According to Bohr each `quantum phenomenon' possesses an element
of "wholeness", in the sense that the experimental conditions
determined by the measurement arrangement are an essential part of a
quantum phenomenon: object and measuring instrument constitute an
indivisible whole. This is the quantum postulate.
The quantum postulate has its physical basis in the unavoidable interaction
between object and measuring instrument, disturbing the object when a quantum
mechanical measurement is performed. The non-vanishing of the "quantum of
(inter)action" h (Planck's constant) makes it impossible either to
neglect this interaction or to compensate for it.
According to Bohr, due to the non-vanishing of h, the process of
obtaining knowledge by means of measurement is unanalyzable.
Due to the essential role played
by the measurement interaction the Copenhagen interpretation is sometimes
referred to as an interactional interpretation.
The quantum phenomenon is a rather vague and intuitive notion that is at the
basis of the concepts of completeness (in the restricted
sense), correspondence, and
complementarity,
characterizing the Copenhagen interpretation. In particular its unanalyzability
has been criticized by Einstein, to the extent of qualifying the Copenhagen
interpretation as a "tranquilizing philosophy".
The Copenhagen thesis that quantum mechanics is a complete theory should
better not be interpreted in the sense of completeness in the
wider sense, signifying that quantum mechanics is the `theory of
everything' (both Bohr and Heisenberg contemplated the possibility that quantum
mechanics could break down in new domains of experience), but as having the
somewhat tautological meaning that quantum mechanics is describing all possible
information to be obtained by means of quantum mechanical
measurements that are subject to the quantum
postulate. This will be referred to as completeness in the restricted sense.
Failure to make this distinction is a major source of confusion.
Bohr and Heisenberg must have been well aware of the fact that quantum
mechanics makes only statistical assertions, and, therefore, trivially
does not describe `everything' in the causal way aspired by Einstein. Yet, they
considered quantum mechanics a complete theory, because measurement disturbance
is preventing a completion of the quantum mechanical description.
Due to the Copenhagen completeness thesis the state
vector is interpreted as a description of an individual
particle. Two particles described by identical state vectors are considered
identical even if measurement of an observable yields
different values. In the Copenhagen interpretation values of
quantum mechanical observables cannot be attributed to the microscopic object prior
to, and independent of, measurement (compare). For this reason the
interpretation is often indicated as `probabilistic' rather than as
`statistical', the latter term being more appropriate in case of a classical
ensemble in which each element has well-defined values for all
physical quantities, independent of any measurement. Completeness in the restricted
sense is ontological, not epistemic.
The Copenhagen completeness
thesis is often seen as a tribute to the logical positivist
abhorrence of metaphysical quantities. Thus, the concept of trajectory is
thought not to have a meaning for electrons, because no sharp value of both position
and momentum can be attributed to it. Note, however,
that the main reason for the Copenhagen completeness thesis is the
quantum postulate, and not the
(logical positivist) idea that an
ensemble interpretation of the state vector would
not be opportune because properties distinguishing different objects, described
by identical state vectors, would not be experimentally verifiable.
It should also be realized that, in view of the possibility of an
empiricist interpretation of the quantum mechanical
formalism, a picture of an individual electron as a wave packet flying
around in space (inherent in an individual-particle interpretation) is no
less metaphysical than is the idea of the existence of hidden variables (of course,
this argument does not concern the Copenhagen interpretation as far as this
interpretation is entertained in an
instrumentalist sense).
Critique of the Copenhagen completeness thesis
It is an open question whether `measurement disturbance' is a fundamental
characteristic of all measurement in the microscopic domain, or whether it is
just an artefact of the kind of measurements within the domain of quantum
mechanics.
The Copenhagen interpretation does not exclude the latter possibility, implying
that future experimentation may require theories different from quantum mechanics
(subquantum or hidden-variables theories),
capable of analyzing the measurement procedure more fully than does
quantum mechanics.
