The Bell inequality in quantum mechanics
  • The (standard) Bell inequality

    figure 10
    Although the Bell inequality was first derived for hidden variables theories is it illuminating to see under which conditions this can also be done within quantum mechanics. The Bell inequality has been derived for EPR-Bell measurements, like the ones performed by Aspect et al. (a discussion of the conditions allowing such a derivation is given here). There are four such measurements involved (see figure 10), in which either standard observable A1 or B1 of particle 1 is measured jointly with standard observable A2 or B2 of particle 2. Observables A1 and B1 are chosen to be incompatible: [A1, B1]O; so are A2 and B2: [A2, B2]O. Observables of different particles are compatible. If all observables are dichotomic (i.e. have only two values), their two values being +1 and −1, then the Bell inequality can be expressed in terms of the expectation values <A1A2>, etc. of the correlation observables A1A2, etc. according to

    |<A1A2> − <A1B2>| ≤ 2 + <B1A2> + <B1B2>,
    which is also called the Clauser-Horne-Shimony-Holt (CHSH) inequality. In the Aspect measurements the particles are actually chosen to be photons. The observables correspond to measurements of photon polarization in different directions perpendicular to the direction of motion of the photons. The experiments in an unambiguous way show experimental violation of the Bell inequality.
  • The BCHS inequality
    A more general inequality than the CHSH inequality is the Bell-Clauser-Horne-Shimony (BCHS) inequality. This inequality is expressed in terms of probabilities rather than expectation values. Therefore, it is independent of the (eigen)values of the observables (compare). There is no restriction to dichotomic ones, nor to the values +1 and −1. Denoting the (eigen)values of observable A1 by a1i, etc., the BCHS inequality is given by
    −1 ≤ p(a1i,b2m) + p(b1j,a2k) + p(b1j,b2m) − p(a1i,a2k) − p(b1j) − p(b2m) ≤ 0,
    in which the univariate probabilities (having a single digit as argument) are marginals of the bivariate ones. Under the restrictions, given above, the CHSH inequality can straightforwardly be derived from the BCHS inequality (e.g. Publ. 15).
  • `Bell inequality' as an expression of `classicality'
    The Bell inequality holds true in any classical statistical theory satisfying Kolmogorov's statistical axioms (e.g. Publ. 15). This is in agreement with the idea that `violation of the Bell inequality' marks the typically non-classical features of quantum mechanics expressed by the `existence of incompatible observables' and the notion of `entanglement'.
    On the other hand, it should be noted that `satisfaction of the Bell inequality' does not imply that `everything would be classical': in quantum mechanics the inequality is often satisfied even if some of the observables do not commute (it, actually, requires some care to find a set of observables for which the Bell inequality is not satisfied). By the same token, the Bell inequalities may be satisfied if the state exhibits a certain measure of entanglement (e.g. Werner states0).
    On the other hand, `violation of the Bell inequality' can be seen as a sign of `non-classicality'.
  • Assumptions allowing derivation of the Bell inequality within quantum mechanics
    As is well known, the Bell inequality can be violated by quantum mechanics. Therefore, next to the quantum mechanical formalism additional assumptions will have to be invoked in order that the Bell inequality be derivable within quantum mechanics. Let us consider some of these:
    • i) Existence of a `quadrivariate probability distribution'
      In order to analyze this question I start from a general theorem, derived independently by Fine and by Rastall73.

      A sufficient condition for derivability of the BCHS (and, hence, the Bell) inequality is the existence of a `quadrivariate probability distribution'62 p(a1i,b1j,a2k,b2m) from which the bivariate probabilities can be derived as marginals.

      As a consequence of Gleason's theorem0 (within the domain of quantum mechanics specifying a `probability distribution' as a `linear functional of the density operator'), for the `quadrivariate probability distribution' we should assume here that

      p(a1i,b1j,a2k,b2m) = Tr ρ Mijkm,
      in which Mijkm are elements of a quadrivariate POVM {Mijkm}.
      The importance of this theorem seems to be underestimated, probably because within the standard formalism the existence of such a `quadrivariate probability distribution' is impossible if incompatible standard observables are involved (compare). However, within the generalized formalism such restraint is unwarranted (compare).
