bohr).
    • Although in the EPR problem Einstein did not transcend the `quantum mechanical formalism', his paradigm may have been incompleteness in the wider sense, his conviction being that `the ψ-function is not to be understood as the description of an individual object but rather of an ensemble of such objects', each individual object being characterized by an `element of physical reality'. This has widely been interpreted as implying that the EPR discussion was about the possibility of hidden variables (rather than about the contextuality implied by the correspondence principle).
      It is questionable whether so many physicists, raised in the classical tradition of an `objective description of physical reality', would have chosen Bohr's side if Einstein would have stressed the `objectivity versus contextuality' antagonism rather than the issue of whether the state vector describes an individual particle or an ensemble (compare). As a result of the confusion between `completeness in the wider sense' and `completeness in the restricted sense' Einstein's opposition was widely interpreted as `yielding to the lure of metaphysics' as implied by the assumption of `incompleteness in the wider sense'.
    • Possibility of `additional assumptions (loopholes)'
      It is important to note that the logical reasoning employed in Einstein's reaction to Bohr's answer, viz.
      (a → b) → (not b → not a),
      is valid only if there are no additional features contributing to the realization of b. If there is such an additional feature c, then we have
      ((a and c) → b) → (not b → (not a or not c)),
      and the conclusion `not a' cannot be drawn.
    • In the EPR discussion an additional assumption exists, viz. the assumption of a `realist interpretation of quantum mechanical observables'. Both Bohr and Einstein agreed on that (compare). Hence, instead of `completenessnonlocality', we have
      completeness and realist interpretationnonlocality
      It is logically possible that the assumption of locality does not entail `incompleteness' but rather the impossibility of a `realist interpretation of quantum mechanical observables'.
    • Actually, Einstein's expectation that an assumption of `incompleteness of quantum mechanics' is sufficient to save `locality' in an objectivistic-realist interpretation, does not come true. It is not possible to consider conditional preparation of particle 2 in the EPR experiment as a mere selection of a subensemble, particles 2 having values of quantum mechanical observables already before the particle 1 measurement is carried out (as assumed by EPR). The reason for this is the invalidity of the possessed values principle. The failure of this principle entails the impossibility of a simultaneous existence of all possible EPR elements of physical reality. In this respect Bohr's allegation of `ambiguity of the EPR element of physical reality' was completely justified: since it is impossible to assume that all particle 2 observables had their values already before the measurement, a realist interpretation of quantum mechanical observables in the EPR experiment is virtually forced into a nonlocal contextualism in which the measurement of a particle 1 observable influences the reality of particle 2 in such a way that the correlated observable can obtain a certain value.
      Nevertheless, Bohr's reasoning is not cogent since there is an alternative to Bohr's `contextualistic-realist interpretation', viz. the assumption that this interpretation is not applicable to the EPR experiment because in that experiment no measurement is carried out on particle 2, and hence, contrary to Bohr's assumption, observables of that particle are not well-defined (compare). Due to the `fundamental difference between EPR and EPR-Bell experiments', causing the experimental contexts of these experiments to be fundamentally different (compare), `nonlocal contextuality' is not at all enforced in an `empiricist interpretation': only the local contexts of the particles need to be taken into account.

  • `EPR nonlocality' and experiment
    • Contrary to a widespread belief nonlocality has not been experimentally demonstrated. On the contrary, measurements like the bi-local ones performed by Aspect et al. exhibit quantum mechanical locality in the sense that in correlation measurements like the one of figure 9 the measurement results of one observable (Q1, say) are independent of which observable of particle 2 is measured jointly (e.g. P2). This is sometimes called `parameter independence'; it is consistent with `compatibility of the observables measured jointly' (c.q. the principle of local commutativity of `local quantum field theory').
    • Of course, in case of `parameter independence' there may be `outcome dependence', to the effect that the individual outcome of the measurement on particle 1 depends on the individual outcome of the measurement on particle 2 (and vice versa). `Outcome dependence' could have different causes. It could be a consequence of
      i) a nonlocal interaction, instantaneously influencing one particle if a measurement is performed on the other one;
      ii) a previous interaction of the particles, by which the two-particle object has been prepared in a correlated state of the two particles.
    • From a physical point of view the explanation of `correlation of the measurement results by means of a nonlocal interaction' is extremely implausible. The reason for this implausibility is the experimental corroboration of `parameter independence', implying that the marginals of the joint probability distribution of a joint measurement are independent of whatever happens to the other particle. This means that, if the individual objects of a pair would interact nonlocally, then the effects of this interaction would cancel on the statistical level, so as to become unobservable to any quantum mechanical measurement. This is sometimes referred to as a `peaceful coexistence' (of `individual nonlocality' and `statistical locality'). Although such a `peaceful coexistence' is logically possible, it is highly improbable. It could only be understood on the basis of a kind of cosmic conspiracy, washing out any experimental trace of `nonlocality on the individual level', so as to make it consistent with `locality on the statistical level of the quantum mechanical ensemble'. `Individual nonlocality' is reminiscent of the 19th century world aether, also perfectly hiding itself to any attempt at approaching it in an operational way. Since there is no empirical evidence supporting `individual nonlocality' (not even by the Aspect measurements!), it seems preferable to deal with it in the same way Einstein dealt with the world aether.
