Necessity of the `generalized formalism of quantum mechanics'

Necessity of the
`generalized formalism of quantum mechanics' is directly seen
from the quantum mechanical description of the
interaction of object and measuring instrument. Let
ρ be the `initial density operator of the
microscopic object', and ρ_{a} that of the measuring
instrument. If H is the interaction Hamiltonian, and T the
duration of the measurement interaction, then the final state is
given by (compare)
ρ_{f}(T) =
e^{−iHT}ρρ_{a} e^{iHT}.
Let {E_{am}} be
the `pointer observable of the measuring instrument' (in the simplistic discussion
this is the projectionvalued measure
{θ_{m}> <θ_{m}}).
Then, applying the Born rule to the measuring
instrument, the probability distribution of the observed pointer
positions is given as
p_{m} = Tr_{oa}ρ_{f}(T)E_{am},
where the trace is performed over both Hilbert spaces of
object (o) and measuring instrument (a). By using a wellknown property
of the trace operation (viz. cyclic permutation of the operators) it is now possible to interpret this
expression as a property of the initial state of the microscopic object:
p_{m} = Tr_{o}ρ M_{m},
with
M_{m} = Tr_{a}ρ_{a} e^{iHT}
E_{am}e^{−iHT}.
The operators M_{m} satisfy the
conditions of a positive operatorvalued measure.
 It should be noted that, even if the
pointer observable {E_{am}} is a PVM, then there
is no reason to assume that the set of operators
{M_{m}} is a PVM too. From the point of view of
`quantum measurement theory' PVMs do not have any particular
position within the set of `all possible generalized observables'.
Also from a practical point of view PVMs do not present themselves in
a particularly conspicuous way: hardly any quantum mechanical
measurement that is performed in actual practice is represented
by a PVM (see illustrations). On closer
inspection most measurement procedures turn out to be represented
by POVMs. The special position often attributed to PVMs must
have a different origin. Presumably the main cause of this is the
idea that Hermitian operators describe objective
properties of the microscopic object, whereas generalized
observables are (co)determined by the measurement arrangement
(they generally depend on its parameters). Unfortunately, this
idea, although attractive, is tantamount to the
possessed values principle,
and is untenable.

From the derivation of the POVM {M_{m}}, given above, it can be seen that it is a
representation of the relative frequencies of `pointer positions
of the measuring instrument'. This is consistent with an
empiricist interpretation, in which
an observable is not a `property of the microscopic object', but a
`label of a measuring instrument'. For this reason it does not seem
to be plausible to interpret POVMs in a realist sense. It is an
open question whether such is possible for PVMs (in view
of the impossibility of an objectivisticrealist interpretation
it would at least have to be a contextualisticrealist interpretation).

Some illustrations of the necessity of the generalized formalism

Inefficient photon detection
A realistic photon counter has a quantum efficiency η ≤ 1. As a result such a photon counter does not measure the
number observable N (having the numbers n= 0,1,... as
eigenvalues, and the number states n> as eigenvectors). The
probability that such a counter detects m photons is given by
M_{m} = ∑ _{n=m,m+1,..}(n!/(m!(n−m)!)η^{m}
(1 − η)^{(n−m)}N_{n},
{N_{n}} being the spectral resolution of the number
observable N. It is evident that M_{m} is
not a projection operator. If η =1 then M_{n} = N_{n}. Hence, the
standard formalism only applies to (unrealistic) ideal photon counters.