The discussion on the (in)completeness of quantum mechanics is obscured by
the fact that two different (in)completeness issues are at stake which are not
sufficiently distinguished. These issues are:
i) the (in)completeness idea in the restricted sense,
related to the quantum postulate, which is a strictly quantum mechanical notion,
referring to measurements within the domain of application of quantum mechanics;
ii) (in)completeness in the wider
sense, related to the possibility or impossibility of "hidden variables".
Failure to make a distinction between completeness in the wider and in the restricted
sense has been a major source of confusion.
The correspondence principle is often equated with the existence of a classical
limit. However, with Bohr it has a much stronger meaning, to the extent that
any quantum mechanical measurement should be described in classical
terms, thus constituting a correspondence between quantum mechanical observables
and classical quantities.
At the basis of the correspondence principle is the
quantum postulate, making it impossible to draw a distinguishing line between
microscopic object and measuring instrument. As a consequence a physical quantity
(observable) is only defined within the context of the measurement
serving to measure it. According to Bohr it does not make sense to talk about
the position of a quantum mechanical object when this quantity is not actually
measured.
As a consequence of the correspondence principle quantum mechanical observables
are often interpreted in a contextualistic-realist sense.
The definition of a physical quantity need not be a sharp
one (represented by a number), but may have a certain latitude (represented by
an interval, like the
intervals between the marks of a ruler). According to Bohr it does not make sense
to think about a physical quantity as being more accurately defined than such an
interval; hence, not: the particle has a sharp position somewhere in the
interval!
With Bohr a physical quantity (observable) of a
microscopic object should be considered, like in classical mechanics, as a
property of the object, the possibility of a non-zero latitude
being the main difference between quantum mechanics and classical mechanics.
This implies that with
Bohr observables are not interpreted in an empiricist
sense, but in a realist one. Physical quantities of
microscopic objects are not defined in terms of the observable quantities
of the measuring instrument (as would be necessary to satisfy the requirements of
logical positivism), but are thought to have an existence of
their own. Hence, the correspondence principle is referring to a reality behind
the phenomena, be it a reality that is interacting with a measuring instrument
(contextualistic realism).
Critique of the correspondence principle
The thesis of the classical description of measurement, inherent
in the correspondence principle, is the main cause of the present obsoleteness
of the Copenhagen interpretation. It is remarkable that this interpretation has
been considered in quantum mechanical textbooks during such a long time as the
standard interpretation, without noticing any discrepancy with the quantum
mechanical description of measurement already practised by von Neumann shortly
after the birth of the Copenhagen interpretation. A quantum mechanical treatment
of quantum mechanical measurement is indispensable, both from a
philosophical point of view as well as from a
physical one.
Complementarity is a direct consequence of the correspondence
principle combined with the mutual exclusiveness of certain quantum
mechanical measurement arrangements, viz. those corresponding to incompatible
observables (in the standard formalism of
quantum mechanics corresponding to non-commuting Hermitian operators).
This implies that incompatible standard observables (like position Q and
momentum P) cannot be simultaneously
sharply defined (i.e. they cannot both have zero latitude).
Particle-wave duality
During the early stages of the development of quantum mechanics particle-wave
duality was considered paradigmatic for the notion of complementarity.
In the context of a position measurement position Q is well-defined,
allowing a particle picture of the quantum mechanical object (in which,
however, the particle does not have a sharp momentum). Analogously, in a
momentum measurement momentum P
is well-defined. Then the corresponding wave function is a plane wave. Hence, in
the context of a momentum measurement the microscopic object should be pictured as
a wave. Particle and wave pictures of the microscopic object (sometimes
called a wavicle) were considered as mutually exclusive, but also as mutually
comple(men)ting each other. It was thought to mean that in interference experiments
like the double-slit experiment (testing the wave character of the object) the
particle picture is inapplicable. Critique of particle-wave duality
It was soon realized by Bohr that particle-wave duality is not a good application
of complementarity, because in an interference experiment both particle and wave
aspects are observable. Thus, whereas the interference pattern is a consequence
of the wave aspect of quantum mechanics, is the particle aspect obvious in the same
experiment when it is observed how the interference pattern is built up by
particle-like impacts on the screen supporting the pattern (see for instance Akira Tonomura,
Double-slit experiment with single electrons
Video clip 1).