  • Remarks on `EPR nonlocality'
    • Origin of `EPR nonlocality'
      The ultimate source of the idea of `EPR nonlocality' is Bohr's application of his notion of quantum phenomenon to the EPR experiment, often associated with the the idea of entanglement.
      A form of `nonlocality' is also implied by a `realist individual-particle interpretation of the quantum mechanical formalism', in which it is assumed that in the original EPR experiment of figure 8 an individual measurement on particle 1 instantaneously influences particle 2 so as to cause strong von Neumann projection of its state. According to Einstein the latter `nonlocality problem' can be solved by abandoning the individual-particle interpretation (to be associated with `completeness of quantum mechanics') in favour of a `(realist) ensemble interpretation' (representing `incompleteness of quantum mechanics'), `strong von Neumann projection' representing a transition to a subensemble (compare).
    • Dubious role of the `possessed values principle'
      Problems with respect to the `possessed values principle' cast strong doubts on Einstein's `ensemble solution' of the `EPR nonlocality' problem:70 it is not possible to attribute in an objective sense, independent of any measurement, to each individual particle 2 a well-defined value of both Q2 and P2. Bohr was right when observing that `the ensemble Einstein needs for his reasoning' does not seem to exist (by Bohr this is cast in terms of ambiguity of the definition of the EPR `element of physical reality').
      This need not imply, however, that Bohr's `contextuality solution' (entailing `EPR nonlocality') is the compulsory alternative. Bohr and Einstein shared a realist interpretation of quantum mechanical observables, the only difference being whether a particle possesses a value of an observable objectively (Einstein), or only contextually (Bohr). According to Bohr the mutual exclusiveness of the measurement arrangements of incompatible observables of particle 1 would prevent simultaneous attribution of values to incompatible observables of particle 2. Bohr's solution implies that the `possessed values principle' is circumvented, however, at the expense of introducing `nonlocality'.
      It seems to me that the solutions of Einstein and Bohr, being equally metaphysical, are equally unacceptable: neither Einstein's `element of physical reality' nor Bohr's `nonlocality' has directly observable consequences.
    • There is a third possibility, viz. an empiricist interpretation, in which a particle cannot simultaneously possess values of incompatible observables because it cannot possess values of quantum mechanical observables at all (since observables refer to the `measuring instrument' rather than to the `microscopic object'). A value of an observable of particle 2 is well-defined only if a `measuring instrument for measuring that observable' is actually present (like in the experiment of figure 9). In the `empiricist interpretation' it is necessary to draw a clear distinction between `EPR experiments' and EPR-Bell experiments, in `EPR experiments' quantum mechanical observables of particle 2 being undefined because it is a `preparation' rather than a `measurement' of that particle.
    • It should be noted that, since in the EPR experiment of figure 8 we do not have a measurement of particle 2 but a preparation, there is no objection to a von Neumann projection as such, since it can be interpreted as a conditional preparation (compare Publ. 57; this, of course, is strictly true only for observables having discrete spectra). Because of the strict correlation in the initial two-particle state, and due to the fact that the state of particle 2 is not influenced by the measurement on particle 1, von Neumann projection even turns out to be applicable to EPR as if it were a first kind measurement of the particle 2 observable. This, incidentally, may have contributed to the interpretation of the `EPR experiment' as a `measurement of particle 2', making Bohr unjustifiedly think that it was allowed to apply his quantum postulate also to that particle.
    • Note that a confusion of `preparation' and `measurement', as observed here, is impossible in an empiricist interpretation, because in that interpretation a quantum mechanical observable is a representation of a `measuring instrument/procedure'. In the EPR proposal of figure 8 no measuring instrument for particle 2 is present. If a `measurement result' corresponds to a `final pointer position of a measuring instrument', then that measurement result cannot have been present before the measurement, and the EPR element of physical reality cannot be represented by a `quantum mechanical measurement result'.
    • In the `empiricist interpretation' the transition from the initial (correlated, entangled) state to the final (conditional) one is not a `(realist) change of the state of an individual particle', nor a `transition to a (realist) description of a subensemble', but it is just a transition to a different preparation procedure, viz. a procedure of conditional preparation, in which particles 2 are selected on the basis of the measurement results for particle 1. In order to be able to execute a conditional preparation of particle 2 by means of selection on the basis of the particle 1 measurement result, this latter result must be known to the selector. Hence, no `nonlocality problem in the Bohr/Einstein sense' can arise in an `empiricist interpretation', contrary to what is the case in a `realist individual-particle interpretation' (compare) or even in a `realist ensemble interpretation' (compare). This is an important reason to prefer an `empiricist interpretation'. If such an interpretation is entertained from the outset, then the EPR proposal does not suggest any `nonlocality'.