The doubleslit experiment
figure 4
The example of the inefficient
photoncounter could be disregarded as being inessential, since
it is not unreasonable that a theory be tested using only
ideal measurement procedures. Such disregarding is impossible with
respect to the doubleslit experiment, since that experiment is at
the heart of quantum mechanics, and yet needs a POVM for its
description. This implies that this `paradigm of quantum
mechanics', often used to clarify the meaning of the standard formalism,
is not even within the domain of application of the standard
theory. By hindsight it is understandable that this has caused
quite a bit of confusion.
A simple way to derive a POVM (depending on the precise way the
measurement is carried out there exist different ones) is the following:
Let ψ_{1} and ψ_{2} be two orthogonal wave packets
originating from slits 1 and 2 in screen S, respectively. The general state
is the superposition
ψ =
αψ_{1} +
βψ_{2},
α^{2} + β^{2} = 1,
bringing about an interference pattern on screen B. As a
pointer scale we can take the vertical position at which a particle
hits screen B. Then, restricting ourselves to a twodimensional
description, the relative frequencies of the pointer are given by
p(z) = ∫dx
ψ(x,z)^{2} = α^{2}p_{11}(z) + α^{*}βp_{12}(z)
+ αβ^{*}p_{21}(z) + β^{2}p_{22}(z),
with
p_{ij}(z) =
∫dx ψ_{i}(x,z)^{*}ψ_{j}(x,z), i,j = 1,2.
The POVM is defined by putting
p(z) = <ψM(z)ψ>,
yielding M(z) as a 2x2 matrix with components M(z)_{ij} =
p_{ji}(z). It is straightforward to prove that
M(z)^{2} ≠ M(z). Actually, we have
M(z) = ∫dx (ψ_{1}(x,z)^{2} +
ψ_{2}(x,z)^{2}) P(x,z),
in which P(x,z) is an idempotent 2x2 matrix for each value of x and z.
 Polarization measurements
One way to measure photon polarization is to send the photon through an
analyzing filter (e.g. a nicol), oriented at angle θ in a plane perpendicular to the direction
of propagation, and to detect whether or not the photon has passed
the analyzer. If the photon detector has quantum efficiency
η, then the probability of detecting the photon behind the analyzer is p_{+} =
η <ψE_{+}ψ>, in which ψ> is the polarization state vector of
the incoming photon. Hence, this experiment is represented by the
POVM {M_{+},M_{−}} = {ηE_{+},
I − ηE_{+}}, where {E_{+},E_{−}} (with
E_{−} = I − E_{+}) is the
PVM of the standard polarization observable. Once again it is
seen that the standard formalism only applies to unrealistically ideal detectors.
Note that the POVM {M_{+},M_{−}} of
the nonideal detector is related to the PVM of the ideal one
according to
M_{m} =
∑_{n}λ_{mn}E_{n},
m,n=+/−,
with
λ_{++} =
η, λ_{+−} = 0,
λ_{−+} = 1 − η,
λ_{−−} = 1.

Joint nonideal measurement of incompatible polarization observables
figure 5
In figure 5 a
photon impinges on a partly transparent mirror (transmission coefficient γ)
sending it either to a polarizer in direction θ (if it is transmitted by the mirror) or θ′
(if it is reflected by the mirror). Angles θ and θ′ are
chosen such that the corresponding standard polarization
observables are incompatible. Now there are three possibilities:
the photon is detected either in detector D or in D′, or it is
not detected at all. Here the quantum efficiency is taken to be 1. The
quantum mechanical probabilities are found according to,
respectively,
p_{+} = γ <ψE_{+}ψ>, p′_{+} = (1 − γ)
<ψE′_{+}ψ>, p_{−} = 1− p_{+} −
p′_{+},
yielding {γE_{+},(1 − γ)E′_{+},
I − γE_{+} − (1 − γ)E′_{+}}
as POVM.
It is important to note here the difference of this
experiment with an experiment in which either the standard
observable {E_{+},E_{−}} or
{E′_{+},E′_{−}} is measured, which
can be achieved by taking either γ = 1 or γ = 0.
The present experiment is fundamentally different from these special cases
which both correspond to a `standard observable'. It should
not be overlooked that in these latter experiments the wave packet
is completely sent either one way or the other, whereas in the
present experiment the wave packet is split so as to send
coherent wave information towards both detectors. This
difference makes it possible for general values of
γ to interpret the experiment as a
joint nonideal measurement
of the incompatible polarization observables
{E_{+},E_{−}} and
{E′_{+},E′_{−}} rather than as an
experiment in which either
{E_{+},E_{−}} or
{E′_{+},E′_{−}} is measured.
In order to interpret the experiment as a
joint measurement of two observables we need a
bivariate POVM. This can be realized by considering the
joint probability distribution of the two detectors D and D' rather than the probabilities of the single ones. Since it does not
occur that both detectors are triggered by the same photon, we
straightforwardly find:
p_{++} = 0, p_{+−} = p_{+}, p_{−+} = p′_{+},
p_{−−} = 1 − p_{+} − p′_{+},
from which the bivariate POVM follows as {R_{mn}, m,n=+/−}, with
R_{++} = O, R_{+−} =
γE_{+}, R_{−+} = (1 − γ)E′_{+},
R_{−−} = 1 − γE_{+} − (1 − γ)E′_{+}.
It is interesting to consider the
marginals {∑_{n}R_{mn}} and {∑_{m}R_{mn}}
of this POVM, which are POVMs, too:

