`Individualparticle interpretation' versus `ensemble
interpretation' of quantum mechanics

Interference with the
realist versus empiricist dichotomy
The dichotomy of `individualparticle 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 `individualparticle version' and an `ensemble version' of the `realist interpretation'.
There does not exist an `individualparticle 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 a_{m} 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 `individualparticle versus ensemble' dichotomy.

Individualparticle interpretation of the quantum mechanical state vector

In the `individualparticle 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 `individualparticle 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 space^{0}).
This replacement may be seen as a mathematical implementation of Bohr's account
of `quantum mechanics as a rational generalization of classical mechanics'.
The `individualparticle 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} c_{m}a_{1m}>b_{2m}>
has been felt to constitute a considerable problem
for the `individualparticle interpretation of the state vector'.
Due to the linearity of the Schrödinger equation the state
vector of any interacting (microscopic) twoparticle 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 `individualparticle interpretation' is replaced by an `individualobject 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 `individualparticle interpretation', making obsolete any consequence of the latter
interpretation;
ii) the `state vector of the universe' is a concept typical of an
`objectivisticrealist
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 EPRBell experiments).
As a consequence of application of an `individualparticle 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 `individualparticle 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 dispersion^{0},
causing a wave packet of a single particle, initially
localized in a small region, to spread all over 3dimensional space^{18};

b) in a `doubleslit 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 time^{19}; 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 Nparticle system the wave function is not defined on the 3dimensional
physical space we live in, but rather on a 3Ndimensional
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 p_{m} 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 doubleslit 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 p_{m}
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 individualparticle interpretation endorsing ontological
probability^{20}.
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 `individualparticle interpretation'
I will mainly restrict myself here to the problems of the
`realist individualparticle 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 (socalled `Schrödinger cat state')
ψ_{cat}> =
2^{−½}(ψ_{alive}>
+ ψ_{dead}>).
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'
ψ_{cat}> (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 individualparticle 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 ψ_{cat}> into either ψ_{alive}> or
ψ_{dead}>.
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 ψ_{cat}><ψ_{cat}
are unobservable, and hence
the cat's state might as well be described by the density operator
ρ_{cat} =
½(ψ_{alive}><ψ_{alive}
+ ψ_{dead}><ψ_{dead}),
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 ψ_{cat}> 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 ψ_{cat}> 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
θ_{m}> should correspond to the states
ψ_{alive}> and ψ_{dead}>),
the `final state of Schrödinger's cat' (to be compared with
ρ_{af})
is given by ρ_{cat} rather than by
ψ_{cat}>.
There is no question of any transition from the state ψ_{cat}>
to the state ρ_{cat}
since the cat has never been in the state ψ_{cat}>
(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
`individualparticle interpretation'
and `ensemble interpretations' of the quantum mechanical state vector,
the paradox evaporating if ψ_{cat}> 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 ψ_{cat}>
to either ψ_{live}> or ψ_{dead}> 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 `individualparticle 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 presentday 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 ρ_{cat} 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 highenergy states of
microscopic systems (like e.g.
Rydberg states^{0}),
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 interpretation^{28}
(be it neither in the sense of the statistical interpretation
nor in that of von Neumann's ensemble interpretation)
over an `individualparticle 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 `individualparticle interpretation':

i) Since experimental tests of
quantum mechanics in general require determination of relative frequencies
in an ensemble, the metaphysical nature of the `individualparticle interpretation' is evident.
The `individualparticle 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 phasespace 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 `individualparticle interpretation of the state vector', to the effect that an individual particle
is warranted to have a welldefined 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 ComptonSimon and EPRBell 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
measurement^{32}.

