Individual-particle versus ensemble interpretations of the quantum mechanical state vector (or wave function)

• Individual-particle interpretations of the quantum mechanical state vector
  • The quantum mechanical state vector 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 usually considered identical.
  • Von Neumann's projection postulate is inspired by an individual-particle interpretation of the quantum mechanical state vector, in which the latter is thought to describe an individual object. It is assumed that after the measurement of standard observable A the (individual) microscopic object has the property A=a_m with certainty if a_m is the result obtained in the measurement. The only state vector representing this state of affairs is the eigenvector |a_m> of the measured observable A. Since the state vector is thought to describe the individual object, there must have taken place a discontinuous transition (von Neumann projection).
  • The Schrödinger cat paradox
    The paradigm of the problems of an individual-particle interpretation is the Schrödinger cat paradox, in which a cat is confined in a cage, together with a radioactive atomic nucleus having a probability 1/2 to decay within the next hour. If the nucleus decays, a contraption is set into motion causing the cat to die. In a quantum mechanical description this might yield a superposition of a living and a dead cat (so-called Schrödinger cat state) :
    

    |y> = 2-½(|yliving> + |ydead>)

    Since the cat is a macroscopic object, observation of it could only yield either a living or a dead cat. Hence, in an individual-cat interpretation the state vector should make a transition from |y> to either |y_living> or |y_dead> if the cat is observed. Problems with this interpretation are:
    i) What is the physical meaning of the Schrödinger cat state (sometimes called a state of "suspended animation")?
    ii) How can mere observation bring about the projection to either |y_living> or |y_dead>? What would happen to the cat if no observation is performed at all?
  • Schrödinger's cat paradox is in the first place a problem of a realist individual-particle interpretation of the state vector. It was originally directed against the Copenhagen interpretation. It seems to be misdirected, however, insofar as this interpretation is instrumentalistic with respect to the state vector. This may be one reason why Bohr does not seem to have taken this problem too seriously. Nevertheless, possibly due to a tendency to interpret the state vector in a realist way (to be observed also in the Copenhagen interpretation) the paradox has gained considerable importance in the form of the so-called "measurement problem", the cat being comparable to a (macroscopic) measuring instrument.
  • A related consequence of a realist version of an individual-particle interpretation of the quantum mechanical state vector is that a microscopic object must split if the state vector does so. For instance, in neutron interference experiments of the type considered in Publ. 27 this would imply that a neutron traversing a neutron interferometer does so while being split into two halves, each of which taking a different path. Since this is in disagreement with all empirical data (strongly suggesting that each neutron follows either one path or the other) a realist individual-particle interpretation of the quantum mechanical state vector is unattractive (as is the "suspended animation" interpretation of the Schrödinger cat state referred to above). It is quite remarkable that nevertheless this interpretation is widely entertained. This may be due to the popular idea of particle-wave duality, having been developed in the Copenhagen interpretation during the early stages of the development of quantum mechanics, but being obsolete by now.