{∑_{n}R_{mn}}
= {γE_{+},
I − γE_{+}}, 


{∑_{m}R_{mn}}
= {(1 − γ)E′_{+}, I − (1 − γ)E′_{+}}.


It turns out that the marginal POVMs can be interpreted as describing nonideal versions of the
measurements of the standard observables
{E_{+},E_{−}} and
{E′_{+},E′_{−}}, respectively.
Note that in the limits γ = 1 and γ = 0 we obtain for the marginal POVMs:
γ = 1: 
{∑_{n}R_{mn}}
= {E_{+}, E_{−}},
{∑_{m}R_{mn}}
= {O, I}, 

γ = 0: 
{∑_{n}R_{mn}} = {O, I},
{∑_{m}R_{mn}}
= {E′_{+}, E′_{−}}.


Hence, as is to be expected because in these limits all photons go one way or the other,
only one standard observable is measured (either {E_{+}, E_{−}} or
{E′_{+}, E′_{−}}), no information being provided by the other observable
({O, I}).
Evidently, the standard formalism is contained in the generalized one as a limiting case.
The example demonstrates that, contrary to what is possible in the
standard formalism, in the
generalized formalism
it is possible to obtain information about different incompatible
standard observables in one single measurement arrangement. The
aspects of complementarity,
exhibited by this experiment, are dealt with in the general
treatment of the Martens inequality.
 Nonideal measurements

Generalizing the examples given above, a generalized
observable {M_{m}} is said to represent a
nonideal measurement of observable {N_{n}}
if a nonideality matrix (λ_{mn}) exists such that
M_{m} =
∑_{n}λ_{mn}N_{n}, λ_{mn}≥
0, ∑_{m} λ_{mn}= 1.
figure 6
The quantities λ_{mn} are
conditional probabilities, relating the `probability distribution
{<M_{m}>} that is actually measured' to the
`probability distribution {<N_{n}>} that
would have been obtained if a measurement of
{N_{n}} had been performed instead of
{M_{m}}'. The nonideality matrix (λ_{mn}) is a socalled
`stochastic matrix'^{75}. It can be seen as a representation of a
transmission channel, {N_{n}} representing the
input channels, {M_{m}} the output channels, and
the nonideality matrix (λ_{mn}) describing crossing of
signals between subchannels (see figure 6).
If there are no crossings, then the stochastic matrix (λ_{mn}) equals the unit matrix
(δ_{mn}). This corresponds to
the situation that the POVM {M_{m}} is
representing an ideal measurement of the POVM
{N_{n}}.

The observable {∑_{n}λ_{mn}N_{n}} can be interpreted as
representing a nonideal way of measuring the observable {N_{n}}.
Note that the quantities λ_{mn} are determined
by the experimental measurement procedure; they are independent of the quantum mechanical state of
the microscopic object. These quantities determine the accuracy observable {N_{n}}
is measured with.
It can also be seen as an application of Bohr's notion of latitude,
observable {N_{n}} in the `measurement procedure represented by POVM
{∑_{n}λ_{mn}N_{n}}' being defined with a certain `latitude determined by the nonideality matrix'.
The `measure of inaccuracy or nonideality POVM {N_{n}} is measured with' is determined by the `deviation of the nonideality
matrix (λ_{mn}) from the unit
matrix (δmn)'. There exist
several quantities that can play the role of `latitude/measure of inaccuracy or nonideality'
(section 7.8 of Publ. 52).
A particularly useful measure is the average row entropy, defined as
J_{(λ)} =
−(1/N)∑_{mm′} λ_{mm′}
ln(λ_{mm′}/∑_{m′′}λ_{mm′′}),
ranging from 0 (if λ_{mm′} = δ_{mm′}; i.e. the ideal case) to ln(N)
(if λ_{mm′} =1/N; i.e. the most
nonideal case, in which the nonideal measurement is completely
uninformative).