By Einstein these problems have been employed to challenge
the Copenhagen assumption of completeness of quantum mechanics (being tantamount to
the `individualparticle 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 individualparticle 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 tradeoff 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 `individualparticle 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 premeasurement, a problem of the `individualparticle
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} p_{m}
ψ_{m}><ψ_{m}), thus
in a miraculous way changing the physical nature of the object^{31}
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 noninteracting), rather than to an `individual object (for instance, one electron)'.
Remember that N need not be infinite.
Since probabilities p_{m} can
be experimentally compared only with relative frequencies
(an individual microscopic object yielding only one single
measurement result a_{m}), 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
`individualparticle interpretation', this reason presumably being decisive even if the latter
interpretation were not problematic also for other reasons.
Attribution of p_{m} 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 `doubleslit 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 particlelike objects'.
Only the `fully builtup 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 doubleslit 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 theory^{56}.
It may have contributed to the Copenhagen interpretation's choice in favour of the
individualparticle 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 abovementioned
observational connection between the state vector
and an ensemble, it is evident that
the `individualparticle interpretation of the state vector' "multiplies meanings" in stark contradiction to
Occam's razor principle^{0}.
In my view therefore the `ensemble interpretation' is preferable over the `individualparticle interpretation', wavelike
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 N_{m}',
and, on the other hand, the `epistemological meanings of the quantum
mechanical quantities p_{m}' with which the limits of
the relative frequencies N_{m}/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 p_{m} 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 `individualparticle
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
`individualparticle parlance' like "determining the state a particle
is in," or "a particle jumping from one state to another." In a fullblown
`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> = ∑_{m}c_{m}a_{m}>
might be considered as a description of an ensemble, consisting of `subensembles described by the vectors
a_{m}>',
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 a_{m} of observable A' might be interpreted as
`finding the (individual) particle in the subensemble described by a_{m}>'.
Note, however, that this interpretation ignores the question of whether there is
a distinction between the ensembles described by the state vector ψ> = ∑_{m}c_{m}a_{m}>
(or the density operator ψ><ψ) and the density operator
∑_{m}c_{m}^{2}a_{m}><a_{m},
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 a_{m} 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 `objectivisticrealist ensemble
interpretation' it seems natural to explain a measurement result a_{m} 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 welldefined values.
In a `contextualisticrealist
ensemble interpretation' a welldefined 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
real^{6} 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 KochenSpecker theorem^{0}
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 a_{m}'.

Reservations based on the `possessed values principle'
as regards the feasibility of ensembles apply only
to the objectivisticrealist interpretation in which quantum mechanical measurement
results are considered to be objective properties of microscopic
objects. Such reservations are futile in the
contextualisticrealist 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'
(objectivisticrealist interpretation),
but either as `properties of the microscopic object, possessed within the context of a measurement'
(contextualisticrealist interpretation)
or as `properties of the measuring instrument'(empiricist interpretation).
It does not assume that all observables have a welldefined 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 b_{n} 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 a_{m}
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 a_{m} 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 `subrigid body
properties' of the interactions between its atoms (which no longer are considered
to be metaphysical after we have overcome the
Machian reluctance^{0}
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
KochenSpecker theorem^{0}
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 a_{m}'.
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 a_{m}).

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
KochenSpecker 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 KochenSpecker 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 `individualparticle interpretation'
adopted. It is highly questionable whether, in view of presentday 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 p_{m} ≠ 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 interpretation^{29}
With respect to pure states von Neumann's interpretation is an
individualparticle interpretation, an individual particle
being described by a quantum mechanical state vector ψ>.
A density operator ∑_{m} p_{m}ψ_{m}><ψ_{m} 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
ψ_{m}>. 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
∑_{m}p_{m}ψ_{m}><ψ_{m} are thought to be just (classical)
inhomogeneous ensembles of `homogeneous ensembles corresponding to pure states ψ_{m}>'.
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 ∑_{m}p_{m}ψ_{m}><ψ_{m}'
is assumed "to be in one of the states
ψ_{m}>", although it is unknown in which one.
However, the `ignorance terminology', translating `von Neumann's ensemble language'
back into the `Copenhagen individualparticle language', derives from the
`ontic versus epistemic' dichotomy, and hence is perpetuating confusion.