  • Role of `strict correlations' in the `EPR experiment'
    In the EPR proposal use is made of a special state vector warranting strict correlations (if measured) between Q1 and Q2, and between P1 and P2, respectively. Such strict correlations are unexpected if each of the particles would behave, independently of the other one, according to the irreducible indeterminism implicit in Jordan's assertion. This poses to the `Copenhagen interpretation' a `problem of explanation', addressed in the `EPR challenge'.
    • Two possible explanations of `strict correlations'
      Ignoring "mystical" solutions like the one presented by Zukav in his book The dancing Wu Li masters0 (exploiting the idea of a certain `cosmic harmony', necessary to be able to understand strict correlations in EPR-Bell experiments if Jordan's assertion would be applicable), two different "physical" explanations of `strict correlations' use to be contemplated:
      i) the possessed values principle, to the effect that `strict correlation' is a result of its already having been created in the `process of emission of the EPR particle pair' (the individual measurement results being supposed to be created already at that stage), correlation being preserved while the particles fly apart and the observables are finally measured in faithful measurements (Einstein's explanation by determinism);
      ii) nonlocal influences, strict correlation being assumed `not to have been there beforehand', but to come into being by means of strong von Neumann projection, the measurement of particle 1 in a nonlocal way realizing the value of the (correlated) observable of particle 2 (compare e.g. Publ. 57). This is the Copenhagen interpretation's `explanation by nonlocal influencing'.
    • Both solutions have their drawbacks. Thus, as we know now, the `possessed values principle' cannot be maintained (also here). So, Einstein's explanation does not work properly. For the majority of physicists this may have been reason enough to reject (yet too hastily) Einstein's proposal, and accept the Copenhagen solution.
      However, also the Copenhagen solution is not very attractive. Thus, by Einstein the Copenhagen interpretation's `explanation by nonlocal influencing' is disqualified as ``spooky action at a distance.'' `Nonlocal influence on the (experimentally relevant) relative frequencies of measurement results' being unobservable, `EPR nonlocality' is as metaphysical as were in Copenhagen eyes Einstein's `elements of physical reality'.
      Note that at the time of the EPR discussion `no convincing reason to make a choice between the two (metaphysical) alternatives' was available (the Kochen-Specker theorem having been derived much later). The only `reason for the general rejection of Einstein's explanation at the time of the EPR discussion' I can think of is the large influence of logical positivism/empiricism, combined with the fact that the aspect of `nonlocality' remained relatively unnoticed for a long time (the latter becoming fashionable only as a result of Bell's work).
    • Things changed as a result of the Kochen-Specker theorem, which theorem implies that Einstein's idea of `quantum mechanical measurement results as elements of physical reality' cannot be maintained. Taking this into account a simplistic application of classical logic to the dichotomy of `determinism versus nonlocality' might seem to result in a conclusion of `EPR nonlocality'.
      Nevertheless, there are physicists who, as a result of the wide acceptance of this latter conclusion by the physics community, have a bad conscience, and who look for loopholes0 to escape from it. Such a loophole may be found in the form of an additional assumption, not explicitly mentioned but nevertheless liable to yield an alternative explanation of experimental results. A well-known loophole is the `efficiency loophole', taking into account the `inefficiency of detection processes'. However, as a result of a general belief in `nonlocality' such loopholes are often considered to be tiny, and it is generally expected that they will all be closed some day to leave `nonlocality' as the only explanation.
    • In my view this expectation is unwarranted, however, an `explanation by determinism of strict correlations' not at all having been demonstrated to be impossible. Admittedly, in view of the Kochen-Specker theorem it is evident that such an explanation cannot be given by quantum mechanics. But perhaps we should not ask such explanations from quantum mechanics, because this theory presumably is unable to yield it, analogous to the inability of the `classical theory of rigid bodies' to explain -rather than describe- the `rigidity of a billiard ball' (compare). Perhaps Einstein's `elements of physical reality' are really there to explain measurement results by determinism (as well as the `strict correlations' they may exhibit), be it that these `elements' may not be described by quantum mechanics but by some subquantum theory (compare). Like an `atomic theory of the solid state' is able to explain the `rigidity of a billiard ball', might `strict correlations' have to find their explanation within a subquantum mechanical theory through the existence of `subquantum elements of physical reality'.
