`Individual-particle interpretation' versus `ensemble interpretation' of quantum mechanics
  • Interference with the realist versus empiricist dichotomy
    The dichotomy of `individual-particle versus ensemble interpretation' interferes with the dichotomy of `realist versus empiricist interpretation'. Under the `realist interpretation' arrow a in the interpretational mapping may point from the theory either to an `individual object existing in reality' or to an `ensemble of objects existing in reality' (compare). Hence, a realist interpretation may be thought to refer either to an `individual object' or to an `ensemble'. There, hence, exists an `individual-particle version' and an `ensemble version' of the `realist interpretation'.
    There does not exist an `individual-particle version' of the `empiricist interpretation of quantum mechanics' since in the interpretational mapping arrow b can map state vector and density operator only to an ensemble. Indeed, one and the same `quantum mechanical preparation procedure', on being repeated, in general prepares different individual objects (yielding different measurement results on measurement), constituting the ensemble.
    In the `empiricist interpretation' a quantum mechanical observable is mapped to a `quantum mechanical measurement procedure' corresponding to an `ensemble of individual measurements'. It does not make sense to map an observable (represented by a Hermitian operator or a POVM) to an `individual measurement' (the latter being characterized by a single measurement result am or m of that observable). The `empiricist interpretation' is an `ensemble interpretation'.
    Most problems and paradoxes of quantum mechanics, being induced by a `realist understanding of the quantum mechanical formalism', evaporate in an `empiricist interpretation'. Unfortunately, the `realist interpretation' is the `standard interpretation' (silently) applied in most quantum mechanics textbooks as well as the scientific literature. For this reason it is necessary to pay due attention to that interpretation here, and discuss its interplay with the `individual-particle versus ensemble' dichotomy.
  • Individual-particle interpretation of the quantum mechanical state vector
    • In the `individual-particle interpretation' the quantum mechanical state vector |ψ> is thought to refer to an `individual microscopic object' (for instance, one single electron produced by a cyclotron). Objects described by identical state vectors are considered identical, or identically prepared.
      The `individual-particle interpretation' is a natural generalization of the usual interpretation of classical mechanics, a point (q,p) in a `classical phase space' (representing a classical state) being replaced by a quantum mechanical state vector |ψ> (a point in a Hilbert space0). This replacement may be seen as a mathematical implementation of Bohr's account of `quantum mechanics as a rational generalization of classical mechanics'. The `individual-particle interpretation' is at the basis of the Copenhagen thesis of the completeness of quantum mechanics.
    • The existence of entangled states like the EPR state
      |Ψ> = ∑m cm|a1m>|b2m>
      has been felt to constitute a considerable problem for the `individual-particle interpretation of the state vector'. Due to the linearity of the Schrödinger equation the state vector of any interacting (microscopic) two-particle system will evolve into an entangled state even if it was initially a (direct) product of the state vectors of each of the particles. Hence, it is not possible to attribute a state vector to each of the particles separately after they have interacted.
      The problem has given rise to a certain change of interpretation, to the effect that the `individual-particle interpretation' is replaced by an `individual-object interpretation', the object being the `particle pair'.
      The impossibility of separately attributing state vectors to particles in an `entangled state' has been interpreted as evidence of a certain `inseparability of particles that have interacted in the past', the particles in some sense losing their individuality, and a state vector being attributable only to the `universe as a whole' (since all particles presumably have interacted in the past).
      For two reasons I shall not consider such "individual universe" interpretations any further:
      i) there are convincing reasons to prefer an `ensemble interpretation' over an `individual-particle interpretation', making obsolete any consequence of the latter interpretation;
      ii) the `state vector of the universe' is a concept typical of an `objectivistic-realist interpretation of the mathematical formalism of quantum mechanics', which interpretation I consider obsolete and `to be replaced by an empiricist interpretation'. An objection to an `empiricist interpretation' that it would require `measuring instruments outside the universe' can be countered by turning the argument upside down, to the effect that in the `empiricist interpretation' the universe as a whole is assumed to be outside the domain of application of quantum mechanics because we do not dispose of measurement procedures to test that theory's `predictions with respect to the universe'.
    • Nevertheless, for a system containing a small number of particles the notion of `entanglement' is relevant even in the ensemble interpretation since it is possible to experimentally test the presence of cross terms by means of measurements of correlation observables (for instance by means of EPR-Bell experiments). As a consequence of application of an `individual-particle interpretation' the nonclassical character of the correlations experimentally found in these experiments has widely been interpreted as evidence of inseparability or even nonlocality, even though a more plausible account is available on the basis of an (empiricist) ensemble interpretation (which interpretation during a long time has been frustrated by i) the Copenhagen completeness claim, ii) internal problems of the ensemble interpretation, caused by too classical implementations of that idea, compare).
    • Realist and instrumentalist versions of the `individual-particle interpretation' have been proposed:
      • i) Schrödinger's realist proposal to consider a microscopic object as a wavelike mass distribution, described by the modulus squared of the wave function, should be mentioned here, as well as its failure. Reasons contributing to abandoning this interpretation are:
        • a) wave packet dispersion0, causing a wave packet of a single particle, initially localized in a small region, to spread all over 3-dimensional space18;
        • b) in a `double-slit experiment' it would seem that with the wave function also the mass distribution is split into two parts, thus seemingly allowing a particle to pass through both slits at the same time19; this, however, is in disagreement with all empirical data yielding a detection of the whole particle in one beam or the other, no `half particle' ever having been found in such experiments;
        • c) for an N-particle system the wave function is not defined on the 3-dimensional physical space we live in, but rather on a 3N-dimensional configuration space 0.
      • ii) As an instrumentalist alternative with respect to the wave function should be mentioned Bohr's version of the Copenhagen interpretation, stressing that the wave function or state vector should not be seen as a description of the microscopic object itself, but as `just an instrument for calculating the quantum mechanical probabilities pm when measuring a quantum mechanical observable'. For this reason the wave function is often referred to as a `probability wave', quantum mechanical probabilities being interpreted in a probabilistic sense. The vagueness of the term `probability wave' provides an instance of the vagueness of the `instrumentalist interpretation'.
      • iii) Since the `probability wave' can be changed by changing the experimental arrangement (for instance, by shutting one of the slits in a double-slit experiment) it seems that it refers to something liable to being physically influenced; it therefore should describe an `entity existing in physical reality'. This idea has promoted a tendency to turn the `Copenhagen interpretation' into a realist (or "ontic") one, purporting to imply an `ontological significance' of the probabilities pm as opposed to the `epistemological significance' of `expressing partial knowledge' as assumed in a statistical (or "epistemic") interpretation. I shall refer to this interpretation as a realist individual-particle interpretation endorsing ontological probability20. It may be seen as an attempt to take an ontological position halfway between Schrödinger's realism and the most extreme instrumentalist versions of the `Copenhagen interpretation', this position endorsing ontological indeterminism.
    • Critique of the `individual-particle interpretation'
      I will mainly restrict myself here to the problems of the `realist individual-particle interpretation endorsing ontological probability'. These problems are illustrated by
      • The Schrödinger cat paradox
        The Schrödinger cat paradox is about a cat, confined in a cage together with a radioactive atomic nucleus having a probability 1/2 of decaying within the next hour. If the nucleus decays, a contraption is set into motion causing the cat to die, if not then the cat stays alive. This process is assumed to result in a superposition of a living and a dead cat (so-called `Schrödinger cat state')
        cat> = 2−½(|ψalive> + |ψdead>).
        As originally conceived by Schrödinger, the cat is acting as a macroscopic measuring instrument for registering the decay of a microscopic particle. The `cat paradox' is about the physical meaning of the `Schrödinger cat state' |ψcat> (sometimes called a state of "suspended animation"), in which the cat is thought to be neither alive nor dead, but just to possess (ontological) probabilities to manifest itself as `alive' or `dead' if it is observed. It is felt as paradoxical that in `realist individual-particle interpretations' it is impossible to attribute the value `alive' or `dead' to a cat described by a `Schrödinger cat state', but that it seems necessary to invoke observation to change |ψcat> into either |ψalive> or |ψdead>.
        It was Schrödinger's goal to exhibit the strange consequences of the Copenhagen idea of quantum jumps, however without being able to convince the physics community of his own ideas.
      • Remarks on the `Schrödinger cat paradox':
        • In the `Schrödinger cat paradox' the cat is treated as a quantum mechanical object. This probably is the reason why Bohr does not seem to have taken Schrödinger's paradox too seriously (cf. Publ. 52, section 3.1.2). According to Bohr's correspondence principle (strong form) `macroscopic measuring instruments' (including cats if they act as such) must be treated in classical terms, the quantum mechanical wave function just yielding a "symbolic" (instrumentalist) representation being `equivalent to the classical one as far as the observed phenomena are concerned'. Bohr's classical account is sometimes implemented into the mathematical formalism of quantum mechanics by means of the observation that in Schrödinger's measurement arrangement the cross terms in the density operator |ψcat><ψcat| are unobservable, and hence the cat's state might as well be described by the density operator
          ρcat = ½(|ψalive><ψalive| + |ψdead><ψdead|),
          referring only to the two states of the cat that are relevant to actual observation.