Heisenberg's disturbance theory of measurement
According to Heisenberg the meaning of complementarity is that a measurement of a standard observable
(e.g. P) disturbs a simultaneously performed measurement of an incompatible observable (e.g. Q).
Heisenberg disturbance can explain the inapplicability of counterfactual definiteness
in quantum mechanics: if a P measurement disturbs Q, then a simultaneous measurement of P and
Q must yield a different measurement result for Q than an undisturbed measurement of Q would do.
With Heisenberg `disturbance' must be taken in a preparative (or predictive) sense, the interaction of object
and measuring instrument being responsible for the disturbance. This corresponds to the impossibility that
the final object state simultaneously be an eigenvector of both P and Q. Critique of Heisenberg's disturbance theory of measurement
Heisenberg's disturbance theory of measurement is based on the restrictive view of measurement symbolized in
figure 3, in which a measurement is tantamount to a preparation of a final
state of the microscopic object. For such measurements Heisenberg's preparative `measurement disturbance'
merges with the determinative notion of `disturbance of the
information on the initial state of the object' yielded by the measurement procedure.
In more general measurements preparative (predictive) aspects should be carefully distinguished
from determinative (retrodictive) ones, thus giving rise to two different kinds of complementarity,
one for preparation, and one for the simultaneous measurement of incompatible observables.
Due to a restriction to the standard formalism (according to which
simultaneous measurement of incompatible observables is not even possible)
there has been quite a bit of confusion with respect to the latter interpretation of complementarity. However,
Heisenberg disturbance can be given an unambiguous determinative (retrodictive) meaning on the basis of the
generalized formalism (compare the
Martens inequality). The impossibility of a simultaneous sharp
definition of incompatible standard observables can be interpreted as a limitation on the
simultaneous measurement of such observables in a much more general way than envisaged by Heisenberg.
The Heisenberg-Kennard-Robertson inequalities
In the standard formalism the complementarity principle is thought to be
expressed by the Heisenberg-Kennard-Robertson inequalities (the so-called
uncertainty relations) of standard observables A and B,
DADB³½|<[A, B]->|,
in which DA
and DB
are standard deviations of measurement results.
in which H{Em}(r) =
-Sm
pm ln pm, pm = Tr Emr,
{Em} the spectral resolution of standard observable A (and analogously for B).
An advantage of the entropic uncertainty relations is that they are independent of the (eigen)values of
the observables, and, therefore, can be used in an empiricist interpretation
(compare).
Critique of complementarity
The identification of Bohr's latitudes, figuring in the
correspondence principle, with the standard deviations of the
Heisenberg-Kennard-Robertson inequalities has been a considerable source of
confusion. First, both latitudes can have a finite value, implying that knowledge is obtained on both
(possibly incompatible) observables A and B. According to the
standard formalism of quantum mechanics this is impossible, however.
Second, contrary to the idea of a latitude, in a standard deviation every
individual object is yielding a sharp measurement result. Third, the
Heisenberg-Kennard-Robertson inequalities are not in any obvious way related to
the simultaneous measurement of the observables that are involved. On the contrary,
the inequalities can be tested by separate (i.e. non-simultaneous)
measurements of the observables.
It is also important to note that Heisenberg and Bohr differed considerably
in their understanding of the meaning of the uncertainty relations. Whereas with
Bohr these are valid within the context of the measurement (i.e. during
the measurement), with Heisenberg they are referring "to the future" (i.e.
after the measurement).
Both Bohr and Heisenberg differ from the textbook understanding in which the
Heisenberg-Kennard-Robertson inequalities are taken in the initial state (i.e.
before the measurement).
It is important to distinguish two different sources of complementarity
(cf. Publ. 47), viz.
preparation and measurement, each being subject to its own
restriction with respect to the simultaneous consideration of incompatible
observables. Whereas textbooks and Heisenberg
refer to preparation (Heisenberg differing from the textbook interpretation
by referring to preparation by means of a measurement), is only
Bohr's interpretation strictly related to the simultaneous measurement of
incompatible observables. Complementarity due to preparation is a restriction on
our ability of preparing a quantum mechanical state. It can be expressed by
means of the Heisenberg-Kennard-Robertson inequality in terms of standard deviations,
obtained in ideal measurements of the standard observables,
or in terms of entropic quantities of the probability distributions
of such measurements.