• Ensemble interpretations of the quantum mechanical state vector
  • The quantum mechanical state vector is thought to refer to an ensemble of identically prepared microscopic objects (for instance, a beam of electrons, produced by a cyclotron without knob settings being varied, and assuming the electrons to be non-interacting). Strictly speaking the ensemble should consist of an infinite number of individual objects; however, as a practical criterion it is sufficient to require, like in classical physics, that the number is large enough to allow for stability of the relative frequencies of measurement results when the number is varied.
  • There are two reasons why an ensemble interpretation might be preferred over an individual-particle one:
    i) The quantum mechanical formalism only provides information on probability distributions and averages obtained from these. Comparison of theoretical quantities with individual measurement results does not make sense due to lack of repeatability of the latter.
    ii) Thus, in an ensemble interpretation the Schrödinger cat paradox could be considered solved by interpreting the Schrödinger cat state as a description of an ensemble of living and dead cats. The transition from state |y> to one of the states |y_living> or |y_dead> could be seen as merely describing a selection of a subensemble. Observation of a cat in one of these states could be explained by assuming that the cat was already in that state before.
  • In an ensemble interpretation of the quantum mechanical state vector particles need not split when the state vector does so (in neutron interference experiments parts of a split state vector can tentatively be thought of as describing subensembles of neutrons following different paths in the interferometer).
  • The existence of entangled states |Y_f> is a considerable problem for an individual-particle interpretation of the state vector. Due to the linearity of the Schrödinger equation the state vector of any interacting two-particle system will evolve into an entangled state, even if it was initially a product of the state vectors of each of the particles. Hence, after having interacted it is not possible any more to attribute a state vector separately to each of the particles. This has been interpreted as evidence of a certain non-separability of such particles. This problem evaporates in an ensemble interpretation, which allows a separate description of the ensembles of the two particles by means of the density operators r_of = Tr_a |Y_f> <Y_f| and r_af = Tr_o |Y_f> <Y_f|, obtained by partial tracing over the degrees of freedom of measuring instrument (a) and object (o), respectively. The existence of r_of and r_af is consistent with the operational possibility to deal with object and measuring instrument as separate entities.
  • When urged, physicists who use to think about electrons as wave packets flying around in space, will generally admit that it is preferable to think about a wave packet or state vector as referring to an ensemble rather than to an individual particle (unless one is convinced that quantum mechanics should refer to individual objects, because that are the "real" things, ensembles just being mathematical abstractions; in that case one may prefer a more or less equivalent terminology in which a wave packet is only yielding statistical, or even probabilistic, assertions about individual objects). Unfortunately, in the quantum mechanical literature ensemble and individual-particle interpretations are not always clearly distinguished. Awareness of the fact that only relative frequencies measured in an ensemble have physical relevance is often combined with individual-particle talk, like "determining the state a particle is in," or "a particle jumping from one state to another." In an ensemble interpretation these latter phrases are meaningless.
  • An empiricist interpretation of the quantum mechanical state vector is necessarily an ensemble one, because one and the same quantum mechanical preparation procedure may yield different individual objects, to be regarded as different elements of an ensemble.
  • In an objectivistic-realist ensemble interpretation it is natural to assume a principle of `possessed values', stating:
    values of the quantum mechanical observables may be attributed to the object as objective properties the object possesses independent of observation.
  • Counterfactual definiteness
    The `possessed values' principle would provide an explanation of individual quantum mechanical measurement results if it is combined with a principle of `faithful measurement', stating:
    a measurement result is numerically equal to the value the observable had immediately preceding the measurement.
    Combining the two principles entails counterfactual definiteness, meaning that within the context of observable A it is legitimate to attribute to the microscopic object the measurement result of an incompatible observable B that would have been obtained if B had been measured instead of A.
  • There exist strong indications, if not proofs, that, due to the incompatibility of quantum mechanical observables, counterfactual definiteness is not a possible assumption, nor is the `possessed values' principle:
    i) there do not exist joint probability distributions of incompatible standard observables, at least 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 Bell inequalities that may be in disagreement with experimental evidence;
    iii) the Kochen-Specker theorem and its recently sharpened versions demonstrate the impossibility that a (small) number of incompatible observables simultaneously have sharp values.

• Homogeneous and inhomogeneous ensembles
  
  • There are different possibilities to entertain an ensemble interpretation of quantum mechanical state vectors and density operators:
    