        • The `Schrödinger cat paradox' might be felt to be misleading because, contrary to Schrödinger's assumption, the state |ψcat> does not play any observational role if the cat acts as a measuring instrument. Indeed, a treatment of measurement based on the Schrödinger cat state |ψcat> completely ignores the interaction between microscopic object and measuring instrument (cat), and therefore is way too simplistic since a treatment as a quantum mechanical measurement would have to take into account that interaction. Applying the theory of measurements of the first kind (in which the pointer states |θm> should correspond to the states |ψalive> and |ψdead>), the `final state of Schrödinger's cat' (to be compared with ρaf) is given by ρcat rather than by |ψcat>. There is no question of any transition from the state |ψcat> to the state ρcat since the cat has never been in the state |ψcat> (at least, in the empiricist interpretation, endorsed here, there is no reason to believe so; see also here).
        • The `Schrödinger cat paradox' epitomizes the distinction between the `individual-particle interpretation' and `ensemble interpretations' of the quantum mechanical state vector, the paradox evaporating if |ψcat> is interpreted as a description of an `ensemble of live and dead cats' rather than as a description of an `individual cat'. It is evident by now that an `ensemble interpretation' is the appropriate way to deal with both state vector and density operator. This reduces the importance of the `Schrödinger cat paradox' to a purely historical one.
          The latter judgment holds equally true for Bohr's epistemological view, probably adopted to circumvent the absurdity of an `ontological understanding of strong von Neumann projection' (to the effect that a transition from |ψcat> to either |ψlive> or |ψdead> would be realized by just looking at the cat). Indeed, Bohr occasionally warned against `Jordan's assertion that a physical quantity would (ontologically) obtain a sharp value by being observed'. Bohr's cautious `restraint from ontological assertions' has probably been induced by a wish to maintain within quantum mechanics the `individual-particle interpretation customary in classical mechanics' (compare), thought to be possible only if that interpretation would be taken in an epistemic sense.
          It seems to me that, unfortunately, this consequence of the classical paradigm is still active in the quantum mechanical literature, continuing to exert its confusing influence when `Bohr's cautious epistemological attitude' has been exchanged for the `ontological attitude customary in present-day quantum physics': whereas Bohr can be silent on the question of whether the cat is "really" `dead' or `alive', is an `ontic interpretation' confronted with that question. Paradox arises, for instance, if a density operator like ρcat allows different representations in terms of eigenvectors of incompatible observables (as is the case, for instance, in the EPR problem), thus seemingly allowing `simultaneous sharp values of incompatible observables'. It is hardly surprising that so diverse views as those of Bohr, von Neumann, Heisenberg and Jordan, brought together under the hospitable roof of the Copenhagen interpretation, have become sources of contradiction and confusion.
        • `Schrödinger cat states' might be useful for studying the `applicability of the superposition principle to mesoscopic systems showing (quasi-)classical behaviour', or to high-energy states of microscopic systems (like e.g. Rydberg states0), which applicability might, for instance, be obstructed by decoherence (e.g. Publ. 49). Such experiments become virtually impracticable, however, if applied to "really" macroscopic objects since they require measurement of microscopic properties of such objects.
        • In the development of my personal views on the meaning of quantum mechanics the problems raised by the `Schrödinger cat paradox' have stimulated my preference for an ensemble interpretation28 (be it neither in the sense of the statistical interpretation nor in that of von Neumann's ensemble interpretation) over an `individual-particle interpretation'. In particular the absurdity of the alleged reliance on observation in order to trigger a transition from a state of `suspended animation' into a state in which the cat is either `dead' or `alive' has contributed to this preference.
          After having realized the misleading character of the `cat paradox', and after having turned to the more "realistic" (though still rather simplistic) account of quantum measurement given here, it is straightforward to see that a simple and plausible implementation of `strong von Neumann projection' (or its generalization to `measurements of the second kind') is provided by an `ensemble interpretation'. This holds true for both the empiricist interpretation (exclusively having an ensemble version) and the realist interpretation, in which, as implied by the theory of conditional preparation, strong von Neumann projection (or its generalization) can be implemented as a transition to the `subensemble of microscopic objects selected on the basis of observation of pointer position m of the measuring instrument' (compare Einstein's application of `selection of subensembles' in the EPR experiment; also Publ. 57). Note that, since the `selection on the basis of measurement result m' can completely be automatized, no `human observer' need be involved after the measurement has started (compare).
      • Additional arguments against the `individual-particle interpretation':
        • i) Since experimental tests of quantum mechanics in general require determination of relative frequencies in an ensemble, the metaphysical nature of the `individual-particle interpretation' is evident. The `individual-particle interpretation of the state vector' is a fruit of classical thinking in which the state vector is considered as the natural generalization of the classical phase-space point (q,p) of `classical mechanics' (compare), rather than a generalization of a state of `statistical mechanics'. As far as I know there does not exist any empirical evidence that the state vector would describe an individual object, or could be experimentally proven to determine an `ontological probability transcending the meaning of a relative frequency in an ensemble' (compare).
        • ii) Von Neumann's strong projection postulate is helpful to the `individual-particle interpretation of the state vector', to the effect that an individual particle is warranted to have a well-defined state vector not only before but also after a measurement. Nevertheless it is also a bone of contention since it implies discontinuous and indeterministic (acausal) behaviour of the state vector during measurement, not described by any quantum mechanical equation of motion.
          Even if measurement indeterminism is not thought to be a real problem (as in the Copenhagen interpretation), then one might still be struck by the strange consequence that in experiments like the Compton-Simon and EPR-Bell ones this indeterminism/acausality is evidently accompanied by a "nonlocal causality" causing distant objects to behave in a highly correlated way notwithstanding there is no quantum mechanical interaction between them, and they, moreover, are assumed not to be conditioned beforehand so as to yield such correlations on measurement32.
        • By Einstein these problems have been employed to challenge the Copenhagen assumption of completeness of quantum mechanics (being tantamount to the `individual-particle interpretation'). Einstein's failure to convince the physics community does not imply that he was not right. However, he did not have the right arguments, neither to prove his own challenge to be right (compare), nor to take the edge off the Copenhagen `charge of metaphysics' by countering with a (presumably more justified) charge with respect to the `equally metaphysical nature of the Copenhagen individual-particle interpretation'. In particular, Einstein's attempt to circumvent the `measurement interaction' rather than taking it into account properly, has been instrumental to his conclusion that there actually is a trade-off between `completeness' and `locality', one of these having to be abandoned. Thus, in the EPR problem `strong von Neumann projection' is constituting a problem to the `individual-particle interpretation' only if the alternative of `nonlocality' is thought to be unacceptable; if locality is thought to be a feature of microscopic reality then an `ensemble interpretation' is the obvious solution.
        • iii) A third objection essentially derives from weak von Neumann projection (or its generalization to `measurements of the second kind'). Taking into account the interaction of microscopic object and measuring instrument by applying the theory of pre-measurement, a problem of the `individual-particle interpretation' is seen to arise. Thus, by the measurement an individual particle (allegedly described by the pure state |ψ>) is transformed into an ensemble of individual particles (described by the density operator ∑m pmm><ψm|), thus in a miraculous way changing the physical nature of the object31 from an `individual object' to an `ensemble'.
          Von Neumann has attempted to remedy this problem of the Copenhagen interpretation by introducing the idea of an homogeneous ensemble (instead of an `individual particle'), thus relaxing the problem by assuming a measurement to yield a transition between ensembles. In my view this attempt has been unsuccessful, however.

  • Ensemble interpretations of the quantum mechanical state vector
    • In order to comply with the definition of probability in terms of `relative frequency' an ensemble need not consist of an infinite number of individual objects (in fact, such a requirement would make the notion of an `ensemble' inapplicable to experimental physics). A certain asymptotic stability for large N is sufficient. Such stability is not a law of nature, but it is a necessary condition for the applicability of quantum mechanics. Its failure would imply the experiment to be outside the domain of application of quantum mechanics, the experiment possibly needing a subquantum theory for its description (compare), allowing to distinguish between individual preparations that are treated as identical by quantum mechanics.