      • Criticism and appraisal of `existence of quadrivariate probability distributions'
        The problematic character of `joint probability distributions of incompatible observables' within standard quantum mechanics is well-known. For this reason it is not surprising that it has been thought that such probability distributions do not exist at all, for instance, because the requirement of `asymptotic stability of the quadrivariate relative frequencies' might not be satisfied if incompatible (standard) observables are involved. If `quadrivariate probability distributions' would not exist, then the above theorem would be inapplicable.
        Note, however, that the theorem is valid not only for probabilities, but, more generally, also for relative frequencies at each single value of N. Hence, it is sufficient that a `quadrivariate relative frequency' exist for each value of N; `asymptotic stability' is not required for application of the theorem.
        Moreover, `quadrivariate probability distributions of incompatible observables' do exist if the restriction to the `standard formalism' is relinquished (compare). Hence, applicability of the theorem is secured at least within the generalized formalism.
    • ii) Compatibility of all (standard) observables that are involved
      Note that the above theorem is satisfied if the four standard observables A1, B1, A2, and B2 that are involved in the Bell inequality are `all mutually compatible'. Then a `quadrivariate probability distribution' exists, viz. (compare)
      p(a1i,b1j,a2i,b2j) = Tr ρE1iF1j E2iF2j,
      in which {Eni} and {Fnj} (n = 1,2) are the (mutually commuting) spectral representations of the four mutually compatible standard observables An and Bn, respectively. For this `quadrivariate probability distribution' the Bell inequality is satisfied for any density operator ρ.
      It is evident from the `derivability of the Bell inequality from compatibility of all observables that are involved' that `incompatibility of observables' is a necessary condition lest the Bell inequality be violated:
      there is no violation of the Bell inequality without incompatibility.
      Experimental violation of the Bell inequality by certain quantum mechanical (standard) observables in EPR-Bell experiments implies that, at least for these observables, neither the `possessed values principle' can hold, nor can there exist a `joint probability distribution of values of these observables'. This result is hardly surprising, since incompatibility is well-known to be at the basis of both the `invalidity of the possessed values principle', and of the `non-existence of joint probability distributions of standard observables'.
    • iii) Existence of `quadruples of measurement results'
      In order that `quadrivariate relative frequencies' (as well as quadrivariate probability distributions p(a1i,b1j,a2k,b2m)) exist there is one important requirement that should be satisfied, viz. the four measurement results a1i, b1j, a2k, and b2m should be attributable to a common quantum event, so as to be liable `to be combined into the quadruple (a1i,b1j,a2k,b2m) as the argument of a relative frequency or probability'.
      Since `quadrivariate probability distributions' cannot exist unless such quadruples exist, we have here a more fundamental possibility to explain the nonexistence of `quadrivariate probability distributions' in case of the Aspect measurements. Indeed, how can the existence of such quadruples be justified if the four measurement results a1i, b1j, a2k, and b2m are not obtained coincidentally in one single experiment but rather by means of four different measurements (cf. figure 10), each measurement probing its own quantum events (e.g. Publ. 41)?
      I am aware of three ways in which `existence of quadruples of measurement results' has been implemented also if incompatible observables are considered:
      iiia) application of the `possessed values principle';
      iiib) application of `counterfactual definiteness';
      iiic) generalized measurements.
      • iiia) Quadruples and the `possessed values principle'
        The Bell inequality can be derived if every element of an ensemble has a well-defined value for the four observables that are involved. Hence, the inequality could be derived within quantum mechanics if the possessed values principle were satisfied (e.g. section 9.4.1 of Publ. 52). From the experimentally found possibility of `violating the Bell inequality if incompatible observables are involved' it follows that the `possessed values principle' cannot be generally valid. Hence, it seems that such derivations, which are characteristic of an `objectivistic-realist interpretation of quantum mechanical observables', are as obsolete as is that interpretation (compare).
        On the other hand, in a `contextualistic-realist interpretation of quantum mechanics' the `nonexistence of quadruples for the standard Aspect measurements' can in a natural way be explained by the fact that in these experiments different contexts are involved. This points into the direction of `objectivity as a cause of derivability of the Bell inequality' (compare).