    • 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 identical. An assumption of homogeneity of an ensemble is tantamount to abandoning any explanation of individual measurement results. 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 homogeneous.
    • Von Neumann's interpretation
      Only ensembles described by state vectors are considered homogeneous. In von Neumann's view a density operator that is not a projection operator describes an inhomogeneous ensemble (mixture) consisting of homogeneous subensembles, each of which being described by a state vector (pure state). Objects from different subensembles are not thought to be identical. We shall refer to such an ensemble as a von Neumann ensemble. It is closely akin to the completeness thesis of the Copenhagen interpretation, which, unfortunately, does not draw a clear distinction between a homogeneous ensemble and an individual particle.
      Critique of von Neumann's interpretation
      Von Neumann's interpretation implies that a homogeneous ensemble may become inhomogeneous just by ignoring certain information. Thus, the ensemble described by the entangled (pure) state vector |Y_f> = S_mc_m |y_m> |q_m> (compare) would be homogeneous. Yet, mixtures like the ones described by the reduced density operators r_of = Tr_a |Y_f> <Y_f| and r_af = Tr_o |Y_f> <Y_f| may be operationally subdivided into distinct subensembles. It would be very counterintuitive if ignorance of all information on one of the objects would make the other object's ensemble inhomogeneous (usually you need more information to distinguish objects that seem to be identical). For this reason the concept of a von Neumann ensemble is unattractive.
    • The `statistical' interpretation
      An ensemble, described by either a state vector or a density operator, is considered 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.
      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 interpretations try to save part of the `statistical' interpretation by trying to circumvent theorems like the Kochen-Specker one by restricting the set of quantum mechanical observables having well-defined values, so as to prevent the theorem's derivability.
    • Explanation by means of subensembles
      Note that 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 divide the initial ensemble into subensembles on the basis of distinct values of a quantum mechanical observable. It is very well possible that for this purpose we have to rely on concepts belonging to some subquantum ("hidden-variables") theory. As a matter of fact, 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, because it is obvious that a pointer position is not a property of the microscopic object.

Quantum mechanics and logical positivism

• According to the philosophical doctrine of logical positivism the theoretical terms of a theory must preferably be defined in terms of observation statements, referring to `phenomena' (the so-called `hard data'), the occurrence of which can be verified by means of direct observation.
• (In)completeness of quantum mechanics
  • (In)completeness in the wider sense
    The logical positivist 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. 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, are deemed metaphysical.
  • The (in)completeness issue meant here will be referred to as `(in)completeness in the wider sense' in order to distinguish it from `(in)completeness in the restricted sense' involved in the Copenhagen interpretation. In this latter issue not the question of "hidden variables" is at issue, but the question whether the measurement arrangement is an essential part of a quantum phenomenon (i.e. the question of contextuality). It is important to note that in the discussion between Bohr and Einstein not the former but the latter completeness idea was at issue. 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 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 a measurement should not be described by the theory to be tested by it. Measurement should be described either by means of observation statements, or in terms of a (pre-)theory, the latter's theoretical terms being defined in the former way. The insight that, in general, a description of a measurement must necessarily make use of the very theory that is tested by it ("theory-ladenness of measurement") has been a cause of the decline of logical positivism as a useful philosophy of science. The `hard data' turned out to be not that hard after all.
  • Bohr's correspondence principle has been interpreted in the past as proof of Bohr's affinity to logical positivism. This is correct in as far as measurement is supposed by Bohr to be exclusively describable in terms of classical mechanics, considered as a pre-theory of quantum mechanics. However, Bohr's correspondence principle is sufficiently tampering with metaphysics (compare) to support Bohr's own assurance that he is not a logical positivist.
  • The insight that the interaction between microscopic object and measuring instrument is necessarily within the domain of quantum mechanics (compare) makes quantum mechanics a paradigm for testing theory-ladenness of measurement.