    • In an `ensemble interpretation' a quantum mechanical state vector is thought to refer to an `ensemble of identically prepared microscopic objects' (for instance, a beam of N electrons produced by a cyclotron, assuming the electrons to be sufficiently distant from each other to be mutually non-interacting), rather than to an `individual object (for instance, one electron)'. Remember that N need not be infinite. Since probabilities pm can be experimentally compared only with relative frequencies (an individual microscopic object yielding only one single measurement result am), the `ensemble interpretation' seems to be the natural interpretation if quantum mechanics has to be an empirically relevant theory. This is the main reason for me to think that the `ensemble interpretation' is preferable over the `individual-particle interpretation', this reason presumably being decisive even if the latter interpretation were not problematic also for other reasons.
      Attribution of pm to an `individual microscopic object' as a `fundamental property in an ontological sense possessed by that object' (as is done in the probabilistic interpretation of the Born rule) is equally metaphysical as is Einstein's attribution of an `element of physical reality' to the microscopic object as an (admittedly metaphysical) `objective property' (compare).
    • Observational evidence supporting the `ensemble interpretation'
      From interference experiments like, for instance, Tonomura's experiment (cf. Video clip 1, presenting the `coming into being' of the interference pattern in a `double-slit experiment') it is evident by now that in interference experiments the observed interference pattern is built up by accumulation of events corresponding to impacts from an `ensemble of particle-like objects'. Only the `fully built-up interference pattern' is described by the state vector or wave function. Hence it is the ensemble that is described by the latter. Therefore this observation supports the `ensemble interpretation of the state vector'.
      In Tonomura's experiment the intensity is so low that it is experimentally demonstrated that in a double-slit experiment interference phenomena are not caused by `cooperation of many particles' (like in water waves) but can be observed notwithstanding at each time only one single particle is present in the interferometer. Although no convincing experimental evidence was extant at that time, this has already been guessed early during the development of quantum theory56. It may have contributed to the Copenhagen interpretation's choice in favour of the individual-particle interpretation of the state vector (as against an `ensemble interpretation'), thus assuming that in interference experiments the quantum mechanical wave function or state vector is describing an individual object (be it in the sense of a probability wave rather than following Schrödinger's realist proposal). This does not seem to contradict observation but, in view of the above-mentioned observational connection between the state vector and an ensemble, it is evident that the `individual-particle interpretation of the state vector' "multiplies meanings" in stark contradiction to Occam's razor principle0. In my view therefore the `ensemble interpretation' is preferable over the `individual-particle interpretation', wave-like phenomena associated with an individual particle (necessary to explain interference) not being assumed to be described by the quantum mechanical wave function but by a subquantum wave which does not necessarily satisfy the Schrödinger equation.
    • Remarks on the `ensemble interpretation':
      • It should be stressed here that a quantum mechanical ensemble does not have a less ontological significance than has an individual object, since both are physically generated by some physical preparation procedure, be it that the procedures are different (one being an N times repeated application of the other). It should be noticed that the experimental failure to comply with the limit N → ∞ reflects the distinction between, on the one hand, the `ontological meanings of the quantities Nm', and, on the other hand, the `epistemological meanings of the quantum mechanical quantities pm' with which the limits of the relative frequencies Nm/N are compared.
        Criticisms of the idea of `quantum mechanical probability as relative frequency' based on the physical unattainability of the limit N → ∞ have their origin in a `realist interpretation of the quantum mechanical formalism', attributing to quantum mechanical quantities like pm an `ontological meaning' analogous to the way in classical physics a `billiard ball is thought to possess the property of rigidity' (compare).
      • Unfortunately, in the quantum mechanical literature `ensemble interpretations' and `individual-particle interpretations' are not always clearly distinguished. Presumably, this is a consequence of von Neumann's interpretation, which combines elements of both interpretations.
        `Awareness of the fact that only relative frequencies, measured in an ensemble, have physical relevance' is often combined with `individual-particle parlance' like "determining the state a particle is in," or "a particle jumping from one state to another." In a full-blown `ensemble interpretation', in which also a state vector is supposed to refer to an ensemble, these latter phrases are meaningless: `state vectors' simply do not refer to `individual particles'.
      • An `ensemble interpretation of the state vector' seems to circumvent in a natural way the objections to von Neumann's projection postulate. Thus, in an `ensemble interpretation' the vector |ψ> = ∑mcm|am> might be considered as a description of an ensemble, consisting of `subensembles described by the vectors |am>', rather than as a description of an `individual particle in the state |ψ>'. `Strong von Neumann projection' might be seen as merely describing a `selection of a subensemble'. An `(individual) determination of a value am of observable A' might be interpreted as `finding the (individual) particle in the subensemble described by |am>'.
        Note, however, that this interpretation ignores the question of whether there is a distinction between the ensembles described by the state vector |ψ> = ∑mcm|am> (or the density operator |ψ><ψ|) and the density operator ∑m|cm|2|am><am|, i.e., the question of the existence of the cross terms. An answer to this question hinges on the possibility of measuring in these states an observable that is incompatible with A. Since there does not seem to exist any theorem of quantum mechanics forbidding such a comparison, and since it is not to be expected that for the two states the same results will be found, we will have to deal with features of quantum mechanics excluding certain types of `ensemble interpretations'. In particular, in order that an `ensemble interpretation of the state vector' be acceptable we shall first have to deal with the `possessed values principle', thwarting an `ensemble interpretation in which quantum mechanical measurement results am are supposed to be classical properties of microscopic objects'. Solution of such problems will often require relinquishing certain properties of ensembles that are too easily borrowed from classical physics.
    • Possessed values principle
      • In an `objectivistic-realist ensemble interpretation' it seems natural to explain a measurement result am by assuming a principle of `possessed values', stating:
        `a value of a quantum mechanical observable may be attributed to the object as an objective property the object possesses independent of observation'. The `possessed values principle' is assumed by EPR to be a necessary condition to be satisfied lest quantum mechanics be a complete theory. In particular it is assumed that position and momentum simultaneously have well-defined values.
        In a `contextualistic-realist ensemble interpretation' a well-defined value is attributed only to a restricted set of observables, viz. those selected by the experimental context actually realized. Thus, in a measurement of standard observable A it would be natural to attribute a value only to that observable (and, possibly, to certain observables compatible with A), leaving observables incompatible with A indeterminate. It is tempting to attribute such a `contextualistic realism' to Bohr's answer to EPR, particularly so, because his idea of strong correspondence (implying a classical description of measurement) suggests that at least the measured quantum mechanical observable may be as objectively real6 as `classical mechanical quantities' usually are supposed to be. It should be remembered, though, that Bohr usually remained on an epistemological level, shunning ontological assertions.
      • There exist strong indications, if not proofs, that, due to incompatibility of quantum mechanical observables, the `possessed values principle' is not a possible assumption within quantum mechanics: thus,
        i) there do not exist `joint probability distributions of incompatible standard observables', which at least is suggesting that incompatible standard observables cannot simultaneously have sharp values;
        ii) the assumption that four (or even three) standard observables simultaneously have sharp values is sufficient to derive a Bell inequality that may be in disagreement with experimental evidence;
        iii) the Kochen-Specker theorem0 and its recently sharpened versions demonstrate the impossibility that a (small) number of incompatible observables simultaneously have sharp values;
        iv) satisfaction of the Bell inequality as a consequence of the `possessed values principle'.
        An important goal of the present account is to support the view that
        the failure of the `possessed values principle' marks the crucial distinction between classical mechanics and quantum mechanics; as stressed already by the Copenhagen interpretation a long time ago this distinction is characterized by the crucial role played within the domain of quantum mechanics by the experimental context. This contextuality is the fundamental reason that Einstein's elements of physical reality cannot be equated to `quantum mechanical measurement results am'.
      • Reservations based on the `possessed values principle' as regards the feasibility of ensembles apply only to the objectivistic-realist interpretation in which quantum mechanical measurement results are considered to be objective properties of microscopic objects. Such reservations are futile in the contextualistic-realist interpretation as well as in the empiricist interpretation of quantum mechanics, where the `possessed values principle' is assumed not to be valid.
    • Counterfactual definiteness
      The assumption of counterfactual definiteness is related to the `possessed values principle', but it is somewhat weaker because measurement results are not considered as `objective properties of the microscopic object possessed prior to measurement' (objectivistic-realist interpretation), but either as `properties of the microscopic object, possessed within the context of a measurement' (contextualistic-realist interpretation) or as `properties of the measuring instrument'(empiricist interpretation). It does not assume that all observables have a well-defined value prior to measurement, but it does assume that, within the context of a measurement of standard observable A, to a standard observable B (possibly incompatible with A, and not actually measured) can be attributed the `value bn that would have been found if, instead of A, observable B would have been measured'.
      Such an assumption would be satisfied if the microscopic object would possess subquantum elements of physical reality determining the measured values in faithful measurements of a (standard) observable if such an observable were actually measured. Thus, contrary to the `quantum mechanical measurement result' the `subquantum element of physical reality' can be thought to be an `objective property of the microscopic object, possessed independent of any measurement to be performed later' (compare). There is no a priori reason why `subquantum elements of physical reality' would have to be denied existence in the same way Einstein's 'quantum mechanical elements of physical reality' turned out to be impossible. So, `counterfactual definiteness' might seem to be a feature to be reckoned with, even though it does not make sense to apply it to `quantum mechanical measurement results' (as a consequence of the latter's contextuality).