        Within quantum mechanics `quadruples of measurement results are to be expected only in a `joint measurement of four observables'. Within the `standard formalism' these observables must necessarily be mutually compatible. Hence, within the domain of application of the standard formalism `derivability of the Bell inequality' is not to be expected if incompatible observables are involved.
      • iiib) Quadruples and `counterfactual definiteness'
        As discussed here `counterfactual definiteness' does not address the objectivistic-realist idea of `quantum mechanical measurement results as objective properties of the microscopic object', but it is referring to `contextualistic-realist' or even `empiricist' ideas in which measurement results are either `properties of the microscopic object, influenced by the measurement' or `post-measurement properties of the measuring instrument'. Hence, the `possessed values principle' is not thought to be applicable.
        Nevertheless, it might be possible to construe for each of the four EPR-Bell experiments of figure 10 quadruples (a1i,b1j,a2k,b2m) by adding to the two `measurement results found in a measurement (e.g. a1i, a2k)' two other results (viz. b1j, b2m) `that would have been found if the other two observables had been measured instead of the actually measured ones'. Once again, it follows from the ensuing derivability of the Bell inequality that such a `counterfactual definiteness' cannot obtain if the Bell inequality is violated.
        Note that the idea of `counterfactual definiteness' is contradicting the idea of Heisenberg disturbance, and, therefore would not even be warranted if reality can be attributed to the latter idea. Since `Heisenberg disturbance' is corroborated by applications of the generalized formalism (compare), it follows that `counterfactual definiteness' can now be considered obsolete too.
      • iiic) Quadruples and `generalized measurements'
        If a measurement has four pointers, and hence is yielding for every quantum event a `quadruple of measurement results', then the Bell inequality is satisfied as a result of the theorem. This is the case, for instance, in a `joint measurement of four compatible standard observables', for which the existence of the quadrivariate probability distribution p(a1i,b1j,a2k,b2m) = Tr ρE1iF1jE2kF2m is warranted by (standard) quantum mechanics.
        However, within the domain of application of the generalized formalism it is possible to consider joint measurements in which incompatible observables are involved. An example is given in figure 11. From the `experimentally obtained quadruples' it is possible to derive relative frequencies that in the asymptotic (large N) limit can be compared with the quadrivariate probabilities predicted by `generalized quantum mechanics'.
        Presently, measurements of the type of figure 5 are performed by an increasing number of groups (compare chapter 8 of Publ. 52), corroborating the generalized theory. As a result of the existence of `quadruples of measurement results', in the `generalized Aspect measurement of figure 11' the Bell inequality must be satisfied.
        Note that in the joint probabilities of these experiments `measurement results' have to be taken in the empiricist sense, implying that here `derivability of the Bell inequality' is neither a consequence of the `possessed values principle', nor of `counterfactual definiteness'.
  • The assumption of `locality'
    • Early developments
      The assumption of `locality' as a `condition allowing a derivation of the Bell inequality' has a long history. The idea of `nonlocality of the quantum world', allegedly following from `experimental violation of the Bell inequality' has its origin both within quantum mechanics and within hidden variables (subquantum) theory.
      • First, within quantum mechanics there was Einstein's trade-off between `(Copenhagen) completeness' and `locality', barring `locality' if quantum mechanics would have to be complete (and vice versa). Indeed, `nonlocality' seemed to be a necessary consequence of the Copenhagen way of interpreting in the EPR experiment the state vector as a `description of an individual object (rather than an ensemble)', allegedly necessitating (nonlocal) strong von Neumann projection (also).
      • Second, within hidden variables theory: although Bell did not endorse the Copenhagen completeness thesis, and, hence, could have relied on Einstein's assumption as regards `locality of an ensemble interpretation of the state vector', he was so impressed by Bohm's nonlocal hidden variables theory that with Bohm he became firmly convinced that the quantum world must be nonlocal notwithstanding `experimental corroboration of the principle of local commutativity'. For these reasons he did not try to prove that `generic hidden variables theories' would be impossible, but he just set out to prove the `impossibility of local hidden variables theories'.