  • Criticism of the assumption of `nonlocality'
    • `Nonlocality' versus `incompatibility'
      It should be realized that `nonlocality' entered the (in)completeness discussion only through the back door. The `(in)completeness discussion' is about the crucial difference between classical and quantum mechanics, statistical applications of classical mechanics being considered to yield incomplete descriptions of reality. The Copenhagen claim was that, as a result of the incompatibility of observables Q and P, in this respect quantum mechanics is fundamentally different from classical mechanics. Hence, `incompatibility of observables' is the crucial issue, distinguishing `(possibly incomplete) classical mechanics' from `(allegedly complete) quantum mechanics'.
    • How `nonlocality' replaced `incompatibility'
      In order to challenge the Copenhagen claim of `completeness of quantum mechanics' Einstein had to deal with the question of `whether well-defined values can be simultaneously attributed to incompatible observables'. In trying to do so he was hampered by the `impossibility of simultaneously measuring incompatible observables' (compare complementarity).
      To circumvent this problem Einstein and co-workers considered the EPR experiment, allegedly `performing a measurement of an observable of particle 2 without interacting with it' by `measuring a compatible observable of a distant particle 1'. It was reasoned that `incompleteness of quantum mechanics' follows from the possibility of in this way `simultaneously attributing to particle 2 well-defined values of the incompatible observables Q2 and P2'. By thus relying on an (alleged) simultaneous measurement of distant particles `nonlocality' entered the scene.
      It should be remembered, however, that the `nonlocality issue' is strongly dependent on which interpretation of quantum mechanics is adopted (for instance, in an `ensemble interpretation' no nonlocality follows from the EPR experiment). It seems to me that the `nonlocality issue' has distracted attention away from the problem that is really at the basis of the EPR problem, viz. incompatibility of observables. By comparing `incompatible observables' in an indirect way by means of their correlations with a `compatible observable' EPR have complicated the issue to such an extent that confusion could arise with respect to the importance of the correlation between `compatible observables' as compared to the `correlation between incompatible observables'.
    • Return to metaphysics?
      The widely held belief in `nonlocality', based on the EPR problem as well as on later developments related with EPR-Bell experiments (in particular, those connected with the Bell inequality), demonstrates a remarkable change with respect to the `physicist's fear of metaphysics' as compared with the `logical positivist/empiricist one rejecting Einstein's elements of physical reality'. Experimental results being consistent with `locality of the phenomena as expressed by a general experimental satisfaction of the quantum mechanical principle of local commutativity (implying that in EPR and EPR-Bell experiments the statistical measurement results obtained for particle 1 are independent of what happens to particle 2, and vice versa) do not suggest any nonlocality. Nevertheless, nonlocality is widely thought to be there, although there is no experimental evidence for it71. It seems that the combined influence of Bohr, Bohm and Bell has been sufficiently strong to outweigh Einstein's opposition against `nonlocality'. It seems to me, however, that this may have rather a sociological origin than a physical one. Although by itself nowadays `taking seriously subquantum reality' is as legitimate as `taking seriously the atomic constitution of a billiard ball', does this not imply that such an endeavour is not restricted by methodological rules. Abiding as long as possible with successful general ideas like the idea of `relativistic locality/causality' seems to be one of these rules, in particular when there is a reasonable alternative (viz. `incompatibility') for giving up an elusive `nonlocality'.
    • Generalized observables and EPR-Bell experiments
      Insight into the problem of `nonlocality versus incompatibility' is gained by considering generalized EPR-Bell experiments in which the two jointly measured observables both correspond to joint nonideal measurements of incompatible standard observables. Such measurements can be used to analyze `(violation of) the Bell inequality' by the Aspect measurements.
      From this analysis it becomes evident that `violation' can be interpreted as a consequence of `local mutual disturbance of the measurement results in a joint (nonideal) measurement of incompatible standard observables of the same particle' rather than as being caused by a `nonlocal disturbance of one particle due to a measurement of a (compatible) observable of the other particle (compare)'.
      It is important to note that in the `EPR experiment' of figure 8 no (local) disturbance of particle 2 by any measuring instrument was allowed. By setting up a measuring instrument for particle 2 the very condition for applying the EPR reasoning has been forfeited. But also Bohr's reasoning becomes obsolete because that instrument can create `its own measurement context' (compare). The important we can learn from analyzing EPR and EPR-Bell experiments is that Einstein's idea of `quantum mechanics as a description of an objective microscopic reality is not feasible. The domain of quantum mechanics is delimited by the impossibility to ignore the essential influence `measurement itself' is exerting on the results obtained.
      If it comes to judging the contest between Einstein and Bohr on the `(in)completeness issue of quantum mechanics, it turns out that with respect to the `influence of measurement' Bohr's insight is unsurpassed. On the other hand, by not sufficiently distinguishing between EPR and EPR-Bell experiments Bohr has contributed to the `nonlocality conundrum' already for such a long time unnecessarily burdening the discussions on the foundations of quantum mechanics.