      However, `counterfactual definiteness' cannot be maintained in the face of `complementarity' as `mutual disturbance in joint measurements of incompatible standard observables' (compare).
    • Explanation by means of subensembles
      The impossibility of the `possessed values principle' does not exhaust our possibility of explaining measurement results by properties objectively possessed by the object preceding the measurement. It only means that it is not possible to explain measurement result am by assuming that the microscopic object possessed that quantity prior to measurement as an objective property. This does not mean that it is impossible to divide the initial ensemble (its elements not yet being in interaction with measuring instruments) into subensembles on the basis of distinct values of a quantum mechanical observable (compare explanation by determinism). However, for this latter purpose we may have to rely on concepts belonging to some subquantum ("hidden variables") theory (i.c. subquantum elements of physical reality).
      The idea that subensembles could be characterized by values of a quantum mechanical observable is characteristic of a realist interpretation. It would not even have occurred to anyone entertaining an empiricist interpretation, since it is obvious that a `pointer position' is not a `property of the microscopic object'. Explanation of quantum mechanical measurement result am on the basis of a `subquantum element of physical reality' would be comparable to explanation of the (empirical) rigidity of a billiard ball on the basis of `sub-rigid body properties' of the interactions between its atoms (which no longer are considered to be metaphysical after we have overcome the Machian reluctance0 to admit the reality of atoms).
    • `Ensemble interpretation' is more general than `statistical interpretation'
      It is often assumed that the notion of `ensemble' encompasses the `possessed values principle' or `counterfactual definiteness', thus maintaining characteristic properties of a `classical ensemble'. In the statistical interpretation ensembles are assumed to be of the latter type. In order to evade problems presented to `ensemble interpretations of this statistical type' (like the Kochen-Specker theorem0 and derivations of the Bell inequality from the `possessed values principle'), it is necessary to conceive of a different notion of `ensemble', to be referred to as a quantum ensemble, in which these classical properties are not assumed to be satisfied.
      More general ensembles than the classical ones can be encountered in the literature, like, for instance, von Neumann ensembles and `ensembles as conceived in the minimal interpretation'. In my view the first one has too much, the second one too little internal structure to encompass all experimental evidence that is available.
      What is the appropriate structure of a `quantum ensemble' is the subject of ongoing research. In particular, both the extension of the domain of application of quantum mechanics when taking into account the generalized formalism, as well as the notion of conditional preparation, throw important new light on the relation between an ensemble and its subensembles.
    • `Ensemble interpretation' and `incompleteness of quantum mechanics'
      • `Ensemble interpretations of quantum mechanics' stem from the idea that the standard formalism of quantum mechanics is incomplete since it does not completely describe the `individual object', in particular not distinguishing between initial states of `individual objects yielding different measurement results am'.
        We should distinguish incompleteness in the restricted sense from incompleteness in the wider sense, both implying that `completing features should be added to the theory', although having different characters.
        In the case of `incompleteness in the restricted sense' it may be postulated that the `possessed values principle' is satisfied, thus adding to the theory `initial values of quantum mechanical observables' (supposed to be equal to the values obtained as measurement results am).
      • In the case of incompleteness in the wider sense it may be assumed that the object initially was in a subquantum state from which the measurement result follows in a deterministic or in a stochastic way.
        Note that, whereas in the `restricted' case it is attempted to resolve the `incompleteness problem' within quantum mechanics itself (by choosing a particular interpretation, viz. the statistical interpretation), in the `wider' case the completion is not thought to be realized by quantum mechanics but by a subquantum or hidden variables theory. Failure to draw a distinction between the two kinds of incompleteness has been a source of much confusion (compare). Thus, following the discussions of the EPR proposal both `ensemble interpretations' were rejected by the majority of physicists as being metaphysical. As far as the EPR proposal is concerned, this rejection has turned out to be correct by the derivation of the Kochen-Specker theorem, demonstrating the impossibility of equating quantum mechanical observables to hidden variables if the latter are supposed to satisfy the rules of classical physics.
        However, this does not disqualify more general `ensemble interpretations' not being based on the (quantum mechanical) `possessed values principle', and therefore able to circumvent `no go' theorems (like the Kochen-Specker theorem and derivations of the Bell inequality from `standard quantum mechanics while presupposing the possessed values principle'). Nevertheless, probably due to the Copenhagen adoption of completeness in the wider sense (as a consequence of not sufficiently distinguishing it from completeness in the restricted sense), generic `ensemble interpretations' were rejected by the physics community and the `individual-particle interpretation' adopted. It is highly questionable whether, in view of present-day experimental evidence, the same choice is still possible.

  • Homogeneous and inhomogeneous ensembles
    There are different possibilities to entertain an `ensemble interpretation of quantum mechanical state vectors and density operators':
    • i) The minimal interpretation
      Any quantum mechanical ensemble (described either by a state vector or a density operator) is considered homogeneous, all elements of the ensemble being considered to be identical or identically prepared. An assumption of homogeneity of an ensemble is tantamount to `abandoning any explanation of individual measurement results' (the latter for different elements of the ensemble not necessarily being equal if probability pm ≠ 1). In the `minimal interpretation' this is even maintained if the state is described by a `density operator that is not a projection operator'.
    • Critique of the `minimal interpretation'
      • Certain ensembles, described by density operators, can be operationally subdivided into distinct subensembles (compare conditional preparation). It does not seem to be reasonable to consider such ensembles to be homogeneous. For this reason I do not think the minimal interpretation to be a useful one.
    • ii) Von Neumann's interpretation29
      With respect to pure states von Neumann's interpretation is an individual-particle interpretation, an individual particle being described by a quantum mechanical state vector |ψ>. A density operator ∑m pmm><ψm| is thought to describe an ensemble (referred to as a von Neumann ensemble) consisting of subensembles, each element of a subensemble being described by the same state vector |ψm>. Unlike in the `minimal interpretation', the `von Neumann ensemble' is thought to be inhomogeneous.
      `Von Neumann projection' has become an important ingredient of the `Copenhagen interpretation'. However, a certain distinction arises when von Neumann introduces the notion of a homogeneous ensemble replacing the `individual particle' of the Copenhagen interpretation. A homogeneous ensemble is thought to consist of particles, all "being in the same pure state |ψ>". The crucial difference with `classical statistical mechanics' is that von Neumann's `homogeneous ensembles' are thought `not to be dispersionless'. Mixtures described by ∑mpmm><ψm| are thought to be just (classical) inhomogeneous ensembles of `homogeneous ensembles corresponding to pure states |ψm>'.
      Although from a pragmatic point of view the introduction of a `homogeneous ensemble consisting of individual particles all being in the same state |ψ>' does not yield any advantage, it appears to lift a fundamental (but largely ignored) inconsistency of the Copenhagen interpretation, to the effect that by a measurement an `individual particle' would be transformed into an ensemble (compare).
    • Von Neumann's influence may have been important in a development of the Copenhagen interpretation from Bohr's instrumentalism into the direction of a realist interpretation of the state vector. The possibility of maintaining `internal consistency of the interpretation' while resolving the `ontic versus epistemic' contrast by introducing `ensembles', may have contributed to von Neumann's acceptance as a member of the Copenhagen family. Indeed, the difference of an `homogeneous von Neumann ensemble' and an `individual particle', described by the same state vector, could be seen as `just a mathematical technicality'.
      On the other hand it has also been a source of confusion. Thus, von Neumann's interpretation of mixtures is often referred to as an ignorance interpretation of states in which an individual element of the `ensemble described by ∑mpmm><ψm|' is assumed "to be in one of the states |ψm>", although it is unknown in which one. However, the `ignorance terminology', translating `von Neumann's ensemble language' back into the `Copenhagen individual-particle language', derives from the `ontic versus epistemic' dichotomy, and hence is perpetuating confusion.
    • Criticisms of von Neumann's interpretation
      • a) If projection operators |ψm><ψm| are not all compatible, then in general the Kochen-Specker theorem will preclude the possibility of looking upon the different states |ψm> as alternative descriptions of individual particles within the same measurement context (since then these states are eigenvectors of incompatible projection operators implying failure of the possessed values principle).