Nonideality, in the sense defined
here, is not an absolute notion but a relative one: it compares
`measurement procedures'. If the measurement of
{N_{n}} would yield undisturbed information about
the microscopic object itself, then the nonideality matrix would
describe the disturbance by the measurement. However, in an empiricist interpretation
this would be questionable even if {N_{n}} corresponds
to a `standard observable', since even the latter are not supposed to describe
microscopic reality. Moreover, the relation between {M_{m}} and {N_{n}}
is possible also if {N_{n}} is a POVM.

Other definitions of `nonideal measurement' may be possible, and perhaps useful.
The present definition has the advantage that it constitutes a
welldefined numerical relation of the measured probability
distribution with the one it purports to `measure in a nonideal
way'. Moreover, relations of this kind turn out to be quite abundant.
If the nonideality matrix (λ_{mn}) is invertible, then the
probability distribution {<N_{n}>} can be
calculated from the actually measured one (at least in principle; in
practice invertibility is frustrated by the fact that a
probability distribution is seldom exactly known over the
whole range of its spectrum).

Joint measurement of generalized observables

Commeasurability of (generalized) observables
Two observables represented by POVMs {M_{m}} and
{N_{n}} are `jointly measurable (commeasurable)' if
a measurement procedure exists, represented by a bivariate POVM
{R_{mn}}, such that
M_{m} = ∑_{n}R_{mn}
and
N_{n} = ∑_{m}R_{mn}.
Note that in general compatibility of the operators of
the POVMs is not required for commeasurability (compare the
example of the joint nonideal measurement of incompatible polarization observables).
However, it can be proven that if
{M_{m}} and {N_{n}} are PVMs,
then, in agreement with the standard formalism, a
`bivariate POVM {R_{mn}} yielding these PVMs as
marginals' only exists if the PVMs are compatible. Evidently, the
impossibility of jointly measuring incompatible
observables is restricted to the domain of application of the
standard formalism.

Joint nonideal measurement of (generalized) observables
A weaker kind of `joint measurability' is defined as follows:
Two observables, represented by POVMs
{M_{m}} and {N_{n}}, are `jointly
nonideally measurable' if a measurement procedure exists,
represented by a bivariate POVM {R_{mn}}, such
that its marginals {∑_{n}R_{mn}}
and {∑_{m}R_{mn}}
represent nonideal measurements of {M_{m}} and
{N_{n}}, respectively. Thus,
∑_{n}R_{mn} = ∑_{m′}
λ_{mm′}M_{m′}, λ_{mm′} ≥ 0,
∑_{m} λ_{mm′}= 1,
∑_{m}R_{mn} = ∑_{n′} μ_{nn′}N_{n′},
μ_{nn′} ≥ 0, ∑_{n} μ_{nn′}= 1.
The two generalized observables
{∑_{m′} λ_{mm′}M_{m′}}
and {∑_{n′} μ_{nn′} N_{n′}} can be seen as being defined in the sense of
Bohr's strong correspondence principle either
as autonomous (generalized) observables, or as defining the observables {M_{m}} and
{N_{n}} with a certain latitude determined by the nonideality matrices
(λ_{mm′}) and (μ_{nn′}), respectively.
It is directly seen that the
joint nonideal measurement of incompatible polarization observables
satisfies this general definition for the special case of `standard observables', `nonideality matrices'
being given by