In assessing the meaning of complementarity the founding fathers of quantum
mechanics have been severely hampered by the fact that only the (restricted) standard
formalism of quantum mechanics was available (or even just being developed). This made
them believe that the inequalities they derived in a rather informal way for the so-called "thought
experiments" should be represented within the mathematical formalism by the
Heisenberg-Kennard-Robertson inequalities (being the only theoretical relations
surfacing). However, the standard formalism is not
able to deal with the simultaneous measurement of incompatible observables.
According to this formalism only compatible standard observables (corresponding
to commuting Hermitian operators) can be simultaneously measured. In order to
describe the simultaneous measurement of incompatible observables (which really is the subject
discussed in the "thought experiments"), a generalization
of the mathematical formalism of quantum mechanics is necessary, allowing to define a concept of
joint (nonideal) measurement (see,
for instance, Publs 25,
26,
27,
31,
35,
47). Complementarity as discussed in
the "thought experiments" actually is about measurement (in)accuracies
that can be described by nonideality measures like the average row entropies
of nonideality matrices figuring in the theory of joint measurement of generalized observables.
The Martens inequalities, rather than the Heisenberg-Kennard-Robertson ones,
must be seen as expressing complementarity in this sense. This is a second kind of complementarity next to
the complementarity described by uncertainty relations like the Heisenberg-Kennard-Robertson inequalities,
or the entropic inequalities. The interpretation of these latter
inequalities as properties of joint measurement has been a formidable source of confusion with respect
to the notion of complementarity.
Copenhagen confusion of preparation and measurement
Most probably the main shortcoming of the Copenhagen
interpretation is its lack of distinction between the notions of preparation
and measurement. This is a consequence of the fact that quantum mechanical
observables are thought not to have values preceding and independent of
measurement (compare), to be revealed by the measurement. As a second
best option it was proposed that, in any case, the observable should possess (in
a realist sense) the measured value after its
measurement (so as to assure that a second measurement of the same observable with certainty yield
the same measurement result; note that this extends the
correspondence definition of an observable from the
interaction phase to the post-interaction phase of the measurement).
If after the measurement the observable has a well-defined value,
then the state vector should be the
corresponding eigenvector of the measured observable. This is the basis of von
Neumann's projection (or reduction) postulate, which has become the characteristic
trait of measurement according to the Copenhagen interpretation.
Hence, in this interpretation a measurement is a preparation of the object
in a certain post-measurement state (so-called measurement of the first kind).
The view of `measurement' as a `preparation' of a post-measurement
state is consistent with Heisenberg's disturbance theory of measurement,
and with his insight that the Heisenberg uncertainty relations
are not relevant to the past (i.e. the initial state) but to the future.
According to his views the meaning of the uncertainty relations is that, due to the
incompatibility of position and momentum, it is impossible to prepare the
object, by means of a simultaneous measurement of the two observables, in a state
described by a simultaneous eigenvector of Q and P (such joint
eigenvectors do not exist!).
The widespread use of the notion of measurement of the first kind in the
literature on the foundations of quantum mechanics contrasts
sharply with the virtual experimental non-existence of such measurements
(compare). It is very unfortunate that, by almost
exclusively dealing with measurements of the first kind, the foundational
discussion has largely lost touch with experimental practice.
The Copenhagen view of what is a quantum mechanical measurement is alien to
the common-sense view that measurement should yield information on the object
as it was preceding the measurement, or, in an empiricist interpretation,
information on the preparation procedure that prepared the object prior to the
measurement. Under the influence of the Copenhagen interpretation this determinative
(or retrodictive) view of measurement has largely been replaced by a preparative
(or predictive) one. It is very fortunate that in recent years the interest in the subject
of quantum information has revived the common-sense idea that the main goal
of quantum mechanical measurement is a determination of the initial state.