      • b) A possible criticism of the idea of an `inhomogeneous von Neumann ensemble', valid even if all projection operators |ψm><ψm| are compatible, is that the representation ρ = ∑mpmm><ψm| need not be unique. It is possible that also ρ = ∑nqnn><φn|, the set {|φn>} of vectors |φn> being different from the set {|ψm>} (this holds, for instance, in the EPR problem). This would imply that an individual element of a von Neumann ensemble must be both in a state described by |ψm> as well as in one described by |φn>. This non-uniqueness has been felt to be a serious problem. It has been used by Einstein to argue against the Copenhagen idea of `completeness of quantum mechanics', which attack has been a reason for Bohr to stress the contextual meaning of the quantum mechanical description.
        The alliance between Bohr and von Neumann has given rise to the idea that the non-uniqueness problem can be solved by changing from an `objectivistic-realist interpretation of the quantum mechanical formalism' to a contextualistic-realist one27. However, in my view this "solution" is insufficient (see e.g. section 6 of Publ. 53).
      • c) An argument against the idea of an `homogeneous von Neumann ensemble' can be derived from the quantum mechanical description of pre-measurement. In this description the transition from the initial state |ψ>|θ0> to the final state |Ψf> = ∑mcmm> |θm> allegedly is a transition between two homogeneous von Neumann ensembles. So far, so good. Yet, inconsistency can be seen to arise from this. Indeed, according to von Neumann the final state density operator ρof of the microscopic object, found as the partial trace
        Traf><Ψf| = ∑m pmm><ψm|,
        should describe an inhomogeneous ensemble. Then, if the ensemble described by |Ψf> were homogeneous, we would have to face the fact that disregarding all information on the measuring instrument (as implied by taking the partial trace Tra) evidently turns the microscopic object's ensemble into an inhomogeneous one. This is highly counterintuitive, however. Usually one needs more (rather than less) information to distinguish objects which seem to be identical at first sight21 (see also section 3.3.2 of Publ. 53). A trick, found in the literature, to the effect that it is assumed that an ensemble described by the density operator ∑m pmm><ψm| might be homogeneous (i.e. a so-called `improper mixture'), does not work since in principle inhomogeneity can be demonstrated by means of conditional preparation.
        It seems that only one way is open toward consistency: the conclusion that during the measurement the object must have been subjected to a miraculous change from an `individual object' to an `ensemble of objects' can only be evaded by assuming that the initial ensemble must have been inhomogeneous too. Von Neumann's idea of `homogeneity of pure states' -and, concomitantly, the Copenhagen idea of `completeness of quantum mechanics as embodied in the individual-particle interpretation'- cannot be maintained.
    • iii) The `statistical interpretation'
      In agreement with the probabilistic versus statistical dichotomy an `ensemble interpretation' satisfying the `possessed values principle' is referred to as the `statistical interpretation' (as opposed to the `probabilistic interpretation'). The statistical interpretation is inspired by `classical statistical mechanics', in which a state is just a statistical distribution of `classical mechanical states', the latter being represented by points in a phase space. The `statistical interpretation' was an important presupposition of Einstein's attempt at proving that quantum mechanics is an incomplete theory by demonstrating that `quantum mechanics is not able to yield a description in which both position and momentum have well-defined values at the same time'.
      In the `statistical interpretation' an ensemble, described either by a state vector or by a density operator, is considered to be inhomogeneous. Elements of an ensemble, -even if identically prepared-, are thought to be distinct, and distinguishable by means of different values of quantum mechanical observables, each element allegedly having a well-defined (sharp) value of every observable.
    • Critique of the `statistical interpretation'
      • The impossibility of the `possessed values principle' entails the impossibility of the `statistical interpretation' in the sense defined above.
        Modal interpretations0 try to save part of the `statistical interpretation' by trying to circumvent theorems like the Kochen-Specker0 theorem by restricting the set of quantum mechanical observables having well-defined values so as to prevent that theorem from being derivable. Thus, in an ensemble described by the state vector |ψ> = ∑mcm|am> modal interpretations assume that within the experimental context of a measurement of (standard) observable A only observables compatible with A (or even just A itself) are well-defined. A reason for such an assumption is that the statistical measurement results pm for A cannot distinguish between the initial states |ψ> = ∑mcm|am> and
        ρ(A) = ∑m |cm|2 |am><am|, the latter corresponding to an ensemble in which each element is thought to have a well-defined value am of A.
        Within the measurement context of A the subensemble of ρ(A) corresponding to am is described by the state vector |am>, thus preventing observables incompatible with A from having well-defined values (however, without necessarily assuming that these subensembles are homogeneous with respect to observables incompatible with A).
    • iv) Quantum ensembles
      In view of the fact that the probabilities pm in the first place are (asymptotic values of) `relative frequencies of measurement results', it is evident that we have to deal with ensembles. Therefore it seems useful to contemplate a more general type of ensemble than the ones corresponding to the `statistical interpretation', the `possessed values principle' not being assumed to be valid, however without assuming homogeneity (in contrast to the `minimal interpretation'). I shall refer to such ensembles as `quantum ensembles'. Such `quantum ensembles' are particularly useful in the empiricist interpretation in which the `possessed values principle' does not make sense (compare). A `quantum ensemble' is prepared by any preparation procedure valid within the domain of application of quantum mechanics (hence, described by a state vector |ψ> or a density operator ρ, which in the `empiricist interpretation' are just labels of preparation procedures).
      By a measurement of standard observable A we get information on the state vector or density operator, represented by the probability distributions pm. For the initial state |ψ> = ∑mcm|am> this information can be encoded in the density operator
      ρ(A) = ∑m|cm|2 |am><am|, which could be interpreted as labeling `preparation procedure |ψ> if this preparation is carried out within the context of a measurement of observable A'.
    • Ontological aspects of quantum ensembles
      • Apart from its epistemological meaning as `representing information on the preparation procedure' (encoded in the initial state), an ontological meaning might be attributed to ρ(A) as in a modal sense preparing a `von Neumann ensemble'. In view of a criticism of von Neumann's interpretation ρ(A) should not be interpreted as describing an `ensemble prepared by the measurement (in the sense of conditional preparation)', but as an `alternative description of the initial state |ψ>, valid within the experimental context of the measurement arrangement'. In this modal (ontological) sense the state ρ(A) is referred to as a contextual state. The `reality of the contextual state' might be compared with the `reality of the rigidity of a billiard ball within experimental contexts in which the ball can behave as if it were a rigid body' (compare), a difference being that ρA would not refer to the `reality of an individual object' but to the `reality of an ensemble'. The `contextual state' ρ(A) might symbolize the possibility of `explanation by means of subensembles', in the sense that it is providing a quantum mechanical representation of `reality as it is within the context of the measurement of observable A', taking into account only differences related to that very observable.
        In the quantum decoherence program it is attempted to explain the (ontological) transition from |ψ> to ρ(A) on the basis of a (thermodynamical) irreversibly stochastic interaction with the environment, be it that in general the crucial importance of the measurement arrangement within the environment is not taken into account, and the state ρ(A) is considered as the `final state of the microscopic object' rather than as a `(contextually valid) initial state'. Note also that decoherence does not explain the value of an individual measurement result am, thus failing to comply with Einstein's demand for explanation.
      • The contextuality involved in the `contextual state' should be a warning that we should be careful not to take this state as a description of a `classical ensemble' in which it is possible to make selections according to arbitrary physical quantities. Although a pure state can be written as a linear superposition of eigenvectors of an arbitrary observable, this cannot imply that an ensemble can be subdivided into subensembles having sharp values for every observable (which would fail as a consequence of the failure of the `possessed values principle').
        Since no empirical data seem to disagree with the assumption that `preparation' can be carried out independent of the measurement (to be performed later), it seems safe to contemplate the notion of a `quantum ensemble of individual (identical) preparations', to be described by a state vector |ψ> or a density operator ρ, without assuming the ensemble to have any internal structure not implied by the mathematical formalism of quantum mechanics. These `quantum ensembles' should be distinguished from the `ensembles of measurement results' as encountered in the relative frequencies pm of individual results am of a measurement of observable A, the latter exhibiting a classical (Kolmogorovian0) nature as long as it is not tried to combine probability distributions of incompatible standard observables into one single joint probability distribution (as is sometimes done in derivations of the Bell inequality).

Ontic versus epistemic interpretation of quantum mechanics
  • Sometimes a distinction is drawn between an `ontic interpretation of quantum mechanics' and an `epistemic one', the latter meaning that not reality itself is described (contrary to what is assumed in an `ontic interpretation') but that the theory is yielding `just a description of our knowledge of that reality'.
    An `epistemic interpretation' may have been attractive to `those worrying about discontinuous quantum jumps' by the reassuring thought that not `reality itself' need behave discontinuously, but that `just our knowledge' may be affected in a discontinuous way by a measurement. In particular, von Neumann's strong projection postulate, implying discontinuous behaviour of the state vector, may in this way have been made acceptable to `Bohr's cautious attitude with respect to ontological assertions'.