        Bell's "success" in this endeavour convinced him that the assumption of `locality of hidden variables theory' is crucial for being able to prove the Bell inequality to be necessarily satisfied. For the majority of physicists, by the 1960s still endorsing the `Copenhagen interpretation' (but presumably mostly in the realist sense of quantum mechanics textbooks, compare), the `apparent support of EPR-nonlocality by Bell's result' has probably been reason enough to accept the idea that `reality behind the quantum mechanical phenomena must be nonlocal'.
        Even today `corroboration of violation of the Bell inequality by the standard Aspect measurements (cf. figure 10)' is widely interpreted as `experimental evidence of violation of locality', although it is realized that this `nonlocality' is unobservable by means of quantum mechanical measurements. For this reason it has been conjectured that the `nonlocality believed to obtain in EPR-Bell experiments' is another type of `nonlocality' than the one expressed by parameter dependence. It is referred to as `hidden nonlocality', `inseparability' (because of its alleged relation to `entanglement', compare), or even by fancy names like `passion-at-a-distance (as opposed to action-at-a-distance)', or `peaceful coexistence'. It is widely believed that
        `there is no violation of the Bell inequality without EPR-nonlocality'.
        Note, however, that this belief is unwarranted, because, as argued here and here, `derivability of the Bell inequality' may have quite a different reason than `nonlocality' (e.g. Publ. 43).
    • Later developments
      • Within the standard formalism
        It was realized only later that, under certain assumptions (like iiia) or iiib)), the `Bell inequality' can be derived within quantum mechanics without reliance on subquantum theories, even though `possible experimental violation of the Bell inequality' makes these assumptions questionable. Indeed, assumptions iiia) and iiib) are inspired by the classical paradigm (`quantum mechanical measurement results' being treated as if they are independent of the way they are measured), and, for this reason, do not seem to be very attractive.
        On the other hand, the `theorem on the derivability of the Bell inequality from the existence of a quadrivariate probability distribution' is useful also here. However, within the standard formalism it is applicable only if all observables that are involved are mutually compatible (compare). Yet, we can learn from it something very important, viz. that `violation of the Bell inequality' necessarily involves incompatible observables (compare). This implies that, in agreement with the principle of local commutativity, quantum mechanics does not suggest that `violation of the Bell inequality by standard Aspect measurements (in which two compatible observables are measured jointly)' would be caused by any `nonlocal disturbance'. On the contrary, since `incompatibility of observables' can only obtain if observables are measured in the same region,
        `violation of the Bell inequality' is a local13 affair.
        Since the `standard formalism of quantum mechanics' is not able to deal with `joint probability distributions of incompatible observables', this is as far as conclusions can be drawn from the standard formalism. It is not equipped to deal with `possible causes of violation of the Bell inequality as a result of joint measurement of incompatible standard observables (compare)'.
      • Within the generalized formalism
        The generalized formalism has more resources in this respect, and is able to deal with `quadrivariate probability distributions of incompatible observables' (compare). By applying the `generalized formalism' to a generalized Aspect measurement it is possible to study simultaneously in both arms of the interferometer `mutual (Heisenberg) disturbance in a joint measurement of incompatible standard observables (as expressed by the Martens inequality)'. It is then not only seen that `unobserved observables may have values' (compare), but even how these values change if the measurement arrangement is changed. It is then important to note that `Heisenberg disturbance of measurement results in one arm of the interferometer' is dependent only on `what changes are made to the arrangement in that very arm'. In the standard Aspect measurements `violation of the Bell inequality' is not caused by `disturbance of the measurement results in one arm of the interferometer by a measurement carried out in the other arm', but it can be seen as a consequence of the `dependence on the measurement arrangement of local Heisenberg disturbances' in `joint nonideal measurements of incompatible standard observables' carried out simultaneously in both arms of the interferometer.
        Hence, if prejudices are abandoned that were caused by restricting oneself to the `standard formalism of quantum mechanics', `violation of the Bell inequality' may be seen to have a local explanation, viz. dependence of `Heisenberg disturbance' on the `measurement arrangement that is present at the position of each measurement'. The quadrivariate probability measured in the generalized Aspect measurement has been obtained under the influence of a cumulative effect of `disturbances separately realized in both arms of the interferometer'. Only by considering `measurements within the domain of application of the generalized formalism is it possible to establish `Heisenberg disturbance' as an alternative explanation of `violation of the Bell inequality', making obsolete explanations based on an alleged, unobservable `nonlocality'.