Criticisms of von Neumann's interpretation

a) If projection operators ψ_{m}><ψ_{m} are not all compatible, then
in general the KochenSpecker theorem will preclude the possibility of looking upon
the different states ψ_{m}> 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 ψ_{m}><ψ_{m} are compatible, is that
the representation ρ = ∑_{m}p_{m}ψ_{m}><ψ_{m}
need not be unique. It is possible that also
ρ = ∑_{n}q_{n} φ_{n}><φ_{n},
the set {φ_{n}>} of vectors
φ_{n}> being different from the set
{ψ_{m}>} (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 ψ_{m}> as well as in one described by
φ_{n}>. This nonuniqueness 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 nonuniqueness problem
can be solved by changing from an `objectivisticrealist interpretation
of the quantum mechanical formalism' to a contextualisticrealist
one^{27}.
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 premeasurement.
In this description the transition from the initial state ψ>θ_{0}>
to the final state Ψ_{f}> =
∑_{m}c_{m} ψ_{m}>
θ_{m}> 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 ρ_{of} of the microscopic object,
found as the partial trace
Tr_{a} Ψ_{f}><Ψ_{f} =
∑_{m} p_{m} ψ_{m}><ψ_{m},
should describe an inhomogeneous ensemble. Then, if the ensemble described by
Ψ_{f}> were homogeneous, we would have to face the
fact that disregarding all information on the measuring instrument
(as implied by taking the partial trace Tr_{a})
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 sight^{21} (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} p_{m}ψ_{m}><ψ_{m}
might be homogeneous (i.e. a socalled `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 individualparticle 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 welldefined 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 welldefined (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 interpretations^{0}
try to save part of the `statistical
interpretation' by trying to circumvent theorems like the
KochenSpecker^{0}
theorem by restricting the set of quantum mechanical
observables having welldefined values so as to prevent that
theorem from being derivable. Thus, in an ensemble described by the state vector
ψ> = ∑_{m}c_{m}a_{m}>
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 welldefined.
A reason for such an assumption is that the statistical measurement results p_{m} for A cannot
distinguish between the initial states ψ> =
∑_{m}c_{m}a_{m}> and
ρ^{(A)} =
∑_{m} c_{m}^{2} a_{m}><a_{m}, the latter corresponding to an
ensemble in which each element is thought to have a welldefined value a_{m} of A.
Within the measurement context of A the subensemble of ρ^{(A)}
corresponding to a_{m} is described by the state vector a_{m}>,
thus preventing observables incompatible with A from having welldefined 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 p_{m} 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 p_{m}.
For the initial state ψ> =
∑_{m}c_{m}a_{m}> this information
can be encoded in the density operator
ρ^{(A)} =
∑_{m}c_{m}^{2}
a_{m}><a_{m}, 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
ρ_{A} 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 a_{m}, 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
p_{m} of individual results a_{m} of a measurement of observable A,
the latter exhibiting a classical
(Kolmogorovian^{0})
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 individualparticle 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, epistemic^{12}.
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 N_{m}/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(nm) 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 `individualparticle' 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 socalled "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 antirealism
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 antirealism, in the sense that `unobservables' would not exist at all. `Unobservability' may be
a timebound quality, dependent on the stateoftheart 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 `antirealist 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 realism^{0},
(which is rather opposing the Copenhagen reliance on the notion of `measurement', and is aspiring at a return to an
objectivisticrealist 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 pretheory, 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 (theoryladenness 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 premeasurement 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
pretheory 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
contextualisticrealist rather than an objectivisticrealist sense; this
is evident from his not distinguishing EPR and EPRBell 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 `theoryladenness of measurement',
demonstrating that the logical/positivist/empiricist requirement of `theoryindependence 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 pretheory 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 socalled `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 operationalism^{0}.
Note, however, that `Bridgman's operationalism' partakes in an
instrumentalism
I do not support, thus deviating from the `empiricist interpretation' I am proposing^{41}.
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 discussions^{0}
on `(scientific) realism versus empiricism/instrumentalism' sealing the fate of
logical positivism/empiricism^{42}.