  • Note, however, that the notion of `epistemic interpretation' has also raised criticism, because `quantum mechanics as a description of our knowledge' seems to imply that quantum mechanics is part of psychology rather than physics. According to these critics `reality itself' is the proper object of interest to a physicist. Hence his theories should allow `ontic interpretation'. An epistemic interpretation could ignore `objectively existing quantum jumps' (compare the realist individual-particle interpretation endorsing ontological probability) possibly being explained away by stressing the subjectivity of knowledge.
  • In my view the `ontic versus epistemic' dichotomy is potentially misleading because it is confounding ontological and epistemological issues. Independent of its interpretation, any physical theory is a `representation of our knowledge about a certain physical domain', and, hence, epistemic12. Different observers may have different knowledge about the same object, and, accordingly, will have different theoretical representations (like e.g. representations of a billiard ball either as a `rigid body', or as `consisting of atoms').
    On the other hand, as far as an interpretation of quantum mechanics is a mapping of its mathematical formalism into reality, any interpretation of that theory is `ontic (in the sense that the theory is thought to describe some part of physical reality)'.
  • The `ontic versus epistemic' dichotomy has been particularly harmful by being unjustifiedly equated with the `individual particle versus ensemble' dichotomy, an `ensemble interpretation' then being seen as a `description of our (lack of) knowledge about an individual particle' (compare). This misconception is a consequence of the idea that, contrary to an `individual object', an `ensemble' would not be a physical object. However, far from being `just psychological', ensembles are experimentally dealt with in a routinely fashion by `repeating preparation and measurement of individual objects a large number of times under conditions warranting the existence of a stable value of the relative frequencies Nm/N' (note that N is not required to be infinite, it is sufficient that the relative frequencies asymptotically reach stability as N is increased). Hence, `quantum mechanical ensembles' are no less `parts of physical reality' than are the `individual particles they consist of'. Ensembles have their own properties, in an `ensemble interpretation' supposed to be described by the `mathematical formalism of quantum mechanics'.
    As can be seen from the theory of conditional preparation, `subjectivity of knowledge' is no problem in the `ensemble interpretation': conditional probabilities p(n|m) obtained after an observer has selected the subensemble corresponding to pointer position m, will in general be different from probabilities p(n) obtained by an observer ignoring the pointer. This epistemological difference is unproblematic because it is based on an ontological difference: the `subensemble obtained by the first observer' and the `whole ensemble taken into account by the second one' are ontologically different objects.
  • The dichotomy of `ontic versus epistemic interpretations' can be seen as an (unsuccessful) attempt to remedy the vagueness of the realism versus instrumentalism dichotomy by stressing the `ontic character of reality' as well as the `epistemic character of the reference of the instrumentalist interpretation to measurement results' (the latter often equated with `sensations stored in our minds').
    However, as seen from the possibility of an empiricist interpretation, measurement results have an ontic character too (viz. as pointer positions of measuring instruments). The `ontic versus epistemic' dichotomy does not distinguish between realist and empiricist interpretations, which are both ontic in the above sense (although mapping into different parts of reality).
  • It seems to me that the confusing dichotomy `ontic versus epistemic' is a consequence of the halfhearted way in which the essential role of measurement was acknowledged by the founding fathers of quantum mechanics, not taking the step toward an `empiricist interpretation', and, hence, lacking the incentive to realize that an ontological assertion about a `final pointer position of a measuring instrument' has an epistemological dimension by expressing `knowledge about the microscopic object'.
    In order to avoid misunderstanding I shall avoid reference to the `ontic versus epistemic' dichotomy. Instead, the distinction between `individual-particle' and `ensemble' interpretations of the quantum mechanical state vector is thought to be important, an `ensemble' considered to be `as real an object' as is an `individual particle'. Like the `pointer position' the ensemble has both an ontological as well as an epistemological dimension since it represents both a `real object' as well as `(statistical) knowledge on an individual element of the ensemble'. It seems appropriate to stress the ontological dimension when `interpretation of quantum mechanics' is the subject of discussion.
Quantum mechanics and logical positivism/empiricism
  • `Logical positivism/empiricism' and `completeness of quantum mechanics'
    • The logical positivist/empiricist abhorrence of metaphysics, and the ensuing inclination to use Occam's razor, has strongly contributed to the idea that quantum mechanics is a `complete theory', in the sense that only the `quantum mechanical probability distributions' have physical relevance. Since individual measurement results are not reproducible, they are thought to be physically irrelevant. Logical positivism/empiricism deems metaphysical so-called "hidden variables" theories (or `subquantum theories'), meant to predict individual measurement results by a `more precise specification of the state of a microscopic object than the quantum mechanical one'. It has given rise to the idea of anti-realism to the effect that only `observable quantities' would have an ontological meaning.
      Logical positivism/empiricism has had considerable influence in the development of quantum mechanics, in particular on the terminology denoting the `physical quantities of quantum mechanics' as `observables'. Note, however, that we should not make the mistake to identify logical positivism/empiricism with anti-realism, in the sense that `unobservables' would not exist at all. `Unobservability' may be a time-bound quality, dependent on the state-of-the-art of experimental physics: vibrations of a billiard ball may become observable as new measurement procedures become available. To logical positivism/empiricism probably no justice would be done by attributing to it a `denial of the possible existence of such vibrations rather than claiming that no scientific account is possible as long as these are not experimentally demonstrated'. Unfortunately, within quantum physics there has been a development towards an `anti-realist attitude' (see, for instance, the `unobservability solution' of the "measurement problem"), often not sufficiently distinguished from `logical positivism/empiricism'50, but actually reflecting the influence of the philosophy of scientific realism0, (which is rather opposing the Copenhagen reliance on the notion of `measurement', and is aspiring at a return to an objectivistic-realist interpretation of the quantum mechanical formalism).
    • (In)completeness in the wider sense should be distinguished from the `(in)completeness in the restricted sense involved in the Copenhagen interpretation'. It is important to note that in the discussion between Bohr and Einstein not the wider but the restricted notion of completeness was at stake. It, therefore, is a (widespread) misunderstanding that Bohr's "victory" over Einstein in the (in)completeness debate (if there was a victory at all) can be seen as a victory of (logical) positivism (compare).
    • It is undeniable that the empiricism of logical positivism/empiricism has influenced Heisenberg in developing matrix mechanics as a theory ``completely'' describing all observations within the microscopic (atomic) domain. Yet, there is no reason to assume that quantum mechanics is the theory of everything, not even of `everything observable'.
  • Theory-(in)dependence of measurement
    • In order to prevent circularity, according to logical positivism/empiricism a measurement should not be described by the very theory to be tested by it: measurement/observation should be described either by means of `observation statements', or in terms of a pre-theory, the latter's `theoretical terms' being defined by means of `observational terms'.
      The insight that, in general, a description of a measurement (observation) must necessarily make use of the very theory that is tested by it (theory-ladenness of measurement/observation) has been a cause of the decline of logical positivism/empiricism as a useful philosophy of science. The `hard data' turned out to be not that hard after all because they need the `theory to be tested' for their definition. This also applies to quantum mechanics, it nowadays being accepted that quantum mechanical measurement should be described by quantum mechanics, at least as far as the pre-measurement phase is concerned.
    • Bohr's strong correspondence principle has been interpreted in the past as proof of Bohr's affinity to logical positivism/empiricism. This is correct in as far as it is supposed by Bohr that `what can be told about a measurement' should be exclusively cast in terms of classical mechanics, the latter theory being considered as a pre-theory for quantum mechanics, `classical quantities' being considered to be directly observable and hence unproblematic. However, Bohr's `strong correspondence principle' is sufficiently tampering with metaphysics to support his own assurance that he is not a logical positivist: for Bohr classicality seems to mark not only the essential characteristic of `observation/measurement' (warranting an objective existence of the measurement results), but he transcended logical positivist/empiricist boundaries by `attributing quantum mechanical measurement results as properties to the microscopic object' (be it in a contextualistic-realist rather than an objectivistic-realist sense; this is evident from his not distinguishing EPR and EPR-Bell experiments).
      By emphasizing the `important influence of measurement within the microscopic domain' Bohr even started a development undermining logical positivism/empiricism, since `taking seriously the necessity of a quantum mechanical description of (pre-)measurement' turned out to be an important instance to test the philosophical issue of `theory-ladenness of measurement', demonstrating that the logical/positivist/empiricist requirement of `theory-independence of measurement' cannot be upheld. Note, however, that Bohr, although starting this development, perhaps even has hampered it by his emphasis on the `use of classical mechanics for describing quantum measurement', thus perhaps unwittingly stimulating the idea of `classical mechanics as a pre-theory for quantum mechanics' (which has been advanced as a `refinement of logical positivist/empiricist ideas'). The insight that the `interaction between microscopic object and measuring instrument' necessarily is `within the domain of quantum mechanics' may have contributed to the present obsoleteness of logical positivism/empiricism.