(λ_{mm′}) = 

(μ_{nn′}) = 
 

figure 7
The two nonideality matrices (λ_{mm′}) and (μ_{nn′}),
given above, exhibit a certain `complementarity', in the sense that if one matrix equals
the unit matrix (for either γ = 1 or
0), then the other one corresponds to a particularly bad
transmission channel (cf. figure 6),
yielding no information at all about the initial probability distribution.
Taking as nonideality measures of these matrices the average row entropies we find

J_{(λ)} = ½[(2 − γ)ln(2 − γ) −
(1 − γ)ln(1 − γ)],  

J_{(μ)} = ½[(1 + γ)ln(1 + γ) −
γ ln(γ)].  
In figure 7 a parametric plot of these quantities is given (curved line, parameter γ).
It is seen that if the measurement is ideal with respect to one PVM, then it is completely
uninformative with respect to the other (incompatible) one (and vice versa).
In the following this is demonstrated to be an example of a generic type of measurements, shedding new light on
the implementation of the notion of `complementarity' within the mathematical formalism of quantum mechanics.

Complementarity in a joint nonideal measurement
of incompatible standard observables^{93}

The Martens inequality
Let us now consider an arbitrary `joint nonideal measurement of two standard observables
{E_{m}} and {F_{n}}'.
Hence we have a bivariate POVM {R_{mn}} satisfying
∑_{n}R_{mn} = ∑_{m′ }λ_{mm′} E_{m′},
λ_{mm′} ≥ 0, ∑_{m} λ_{mm′}= 1,
E_{m}^{2} = E_{m},
∑_{m}R_{mn} = ∑_{n′ } μ_{nn′}F_{n′},
μ_{nn′} ≥ 0, ∑_{n} μ_{nn′}= 1, F_{n}^{2} = F_{n}.
Using the `average row entropies' J_{(λ)}
and J_{(μ)}
Martens (Publs 25, 26) has derived an
inequality for the general case^{76}:
J_{(λ)} + J_{(μ)} ≥
−ln(max_{m,n}<a_{m}b_{n}>^{2}),
a_{m}> and b_{n}> the eigenvectors of the two
incompatible standard observables {E_{m}} and {F_{n}}, respectively.
I shall refer to this inequality (and its generalizations) as the Martens inequality.
 Proof of the `Martens inequality'
Since the derivation of the Martens inequality is not easily accessible in the literature, it is
presented here:
From the equalities
∑_{n}R_{mn} = ∑_{m′}λ_{mm′}E_{m′}
and E_{m} a_{m′}>= δ_{mm′}a_{m′}> it follows that
λ_{mm′} = <a_{m′} ∑_{n}R_{mn} a_{m′}>.
We now make use of the definition of von Neumann entropy to express the average row entropy
J_{(λ)} according to
J_{(λ)} =
(1/N)∑_{m} (Tr ∑_{n} R_{mn})
H_{{Em}}(∑_{n} R_{mn}/ Tr ∑_{n} R_{mn}).
In an analogous way we have
J_{(μ)} = (1/N)∑_{n} (Tr ∑_{m} R_{mn})
H_{{Fn}}(∑_{m} R_{mn}/Tr ∑_{m} R_{mn}).
It is now important to note that the `arguments of the von Neumann entropies', although not being related
to `preparation' but to `measurement', both have the mathematical properties of density operators.
This enables to apply the following wellknow property of von Neumann entropy:
H_{{Em}}(ρ) ≥
∑_{k} r_{k} H_{{Em}}(ρ_{k}),
ρ = ∑_{k} r_{k} ρ_{k}, r_{k} ≥ 0,
∑_{k} r_{k} = 1,
valid if the ensemble described by ρ
consists of subensembles described by the density operators ρ_{k}.
Application of this inequality with
ρ = ∑_{k} (R_{mk}/Tr ∑_{k′} R_{mk′}),
r_{k} = Tr R_{mk}/Tr ∑_{k′} R_{mk′},
ρ_{k} = R_{mk}/Tr R_{mk}
yields
H_{{Em}}(∑_{k}R_{mk}/
Tr ∑_{k′} R_{mk′}) ≥

∑_{k} (Tr R_{mk}/
Tr ∑_{k′} R_{mk′})
H_{{Em}}(R_{mk}/ Tr R_{mk}).