The "measurement problem"

• The so-called "measurement problem" stems from a quantum mechanical description of the measurement process for a standard observable A. In particular, the final state of the (pre)measurement process, |Y_f> = S_mc_m |y_m> |q_m>, is felt to be problematic because it is a linear superposition of states of a macroscopic object.
• Possible solutions to the "measurement problem"
  
  • Solution by means of von Neumann ensembles
    For measurements of the first kind the "measurement problem" might tentatively be solved by invoking the concept of a von Neumann ensemble. Thus, calculating for first kind measurements from |Y_f> the reduced density operators of microscopic object r_of and measuring instrument r_af, we find, respectively,

    	rof = Sm|cm|2 |am><am|
    	raf = Sm|cm|2 |qm><q m|

    suggesting a von Neumann ensemble of microscopic objects, each of which "being in one of the states |a_m>", and a similar ensemble of measuring instruments each "being in a well-defined pointer state |q_m>", with relative frequencies |c_m|² for both ensembles. The possibility of interpreting the final state of a measurement in this way explains the great popularity of first kind measurements.
    Critique of the solution by means of von Neumann ensembles
    Apart from the dubious quality of the concept of a von Neumann ensemble (compare), there is still a second reason not to be satisfied with this solution. The solution only works for measurements of the first kind, whereas, unfortunately, virtually all measurements are of the second kind. Since, then, in general, <y_m|y_m'> ¹ d_m,m' we get for the measuring instrument the reduced density operator

    raf = Sm,m'< ym |ym'>|qm'><qm|

    The existence of the nondiagonal (`cross') terms with m ¹ m' prevents an interpretation of the final state of the measuring instrument as a von Neumann ensemble of instruments having well-defined pointer positions. Of course, a representation of r<sub>af</sub> in terms of its eigenvectors would yield an opportunity to interpret it, once again, as a description of a von Neumann ensemble. However, in general the eigenvectors will be superpositions of the pointer states |q<sub>m</sub>>. This would once again lead to Schrödinger's cat paradox.
  • The "unobservability" solution
    The `cross' terms in r_af are only relevant if an observable of the measuring instrument is measured that is incompatible with the pointer observable. It is conceivable that density operator r_af yields a correct description of the final state of the measuring instrument, but that it is impossible to observe the `cross' terms. Note that within the standard formalism observables that are incompatible with the pointer observable cannot be measured simultaneously with the latter observable. Hence, 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 akin to Copenhagen instrumentalism, r_af being considered just an instrument to predict relative frequencies of final pointer positions.
    Critique of the "unobservability" solution
  
  Since the standard formalism does not exhaust the whole class of possible measurements the "unobservability" solution is questionable. 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 means to observe evidence of the `cross' terms.
  • The decoherence solution
    Notwithstanding their "unobservability" the presence of `cross' terms in r_af has sometimes been felt as an insuperable difficulty. The main reason for this worry is a realist interpretation of the state vector c.q. density operator, requiring that the `cross' terms not only be unobservable, but even be non-existent. Interaction with a stochastically fluctuating environment, allegedly causing the `cross' terms to vanish on the average (random phase approximation), has been invoked to make an interpretation in terms of von Neumann ensembles feasible again. Accordingly, environment-induced decoherence has been proposed as a solution to the "measurement problem".
    Critique of the decoherence solution
    It is being realized by now, however, that the decoherence solution, at least in the form presented above, 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). These states are not mutually orthogonal, however. This seems to make completely obsolete an analysis in terms of orthogonal pointer states |q_m> and corresponding `cross' terms. Although environment-induced decoherence undoubtedly is a very important physical effect, is it questionable whether it must play any fundamental role either in the quantum mechanical theory of measurement or in the description of macroscopic bodies (that, maybe, are not even describable by quantum mechanics at all, compare).
  • 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 one could solve the "measurement problem" analogous to the ensemble solution of the Schrödinger cat paradox.
    Note that, since |Y<sub>f</sub>> is an entangled state, the 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; it may not even be applicable to the measuring instrument, at least not as far as described quantum mechanically in the pre-measurement phase). There is no reason to assume that the `cross' terms are not there. However, these terms are extremely difficult to observe (in the original Schrödinger cat paradox it would require a measurement of an observable of the cat having superpositions of |y<sub>living</sub>> and |y<sub>dead</sub>> as eigenvectors). The study of so-called Schrödinger cat states is a new and challenging field of research, in which it is attempted to get experimental evidence of `cross' terms in objects of ever increasing dimensions.