  • `Empiricist versus operationalist' interpretation of quantum mechanics
    • According to the philosophical doctrine of logical positivism/empiricism, in order to avoid metaphysical statements the theoretical terms of a theory must preferably be defined in terms of `observation statements referring to the phenomena' (the so-called `hard data'), the occurrence of the latter being liable to verification by means of direct observation.
      The alleged `unobservability of microscopic objects and processes' has inspired logical positivism/empiricism to take up a strong interest in quantum mechanics. In particular, the empiricist interpretation of quantum mechanics might seem to be inspired by logical positivism/empiricism. Thus, the reference to operations in the `empiricist account of states and observables' may be reminiscent of Bridgman's operationalism0.
      Note, however, that `Bridgman's operationalism' partakes in an instrumentalism I do not support, thus deviating from the `empiricist interpretation' I am proposing41. For this reason I have evaded reference to `Bridgman's operationalism', even though the operationalist and empiricist interpretations have important elements in common (in particular, the generalized formalism of POVMs is strongly pointing into an operationalist direction).15
      A reason to use the term `empiricist interpretation' (notwithstanding `empiricism' is too much associated with `human observation', the human observer being dispensable within quantum mechanics) might be that reference to `empiricism' is in agreement with (part of) the philosophical discussions0 on `(scientific) realism versus empiricism/instrumentalism' sealing the fate of logical positivism/empiricism42.
  • Quantum mechanics after `logical positivism/empiricism'
    • Realist and empiricist interpretations of quantum mechanics can be seen as reactions to the downfall of logical positivist/empiricist approaches of that theory, however going into different directions.
      `Realist interpretations' try to amend the `empiricist tendencies within logical positivism/empiricism', stressing that the processes described by quantum mechanics are usually microscopic, and, unless a measurement is actually performed, independent of any observation or measurement. The corresponding philosophical view goes under the name scientific realism0. In this view quantum mechanics should preferably be interpreted as an `objective description of an objective quantum reality' (at least as long as the object is not interacting with other entities). It seems to me that the way quantum mechanics is dealt with in modern textbooks is consistent with a `scientific realist view'.
      The `empiricist interpretation', on the other hand, takes seriously the issue of theory-ladenness of measurement/observation, but does not consider the circularity induced by it a vicious one. The theory (i.c. the quantum mechanical formalism) is considered as a mathematical structure30 covering a certain part of reality, viz. the part made visible by our measuring instruments. The `empiricist interpretation' can be looked upon as an heir to logical positivism/empiricism in the sense that quantum mechanics is considered to yield a phenomenological description of (a certain part of) physical reality, and, hence, need not constitute the most fundamental account of microscopic reality. However, the `empiricist interpretation' does not follow logical positivism/empiricism in its fear of the metaphysical, being aware of the possibility that underneath the level of quantum phenomena there may exist a deeper level of reality on which the phenomena may be based.

The "measurement problem"
  • The so-called "measurement problem" is symbolized by Schrödinger's cat paradox: how to account for a `linear superposition of macroscopically distinguishable states' like cat>. Since, when observed, a cat is usually found to be either `alive' or `dead', it essentially boils down to the problem of whether this state can describe an `individual cat' or whether it should be seen as a description of an `ensemble of cats that are either alive or dead'.
    In agreement with the misleading character of the `cat paradox', the "measurement problem" is sometimes considered to be `just a pseudo problem'. Indeed, the problem seems to disappear when `quantum measurement' is recognized as an ordinary quantum mechanical process. Thus, although the final state |Ψf> = ∑mcmm> |θm> of the `pre-measurement process of standard observable A' might seem to be still problematic because it is a `linear superposition of states of a macroscopic object', a solution to the problem seems to be in sight because, at least for measurements of the first kind, the final density operator ρaf = ∑m|cm|2m><θm| of the measuring instrument can be understood as a description of an ensemble of measuring instruments (cats). In that case for consistency also the initial state should be interpreted as referring to an ensemble (compare).
    Note that the "measurement problem" as inspired by `Schrödinger's cat paradox' ignores the quantum mechanical nature of the measurement interaction, thus failing to take advantage of the resources made available by this feature. In particular, weak von Neumann projection is realized by it in a natural way, thus eliminating the so-called cross terms felt to be problematic in the density operator |ψcat><ψcat|. Note also, however, that application of this resource must be carried out with due care because it only works for `measurements of the first kind'.
  • Some proposals to solve the "measurement problem"
    • i) Solution by means of `von Neumann ensembles'
      • The concept of a von Neumann ensemble might be invoked in order to satisfy, next to weak von Neumann projection also strong von Neumann projection. According to `von Neumann's ignorance interpretation of states' each element of the inhomogeneous ensemble of measuring instruments described by the density operator ρaf = ∑m|cm|2m><θm| allegedly is in a well-defined pointer state |θm>. Hence, `strong projection' could be realized by selection of an element of the ensemble. By the same token the final state of the microscopic object is found according to ρof = ∑m|cm|2|am><am|, suggesting an interpretation as an `inhomogeneous von Neumann ensemble of microscopic objects', each element being in a well-defined state |am>.
      • Criticism of `solution by means of von Neumann ensembles'
        However, there are several reasons not to be satisfied with `von Neumann's ignorance interpretation of states' as a solution to the "measurement problem":
        a) the problems with the concept of a `von Neumann ensemble' discussed here, implying that an element of a `von Neumann ensemble' cannot unambiguously be represented by a state vector, thus making von Neumann's interpretation incompatible with an `individual-particle interpretation' (compare);
        b) there is an additional problem stemming from the theory of `measurements of the second kind', to the effect that if <ψmm′> ≠ δm,m′ the reduced density operator of the measuring instrument is given by
        ρaf = ∑m,m′cm*cm′mm′> |θm′><θm|.
        Since virtually all measurements are of the `second kind' this implies that in general the `cross terms' do not disappear. By diagonalizing the density operator ρaf a representation can be found that would be interpretable as a `von Neumann ensemble'. However, then each element of the ensemble would be in a `superposition of the states |θm>' rather than in the states |θm> themselves. This would disqualify the states |θm> as `pointer states'. This actually would mean a return to Schrödinger's cat paradox.
        This may explain the great popularity of `measurements of the first kind' in discussions of the "measurement problem". Taking seriously `measurements of the second kind' makes it impossible to get rid of the "measurement problem" by means of the attractive idea of a `von Neumann ensemble with elements having well-defined pointer positions'. Taking such measurements seriously, I consider this problem to be an indication (next to the possible `ambiguity of the representation' of ρaf for `measurements of the first kind') that the idea of a `von Neumann ensemble' is not a consistent one.
    • ii) The "unobservability" solution
      • Sometimes the unobservability of the `cross terms' in ρaf (with m ≠ m′) is stressed. These terms might refer to some `unobserved, or even unobservable feature of reality', analogous to the existence of unobserved atomic vibrations in a billiard ball, which are unobservable within the domain of rigid body theory.
        `Unobservability of the cross terms' was suggested by the fact that, in order to obtain relevant experimental evidence, a measurement should be performed of an observable of the measuring instrument that is incompatible with the pointer observable. Since also the pointer observable itself is thought to be measured, this would require a simultaneous measurement of incompatible observables, which, according to the standard formalism, is impossible.
        Hence it is conceivable that density operator ρaf yields a correct description of the final state of an ensemble of measuring instruments (c.q. of the `measurement procedure labeled by that density operator') even if containing `cross terms', but that it is impossible to observe these terms while remaining within the domain of application of the `standard formalism'. If the standard formalism would exhaust the whole class of possible measurements, then the presence of the `cross terms' would not violate any observation.
        This solution is also consistent with Copenhagen instrumentalism, ρaf being considered `just a mathematical instrument to predict relative frequencies of final pointer positions |θm>'.
      • Criticisms of the "unobservability" solution
        The "unobservability" solution is based on the impossibility to perform a measurement of an observable incompatible with the pointer observable. This is based on the idea of `mutual exclusiveness of measurement arrangements of incompatible observables' as implied by the Copenhagen idea of complementarity.
        By itself this application of the `complementarity principle' is not completely convincing, however, because not simultaneous measurement of incompatible observables, but rather consecutive measurement of incompatible observables is at stake here, and no principle of quantum mechanics seems to oppose such a procedure. Even if the measuring instrument is a macroscopic object there is no reason to believe that no microscopic property could be measured that is incompatible with the pointer observable, although this would be very difficult34.