    • Methodological remark on `nonlocality versus (local) Heisenberg disturbance'
      `Nonlocality of the quantum world' is not based on any experimental evidence, the latter corroborating `parameter independence of (bi-locally performed) EPR-Bell experiments'. Contrary to an often-heard assertion, `violation of the Bell inequality by the standard Aspect measurements' is not `experimental evidence of nonlocality', because, contrary to Bell's conviction, locality is not a crucial assumption in derivations of the Bell inequality (compare, also Publ. 43).
      Often EPR-nonlocality is assumed to be a subquantum issue, being different from `negation of the locality expressed by the quantum mechanical principle of local commutativity'. It, then, could be doubted whether a `local explanation on the basis of (generalized) quantum mechanics' can be seen as an alternative to Bell's explanation by `nonlocality of the underlying subquantum theory'. It could be surmised that two different mechanisms are at stake, both tending to violate the Bell inequality.
      However, in my view this idea is unattractive for two reasons:
      i) the dubious role played by the assumption of `locality' in derivations of the `Bell inequality' both within quantum mechanics and within subquantum theory (compare). In my view the determining factor allowing `derivation of the Bell inequality' is not `locality' but rather objectivity (within standard quantum mechanics represented by the possessed values principle, within subquantum theory by the assumption of quasi-objectivity);
      ii) the methodological principle that the cause of a phenomenon, described by two different theories at different levels of sophistication, should be ontologically the same in the two theories. Thus, it does not make sense to explain at the macroscopic level the rigidity of a billiard ball by some `macroscopic (nonlocal) influence', whereas at the microscopic level it is explained by `atomic interactions acting at a microscopic scale', unless the macroscopic influence can be reduced to the atomic interactions. By the same token it is sound physical methodology at the levels of description of quantum mechanics and subquantum theory to try to explain `experimental violation of the Bell inequality' by one and the same cause.
      Even though the existence of `nonlocal interactions at the subquantum level, unobservable at the quantum mechanical level of observation' cannot be logically excluded, I therefore prefer trying to understand at the subquantum level the `hard evidence provided by the (generalized) quantum mechanical formalism as applied to joint nonideal measurement of incompatible standard observables' on the basis of ideas provided by that formalism.
      Some ideas pointing into the direction where to look for the above-mentioned `reconciliation of quantum mechanics and subquantum theory' are briefly discussed here. At present it does not seem useful, however, to invest too much effort in such an endeavour because we do not have any experimental clue as to the details of such a subquantum theory, which, moreover, would be really fruitful only if its domain of application would exceed the domain of application of quantum mechanics.
  • Non-entanglement (separability)
    The `Bell inequality' is satisfied for any non-entangled (or separable) state described by a density operator of the type ∑n pnρ1n ρ2n. This follows easily from the existence of a quadrivariate probability distribution reproducing the bivariate probabilities of the standard Aspect measurements for these states, viz.
    p(a1i,b1j,a2k,b2m) = ∑n pnp1niq1njp2nkq2nm,
    in which p1ni = Tr1ρ1nE1i, q1nj = Tr1ρ1nF1j, etc.79 Hence, within quantum mechanics
    there is no violation of the Bell inequality without entanglement.
    Until Werner proved that the Bell inequality is also satisfied by certain inseparable states0 it could be thought that, in agreement with the dichotomy between quantum and classical mechanics, the `Bell inequality' would be drawing a sharp line between entangled and separable states. It is obvious by now78 that the situation is not that simple. Study of `entanglement' (important for applications in `quantum computation' and `quantum information') has become an elaborate subdiscipline, attempts at developing `alternative entanglement criteria (replacing, or at least supplementing, the Bell inequality)' being an important branch. It is felt that, like `incompatibility' is for `measurement', `entanglement' is an additional resource for `preparation', offering new possibilities of experimentation in the above-mentioned fields.