Conditional preparation

• Every quantum mechanical measurement has also a preparative aspect in the sense that it not only prepares the measuring instrument in some final state, but it does so also with the microscopic object. In the final state of the (pre)measurement process, |Y_f> = S_mc_m |y_m> |q_m>, this preparative aspect is represented by the state vectors |y_m>. Such a state vector can be legitimately interpreted as describing the state of the object, conditional on measurement result m having been read off the measuring instrument's pointer. Note that the state vectors |y_m> are completely determined by the interaction between object and measuring instrument. Only in very special cases it may occur that the vectors |y_m> equal the eigenvectors of a standard observable. In that case von Neumann's projection postulate is satisfied. However, in actual experimental practice this is seldom, if ever, fulfilled (compare). In general the vectors |y_m> are not even orthogonal.
• In a conditional preparation by a measurement the state vector undergoes a transition

  |y> ---> |ym>,

  generalizing von Neumann's projection. The necessity of taking |y_m> as the post-measurement state conditional on measurement result m, follows from the fact that a measurement of an arbitrary standard observable B (with eigenvectors |b_n>) performed immediately after the first measurement, is yielding (conditional) probabilities

  p(n|m)=|<ym|bn>|2.

  These conditional probabilities are derivable from the joint probabilities p(m,n) of obtaining the pair of measurement results (m,b_n).
• In an individual-particle interpretation this discontinuous change of the state vector may cause a Schrödinger cat problem that can be tentatively solved by switching to an ensemble interpretation, in which the transition from |y> to |y_m> is interpreted as a selection of a subensemble from the ensemble of microscopic objects, carried out on the basis of the observation of measurement result (pointer position) m. Alternatively, and, from the point of view of an empiricist interpretation more preferably, it is possible to interpret the transition to |y_m> as a transition to a different preparation procedure, the observation of measurement result m and the selection on the basis of this observation being part of this preparation procedure.
• Figure 3
  Whereas von Neumann's projection postulate is inapplicable as a measurement principle, is it, in the generalized form given above, a highly useful and practically applied preparation principle (sometimes referred to as a `preparative measurement' or a filter). That it, yet, is presented so often as a feature of measurement is a consequence of a very restricted view of measurement, in which the states |y_m> correspond to outgoing beams that do not spatially overlap (see figure 3). In that case the measurement result m is determined by the beam the particle is found in after leaving the apparatus. In this case the final position of the microscopic object is taken as the pointer observable. Due to this very special choice of the pointer observable the preparation of the final state of the object is important for the procedure to function as a measurement of a property of the initial state. In general pointer observables are quite different, however. In general they correspond to properties of the measuring instrument rather than the microscopic object.
• Unfortunately, in foundational discussions this very restricted and seldom realized procedure (the Stern-Gerlach measurement of spin is (approximately) of this type) has become more or less paradigmatic of quantum mechanical measurement. In particular, Heisenberg's interpretation of his uncertainty relation hinges on it, this relation being taken to be valid in the final state of the object, and, allegedly, expressing the measure of disturbance of the microscopic object by the measurement. The Copenhagen confusion of preparation and measurement is largely a consequence of this restricted view on quantum mechanical measurement. It has had a considerable confusing effect in the discussion on the foundations of quantum mechanics. This can only be resolved by means of a careful description of quantum mechanical measurement process.
• The quantum Zeno effect
  One example of the above-mentioned confusion is the application of von Neumann projection to the so-called quantum Zeno effect, which allegedly has as a result that the object is frozen in its initial state by continuous observation (``a watched pot never boils''). This can only happen if the interaction with the measuring instrument that performs the observation is such that it has this effect. This means that the interaction of object and measuring instrument must be sufficiently energetic. Thus, it will certainly not be possible to prevent a radioactive nucleus from decaying by merely continuously looking at a 4p counter surrounding the nucleus.