From this it directly follows that
J_{(λ)} ≥ (1/N)∑_{mn}
(Tr R_{mn}) H_{{Em}}(R_{mn}/ Tr R_{mn}).
In a completely analogous way we find
J_{(μ)} ≥ (1/N)∑_{mn}
(Tr R_{mn}) H_{{Fm}}(R_{mn}/ Tr R_{mn}).
It then directly follows that
J_{(λ)} + J_{(μ)} ≥

(1/N)∑_{mn}
(Tr R_{mn}) {H_{{Em}}(R_{mn}/Tr R_{mn}) +
H_{{Fm}}(R_{mn}/Tr R_{mn})}.


Finally we use an inequality derived by Maassen and
Uffink^{80}, viz.
H_{{Em}}(ρ) + H_{{Fm}}(ρ)
≥ −ln (max_{mn} Tr E_{m}F_{n}),
valid for an arbitrary density operator ρ,
once again realizing that the operators R_{mn}/Tr R_{mn} are mathematically equivalent to
density operators.
For the nondegenerate, Ndimensional case we have
restricted ourselves to, the Martens inequality then follows from the condition
that for a bivariate POVM we have ∑_{mn}R_{mn} = I,
and Tr I = N.
 `Martens inequality' versus `entropic uncertainty relation'
The Martens inequality, with
J_{(λ)} the average row entropy of the
nonideality matrix (λ_{mm′}), should be distinguished from the
entropic uncertainty relation
looking deceivingly similar. Yet, they are completely different, the entropic quantity
H_{{Em}}(ρ) having a meaning that is very
different from J_{(λ)}, even though referring to the same standard observable
{E_{m}}.
It is important to note that the quantities involved in the
`Martens inequality' do not depend on the `initial density
operator ρ of the microscopic object', but only on `parameters of
a measurement procedure'. The `Martens inequality' unambiguously is not a property of the state
(density operator) but it is a `property of a generalized observable (representing a
joint nonideal measurement procedure of incompatible standard observables)'. It is a property of
the `measurement procedure' only.
It has the meaning of an inaccuracy relation satisfied by the
`inaccuracies of nonideal measurements of incompatible standard observables that are performed jointly'.
 `Martens inequality' versus
`HeisenbergKennardRobertson inequality'
Note also the similarity of the meanings of the entropic uncertainty
relation and the HeisenbergKennardRobertson inequality,
the standard deviations of the latter inequality too
depending only on the `initial state ρ'.
Both inequalities can be tested in `separate ideal measurements' of the two PVMs,
and, therefore, they are `not properties of a joint measurement' (more accurately:
they are `properties of two different joint measurements', each of which usually being interpreted as
a `measurement of a single standard observable', compare).
This draws a sharp line distinguishing, on the one hand, the `Martens inequality' from, on the other hand,
the `HeisenbergKennardRobertson inequality' and the `entropic uncertainty relation',
thus clarifying a basic confusion in the notion of complementarity
(cf. Publ. 48).
 The `Martens inequality' and `complementarity'
This distinction enables to strengthen
Ballentine's criticism of the `HeisenbergKennardRobertson inequality' by distinguishing
two different notions of `complementarity', viz.
`complementarity of preparation' and 'complementarity of measurement'. As stressed by Ballentine,
the `HeisenbergKennardRobertson inequality' has unjustifiedly been interpreted as a property of a `joint
nonideal measurement of incompatible standard observables': it, on the contrary, is a `property of
a (preparation of a) state of the microscopic object'.
Actually the Martens inequality
is quantifying `complementarity in a joint nonideal measurement
of incompatible standard observables' in the way exemplified by figure 7.
The essential feature, generally valid for any `joint nonideal measurement of incompatible standard observables',
is that there is a forbidden region (shaded area), defined by the `Martens inequality', and
signifying that the nonideality measures J_{(λ)} and
J_{(μ)} cannot both approach zero for
any value of the `parameters specifying the measurement arrangement'.
The Martens inequality is a perfect representation
of the idea of `complementarity' as embodied in the idea of `mutual exclusiveness
of measurement arrangements of incompatible standard observables'
implied by Heisenberg's disturbance theory of measurement,
disturbance of the measurement results of one standard observable ({E_{m}})
being induced by changing the measurement arrangement
so as to also obtain information on an incompatible standard observable ({F_{n}}), and
vice versa.
On the other hand, inequalities like the `HeisenbergKennardRobertson inequality' and the `entropic uncertainty relation'
are best considered as restrictions on our ability to prepare `quantum mechanical states having arbitrarily small dispersions'.
 Remarks on `Martens inequality and complementarity'