    • `Nonlocality' versus `entanglement'
      Comparing the (valid) relation between the `violation of the Bell inequality' and `entanglement', given above, with the (dubious) analogous assertion with respect to `nonlocality' may explain the equally dubious relation often assumed between `entanglement' and `(hidden) nonlocality' (manifest by the costum of equating `entanglement' with `inseparability', compare). `Correlations entailing violation of the Bell inequality' are often referred to as `nonlocal correlations' (thus suggesting that `correlations entailing satisfaction of the Bell inequality' would be `local correlations'), even though `all correlations obtained in bi-local (EPR-Bell) measurements' are nonlocal (even though the principle of local commutativity is satisfied).
      That `entanglement' cannot be equated with `hidden nonlocality' is evident from the fact that, contrary to `hidden nonlocality', `entanglement' has observational consequences within the domain of quantum mechanics. Indeed, independently of whether the state is entangled or not entangled, the quantum state can be experimentally determined by means of quantum tomography0, in which `bi-local measurements of the Aspect type' are carried out (for the case of two spin-1/2 objects this can even be done within the `standard formalism'77; for higher-dimensional spaces `bi-local measurements of generalized observables' are necessary for complete determination, e.g. Publ. 47).
      Also from a preparative point of view it seems improbable that `entanglement' has any relation to `nonlocality'. Indeed, in the Aspect measurements the entangled state is there already when the particles are close together. In case they previously were not entangled, entanglement may very well be brought about by means of local interactions in the process of pair formation.

  • The Bell inequality in the generalized formalism
    By generalizing the quantum mechanical formalism so as to encompass experiments that can be interpreted as `joint measurements of incompatible observables', it is possible to derive the Bell inequality also within an empiricist interpretation.
    • A generalized Aspect measurement
      An example is the following generalized Aspect measurement, which is a variation of Aspect's switching experiment. In this latter experiment the experimental arrangements of the four polarization observables were set up simultaneously, be it in different places, so as to avoid mutual disturbance: by means of a switching mechanism photon n (n = 1,2) is directed either toward the measurement arrangement for An or Bn. Hence, depending on the switches, one of the experiments of figure 10 is performed.
      figure 11
      In the generalized Aspect measurement considered here the photon switches are replaced by partly transparent mirrors (see figure 11), having transparencies γn, 0 ≤ γn ≤ 1, n = 1,2. So, in a sense the Aspect switching experiment is a special case of the generalized Aspect measurement, the values of γn being restricted to 0 or 1, and being switched (pseudo-)randomly between these values. In the generalized experiment the values of γn are kept stationary. In this experiment there are four detectors (pointers), viz. D1, D′1, D2, and D′2. Since it is possible to determine for each detector and for each individual photon pair whether it has detected a photon or not, it is possible to construe a `quadrivariate probability distribution' p(a1i,b1j,a2k,b2m) of the responses of the four detectors (each response being, for instance, +1 or −1 as the answer to the `question whether a photon is detected' is `yes' or `no'; note, however, that the choice of the (eigen)values is unimportant: they can just as well be taken as + or −, as was done in the discussion of the joint nonideal measurement of incompatible polarization observables). From the existence of the `quadruples of the measurement results' the Bell inequality can be derived.
    • The POVM of the generalized Aspect measurement can be derived from the POVM of the joint nonideal measurement of two polarization observables (which, actually, coincides with the present arrangement as far as measurement of one photon is concerned, compare figures 5 and 11). Thus we get as a POVM
      Rijkm = R(1)ij R(2)km,
      in which R(1)ij and R(2)km are the POVMs of the measurements of photons 1 and 2, respectively, given by (for n=1,2):
      R(n)++ = O, R(n)+− = γnE(n)+,
      R(n)−+ = (1 − γn)E(n)+, R(n)−− = 1 − γnE(n)+ − (1 − γn)E(n)+.
      By taking the product of the two POVMs of the measurement procedures it is expressed that these measurement procedures, in agreement with the principle of local commutativity, do not influence each other.
      The `quadrivariate probability distribution' of the generalized Aspect measurement is
      p12)(a1i,b1j,a2k,b2m) = Tr ρ R(1)ij R(2)km.
      Taking for granted, analogously to what is usually done in the standard theory, that relative frequencies have a limit for large numbers, the existence of a `quadrivariate probability distribution' is a consequence of the existence of `quadruples of measurement results for each individual photon pair'. Derivability of a Bell inequality from the existence of such a probability distribution is evidence of the existence of a classical statistical Kolmogorovian model for measurement results jointly obtained within a single measurement set-up (compare).