        More generally, since the `standard formalism' does not exhaust the whole class of possible measurements, the "unobservability" solution is questionable to the extent that it may be questioned whether the `cross terms' are really unobservable. Within the domain of application of the generalized formalism it is possible to consider measurements yielding simultaneous information on incompatible standard observables (compare). This may provide experimental means to obtain observational evidence of the `cross terms' (see publs Publ. 47 and Publ. 49). `Generalized measurements' can yield information on `cross terms', which, for this reason, do not seem to be necessarily unobservable. However, their observation necessitates measurement procedures that possibly are more invasive than the measurement procedures described by the `standard formalism' (in order to measure the `cross terms' in |ψcat><ψcat| one might probably have to carry out a measurement inducing a nonzero probability that the cat be killed by the measurement itself).
    • iii) The `decoherence solution'
      • Even within the standard formalism the "unobservability" solution has been felt to be insufficient because of the idea (implicit in the classical paradigm) that a physical theory must yield an objective description of reality. This idea is responsible for the `decoherence solution' (cf. section 3.4 of Publ. 52), looking for a physical mechanism responsible for the eradication of the `cross terms' in ρaf, which should be `not only unobservable, but even non-existent'. Interaction with a stochastically fluctuating environment, allegedly causing the `cross terms' to vanish on the average (random phase approximation), has been invoked for this purpose53. Accordingly, environment-induced decoherence has been proposed as a solution to the "measurement problem". Thus, the transition from the Schrödinger cat statecat> to the state ρcat might be thought to be a `consequence of decoherence'.
      • Criticism of the `decoherence solution'
        It is questionable, however, whether `decoherence' must be considered as a solution to the "measurement problem". Indeed, from the theory of pre-measurement it follows that, at least for `measurements of the first kind', decoherence is not necessary for `washing out the cross terms': for such measurements the quantum mechanical description of the measuring process does not need any additional feature for arriving at final state ρaf = ∑m|cm|2m><θm| of the measuring instrument.
        On the other hand, for `measurements of the second kind' decoherence might seem to yield a solution by removing the off-diagonal terms from the final state ρaf = ∑mm′cmcm′*m′m> |θm><θm′|. However, it seems to do so by affecting the microscopic part <ψm′m> of the expression. This contradicts the idea that `decoherence is effective only in macroscopic objects, leaving microscopic objects largely unaffected'.
        It is being realized by now that the decoherence solution, at least in the form in which pointer states |θm> are supposed to be mutually orthogonal, can hardly be very realistic, because a pointer of a measuring instrument is a macroscopic object, and, hence, must be in some state having (reasonably) sharp values of both position and momentum. Such states could be, for instance, the so-called coherent states (which are the most classical states of a harmonic oscillator). This seems to make completely obsolete an analysis in terms of orthogonal pointer states |θm> and corresponding `cross terms'. `Nonorthogonal pointer states' should be dealt with by means of POVMs of the generalized formalism. By sticking to the standard formalism of quantum mechanics the `decoherence solution' has during a long time pursued an `easy solution' by exploiting an "obvious" explanation of weak von Neumann projection. However, the problem is more involved. Fortunately, this seems to have been realized in recent literature on `decoherence'.
      • The `decoherence solution' hinges on a realist interpretation of the quantum mechanical formalism, not distinguishing between `what is' and `what is observed'. In the empiricist interpretation such a distinction is possible. Thus, analogous to what is going on in a billiard ball, subquantum fluctuations might be present which are `not described by the quantum mechanical formalism', the formalism being interpreted as describing `just the phenomena'. This does not imply that subquantum fluctuations are thought to be unobservable. However, in order to be able to observe them measurement procedures are necessary trandscending the domain of quantum mechanics analogous to the way atomic vibrations in a billiard ball transcend the domain of the `classical theory of rigid bodies', being unobservable within the latter domain without being washed out by any decoherence effect. Atomic vibrations may have observational consequences, for instance, by increasing a billiard ball's temperature.
      • It seems to me that environment-induced decoherence is a very important physical effect, having implications for application of quantum mechanics in the fields of quantum information and quantum computation. It is questionable, however, whether decoherence must play any fundamental role in the `quantum mechanical theory of measurement'. If the thermodynamic analogy has any reality, then quantum mechanics is to be considered as a phenomenological theory, comparable with thermodynamics, which within its domain of application is applicable without ever referring to atomic fluctuations. The domain of application of thermodynamics is determined by the times atomic systems need to relax to equilibrium. In the `statistical mechanics of atomic systems' decoherence is observable only by considering `atomic processes faster than such relaxation times'. Observation of subquantum decoherence will probably require even faster experiments.
    • iv) Solution by adopting an `ensemble interpretation'
      • In its most elementary form the "measurement problem" is just an extension of the Schrödinger cat paradox, the cat being considered as a measuring instrument for "observing" a radioactive nucleus. Then the problem is connected with the individual-particle interpretation of the state vector. Abandoning this interpretation for an ensemble interpretation could be thought to solve the "measurement problem" analogous to the ensemble solution of the Schrödinger cat paradox.
        In the `ensemble interpretation' strong von Neumann projection (and its generalization to more general measurements) may be interpreted as a description of `conditional preparation of a subensemble' obtained by selecting from an `ensemble of individual measurements' a `subset of post-measurement objects corresponding to one and the same pointer position m (compare figure 3)'. This subensemble is then thought to be described by the state vector |ψm>.
      • Abandoning the individual-particle interpretation in favour of the `ensemble interpretation' resolves the dubious aspects of the `ontic versus epistemic dichotomy', the `dubious change of knowledge as exploited in the epistemic interpretation' being replaced by an (ontic) `transition to a different physical object' (viz. from an `ensemble' to a `subensemble'). In my view this solution, although not completely sufficient, is a large step towards solving the "measurement problem".
    • v) Empiricist perspective on the `ensemble interpretation'
      • Note that, since the final state |Ψf> of the pre-measurement process is an entangled state, the final ensemble may exhibit non-classical correlations, and, hence, should not be interpreted as a classical ensemble. In particular, the possessed values principle should not be assumed to be valid (compare).
        More generally, from the discussion of the EPR problem it is seen that, if a `realist interpretation' is maintained, an `ensemble interpretation of the state vector' either perpetuates the problematic `attribution of quantum mechanical measurement results as objective properties to the microscopic object (if the measurement result is considered in an objectivistic-realist sense)', or it introduces an equally problematic `EPR nonlocality (if the measurement result is considered in a contextualistic-realist sense)'.
        In the `empiricist interpretation' this dilemma is evaded because the state vector c.q. density operator of particle 2 in the EPR experiment is not thought to describe a `contextual reality' of that particle, but only to refer to a `conditional preparation procedure' for particle 2, conditional on the measurement result obtained for particle 1 (compare Publ. 57).
        In the `empiricist interpretation' the `reality of the particles' is thought to be described by a subquantum theory, which (submicroscopic) reality is probably not probed by quantum mechanical measurements any better than the `reality of a billiard ball' is probed by `experiments applicable within the domain of application of classical rigid body theory'.
      • `Quantum mechanical preparation procedures' are often described by entangled states like the final state |Ψf> = ∑mcm|am> |θm> of a first kind pre-measurement, or the state |ψ> = ∑m cm|a1m>|b2m> corresponding to `preparation by the source in EPR and EPR-Bell experiments (valid before interaction with the measuring instrument(s))'. Such states give rise to cross terms in the corresponding density operator, which in `realist interpretations' are often thought to be problematic. Such worries are at the basis of the `decoherence solution' as well as the `unobservability solution'.
        In the `empiricist interpretation' (like in the instrumentalist interpretation) there is no necessity of a `decohering mechanism' because there is no reason to deny the existence of `cross terms': these terms are simply not observed by measuring a `standard observable'.
        Note, however, that this does not imply that the `unobservability solution' is a sound alternative. If the `cross terms' are `not observed by standard observables' this does not imply that they are unobservable. Nowadays measurement procedures are available yielding information on the `cross terms', either (remaining within the `standard formalism') by applying `quantum tomography', or (transcending the `standard formalism') by measuring generalized observables, some of which even being able to yield `complete information on the state vector' (compare sections 7.9.4 and 8.4.4 of Publ. 52).
        In macroscopic objects like `measuring instruments' `cross terms' may be extremely difficult to observe. Nowadays the study of so-called `Schrödinger cat states' is a new and challenging field of research, in which it is attempted to obtain experimental evidence of `cross terms' in objects of ever increasing dimensions. Such measurements often transcend the standard formalism (for instance, Publ. 49), the generalized formalism being necessary for their description.
      • It is not unlikely that the `result of preparation' will not be independent of the `preparation procedure'. Hence, `cross terms' might contain information on features of a submicroscopic reality that are "really" there (however, to be described by a subquantum theory), but that are not observed within the domain of application of quantum mechanics (for instance, due to insufficient resolution of the measurement procedures actually used, either standard or generalized). Some ideas how for standard observables this could be implemented into `subquantum theory' are given here, developing the `empiricist' idea that quantum mechanics is just describing `phenomena observed within its domain of application', and need not account for features that are only revealed by `measurement procedures outside that domain', not yet available.