Note that here `Heisenberg disturbance' is not taken in the
`preparative (predictive) sense as conceived
by Heisenberg himself' (i.e.
referring to the future), but in the `determinative (retrodictive) sense
of the empiricist interpretation' (i.e. referring to the initial state).
As we can learn from the
general theory of quantum measurement,
a `nonideality measure' has both an ontological and an epistemological meaning
(compare):
it reflects how a `pointer position
of a measuring instrument' is (ontologically) changed when changing the measurement arrangement
(for instance, by introducing a semitransparent mirror having transmissivity γ),
at the same time (epistemologically) changing the `information obtained
on the initial state of the microscopic object'.

That the `HeisenbergKennardRobertson inequality' has been widely (though unjustifiedly) accepted as a property of a
`joint measurement of incompatible standard observables' can be understood as a consequence of
a number of circumstances:

i) in 1925 quantum mechanics did not come into being equipped with a fullblown
generalized formalism,
nor even with the `welldeveloped standard formalism
nowadays presented in textbooks', but it has grown from discussions on the
`restricted
set of measurements carried out experimentally at that time' as well as on `thought experiments',
in which different notions of correspondence have played important parts, and in which
classical reasonings (in terms of both position and momentum) were applied (be it
that allowance was made for deviation from classical mechanics into the direction of a certain
latitude or uncertainty); as a result not all conclusions then obtained
are still generally valid today;

ii) the inequalities of the `uncertainty principle',
derived in rather informal ways from the `thought experiments',
appeared to be represented by the inequality derived by Heisenberg from the mathematical formalism of quantum mechanics,
thus causing physicists to jump to the conclusion that the inequalities should be identical;

iii) Heisenberg did not apply his inequality to the `initial state of the microscopic object'
(as is usually done in textbooks), but to its `postmeasurement state'
(compare), thus implementing the
possibility of `measurement disturbance' as well as of `an
influence the choice of a particular measurement arrangement may have';

iv) only as late as 1970 it was realized by Ballentine that this is inconsistent with the way the
`HeisenbergKennardRobertson inequality'
is usually presented in textbooks, viz. as a `property of the premeasurement state',
not depending on any parameter of the measurement arrangement for a measurement to be performed later;

v) it was not noticed that there actually are at stake
two different kinds of complementarity, viz.
`complementarity of state preparation' and 'complementarity of measurement', the
`HeisenbergKennardRobertson inequality' representing a `property of the postmeasurement
state of the microscopic object' rather than a `property of the measurement arrangement';

vi) the difference between these kinds of complementarity was not obvious because
it is not unreasonable that the postmeasurement
state be codetermined by the parameters of the measurement arrangement; however, before Ballentine's observation
it was overlooked that the standard deviations involved in the `HeisenbergKennardRobertson inequality'
are not found by means of a `joint nonideal measurement of incompatible observables', but through separate ideal measurements
of these observables;

vii) only after the `Martens inequality' had become available, was it obvious that
the `HeisenbergKennardRobertson inequality'
is only referring to the two extremes of the parameter range determining the generalized observables really
describing a `joint nonideal measurement'; the latter measurements, although already discussed as `thought experiments',
could not be formally treated as long as the `generalized mathematical formalism' had not been developed.
Standing on the shoulders of many giants we now can do better.