    • Relation to the `standard Aspect measurements'
      By taking (γ12) = (1,1), (1,0), (0,1), or (0,0) the generalized Aspect measurement reduces to one of the four `standard Aspect measurements depicted in figure 10'. Hence, even if only two pointers are present in each of these latter experiments, `quadrivariate probability distributions' do exist.
      `Quadruples of measurement results' can be seen to exist also in the limiting cases. Indeed, we find
      γn = 1: R(n)++ = R(n)−+ = O, R(n)+− = E(n)+, R(n)−− = E(n)−,
      γn = 0: R(n)++ = R(n)+− = O, R(n)−+ = E(n)+, R(n)−− = E(n)−.
      Hence, to the `observables that are not measured because there does not come any photon in their direction' should be attributed the value `no photon detected' (note that it is conventional whether this value is represented either by 0, −1, or by any other digit). This implies that in each of the four `standard Aspect measurements' to all of the four standard observables a well-defined measurement result can be attributed:74 "unmeasured" observables may have values! It is easily seen that each of the `standard Aspect measurements' has its own `quadrivariate probability distribution
      p12)(a1i,b1j,a2k,b2m)'. In agreement with the existence of quadruples of measurement results the `Bell inequality' is satisfied by the `generalized Aspect measurement', and, hence, separately by each of the four `standard Aspect measurements' (in which the missing measurement results are obtained according to prescription given above).
    • `Bell inequality' and `Heisenberg disturbance'
      In order to understand why the four `standard Aspect measurements' do not satisfy the `Bell inequality' if their directly observed results are combined, it is useful to realize that the four `quadrivariate probability distributions',
      p(1,1)(a1i,b1j,a2k,b2m), p(1,0)(a1i,b1j,a2k,b2m), p(0,1)(a1i,b1j,a2k,b2m), p(0,0)(a1i,b1j,a2k,b2m),
      are different from each other. This difference can be understood on the basis of Heisenberg's disturbance theory of measurement (taken in a `determinative sense' rather than in `Heisenberg's preparative sense'), to the effect that in each arm of the interferometer `Heisenberg disturbance' obtains in the way discussed here, independently of what happens in the other arm of the interferometer. It is easily verified that the `mutual disturbance of observables A1 and B1' is different for different values of γ1 (and analogously for photon 2). This is expressed by the Martens inequality, applied independently in each arm of the interferometer.
    • In order to derive a Bell inequality for the combined `standard Aspect measurements' it would be necessary to have one single `quadrivariate probability distribution' from which the bivariate ones of the four measurements could be derived as marginals. However, if such a `quadrivariate probability distribution' would exist, the Bell inequalities could not be violated. Hence, experimental evidence is in disagreement with the existence of this special `quadrivariate probability distribution'. The same holds true for the existence of `quadruples of measurement results applicable in all of the four experiments', which also would imply satisfaction of the `Bell inequality' by the combined results of the four `standard Aspect measurements'.
      Due to `Heisenberg disturbance in a joint nonideal measurement of incompatible observables' the existence of such quantities is not to be expected if incompatible observables are involved. Hence, from the point of view of quantum mechanics we do not have any reason to expect that the `standard Aspect measurements' should satisfy the `Bell inequality' if their measurement results are combined in one single expression (like e.g. the CHSH inequality). On the contrary, the different ways `Heisenberg disturbance' is affecting the `bivariate probabilities in an arm of the interferometer when either An or Bn (n = 1 or 2) is measured ideally', makes it utterly plausible that, even if it is assumed that it is possible to reproduce the same physical initial condition in each of the four `standard Aspect measurements' (compare Publ. 41) this does not entail the existence of `unique quadruples of measurement results that could be used as the argument of a quadrivariate probability distribution', and, consequently, could warrant `satisfaction of the Bell inequality'.