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 realism^{0}.
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
theoryladenness 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 structure^{30} 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 socalled "measurement problem" is symbolized by Schrödinger's cat paradox:
how to account for a `linear superposition of macroscopically distinguishable states' like
ψ_{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
Ψ_{f}> = ∑_{m}c_{m} ψ_{m}> θ_{m}>
of the `premeasurement 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 ρ_{af} = ∑_{m}c_{m}^{2}
θ_{m}><θ_{m}
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 socalled cross terms
felt to be problematic
in the density operator ψ_{cat}><ψ_{cat}.
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 ρ_{af} =
∑_{m}c_{m}^{2}
θ_{m}><θ_{m}
allegedly is in a welldefined pointer state θ_{m}>.
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
ρ_{of} = ∑_{m}c_{m}^{2}a_{m}><a_{m},
suggesting an interpretation as an `inhomogeneous von Neumann
ensemble of microscopic objects', each element being in a welldefined state a_{m}>.

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 `individualparticle interpretation' (compare);
b) there is an additional problem stemming from the theory of `measurements of the second kind',
to the effect that if
<ψ_{m}ψ_{m′}> ≠ δ_{m,m′}
the reduced density operator of the measuring instrument is given by
ρ_{af} = ∑_{m,m′}c_{m}^{*}c_{m′}
<ψ_{m}ψ_{m′}> θ_{m′}><θ_{m}.
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
ρ_{af} 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
θ_{m}>' rather than in the states θ_{m}>
themselves. This would disqualify the states θ_{m}> 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 welldefined pointer positions'. Taking such measurements seriously, I consider this problem
to be an indication (next to the possible `ambiguity of the representation' of
ρ_{af} 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 ρ_{af} (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
ρ_{af} 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,
ρ_{af}
being considered `just a mathematical instrument to predict relative frequencies of final pointer
positions θ_{m}>'.

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 difficult^{34}.
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
ψ_{cat}><ψ_{cat}
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', looking for a physical mechanism responsible for
the eradication of the `cross terms' in ρ_{af}, which should be `not only
unobservable, but even nonexistent'. Interaction with
a stochastically fluctuating environment, allegedly causing the `cross terms' to
vanish on the average (random phase approximation), has been invoked for this purpose^{53}. Accordingly,
environmentinduced decoherence has been proposed as a solution to the
"measurement problem". Thus, the transition from the Schrödinger cat state
ψ_{cat}> to the state
ρ_{cat}
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 premeasurement 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
ρ_{af} = ∑_{m}c_{m}^{2}θ_{m}><θ_{m}
of the measuring instrument.
On the other hand, for `measurements of the second kind' decoherence
might seem to yield a solution by removing the offdiagonal terms from the final state
ρ_{af} =
∑_{mm′}c_{m}c_{m′}^{*}
<ψ_{m′}ψ_{m}>
θ_{m}><θ_{m′}.
However, it seems to do so by affecting the microscopic part
<ψ_{m′}ψ_{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 θ_{m}>
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 socalled 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
θ_{m}> 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 environmentinduced 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 individualparticle 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 postmeasurement 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 ψ_{m}>.

Abandoning the individualparticle 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 Ψ_{f}> of the
premeasurement process is an
entangled state, the final ensemble may exhibit nonclassical 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 objectivisticrealist
sense)', or it introduces an equally problematic `EPR nonlocality
(if the measurement result is considered in a contextualisticrealist 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
Ψ_{f}> = ∑_{m}c_{m}a_{m}> θ_{m}>
of a first kind
premeasurement,
or the state ψ> = ∑_{m} c_{m}a_{1m}>b_{2m}>
corresponding to `preparation by the source in EPR and EPRBell
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 socalled `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.