    • `Violation of the Bell inequality' is a consequence of the incompatibility of certain of the observables. Therefore it can be accounted for by Heisenberg disturbances taking place locally in each of the arms of the interferometer. We do not have any reason to assume that there is any influence of the `measurement in one arm of the interferometer' on the `measurement results obtained in the other arm'. In this respect there is no difference between the `standard Aspect measurements' and the `generalized one', in all cases the POVM of the Aspect measurement being a direct product of the POVMs of the measurements performed in each of the arms. This implies that the measurement results for photons 1 and 2 are statistically independent if the initial states of the photons are uncorrelated (i.e. if ρ12 = ρ1ρ2). Any correlation between the measurement results of the two photons must be a consequence of a preparation of the two-photon object, such that ρ12ρ1ρ2. No correlation, either local or "nonlocal", is established by the measurement procedure.

  • Bell inequality and interpretations of quantum mechanics
    Which significance is attributed to `(violation of) the Bell inequality' is dependent on which particular interpretation of the quantum mechanical formalism which is being entertained. Barring the objectivistic-realist interpretation of quantum mechanical observables (which is ruled out because it would entail the Bell inequality to be satisfied for all quantum mechanical observables), we must consider the contextualistic-realist and the empiricist interpretations.
    • `Bell inequality' within a `contextualistic-realist interpretation'
      Contrary to the `empiricist interpretation' the `contextualistic-realist interpretation' allows that in the EPR experiment of figure 8 a value is attributed to an observable of particle 2, even when that particle is not interacting with a measuring instrument. `Nonlocality' is suggested by Bohr's application of the correspondence principle, implying that the contextual meaning of a quantum mechanical observable of particle 2 must be accounted for by referring to the measurement arrangement for particle 1. By overlooking the fundamental difference between EPR and EPR-Bell experiments this idea of `nonlocal contextuality' has proliferated to (allegedly) apply to EPR-Bell experiments too.
      However, there is no reason to believe that this makes sense. Even if the `distant measurement arrangement of particle 1' would contribute to the `context of particle 2', then the contribution of the `measurement arrangement for particle 2 itself' would certainly outweigh this by far. Because of the experimental corroboration of `parameter independence' it seems reasonable to suppose that the influence of the far measuring instrument is negligible. Then, in EPR-Bell experiments the two particles each have their own context. This `local contextuality' can be implemented on the basis of Heisenberg's disturbance theory of measurement, by taking into account a local influence of the measuring instrument for observable A1 on the incompatible observable B1, and vice versa (in an EPR-Bell experiment analogously for A2 and B2). Heisenberg disturbance can explain the non-existence of `quadruples (a1i,b1j,a2k,b2m) which are valid in all of the four standard Aspect measurements', the non-existence of similar `quadrivariate probability distributions', and the `inapplicability of counterfactual definiteness'.
      EPR-Bell experiments do not seem to frustrate a `local contextualistic-realist interpretation of the quantum mechanical formalism'. However, this latter interpretation is hard to maintain in the face of the preparative features of EPR experiments (as opposed to EPR-Bell experiments). This makes the `empiricist interpretation' preferable to the `contextualistic-realist one'.
    • `Bell inequality' within the `empiricist interpretation'
      In the `empiricist interpretation' there is no danger of confusing EPR and EPR-Bell experiments. The `Bell inequality' could only be derived if it were possible to dispose of either `quadruples of measurement results (a1i,b1j,a2k,b2m)', or a `joint probability distribution of the four observables', valid in all of the four standard Aspect experiments. If incompatible observables are involved, then, as a result of `mutual exclusiveness of measurement arrangements', there is no way to construe either of these within the `standard formalism of quantum mechanics'. The Bell inequality simply is not derivable within the standard formalism.
      However, the `generalized formalism of quantum mechanics', being a direct consequence of the empiricist idea to associate a quantum mechanical observable with the `pointer positions of a measuring instrument', has more resources in this respect. As seen both from the general theory and from the example of the generalized Aspect measurement, `joint probability distributions of incompatible observables' turn out to be viable concepts, describing measurement results of real measurements, including the standard Aspect ones. The possibilities provided by the `extension of the domain of quantum mechanics offered by the generalized formalism' to apply the `Bell inequality' to `measurements of the Aspect type' give additional insight into the mechanisms at work in such measurements. It turns out that, far from having to introduce `spooky nonlocality', the experiments can be understood on the basis of good old `Heisenberg disturbance' (generalized from a contextualistic-realist to an empiricist sense).