The Bell inequality in quantum mechanics

The (standard) Bell inequality
figure 10
Although the Bell inequality was first derived for hidden variables
theories is it illuminating to see
under which conditions this can also be done within quantum
mechanics. The Bell inequality has been derived for EPRBell measurements,
like the ones performed by Aspect et al. (a discussion of the conditions allowing such a derivation is given here).
There are four such measurements
involved (see figure 10), in which either standard observable
A_{1} or B_{1} of particle 1 is
measured jointly with standard observable A_{2} or
B_{2} of particle 2. Observables
A_{1} and B_{1} are chosen to be
incompatible: [A_{1},
B_{1}]_{−} ≠
O; so are A_{2} and B_{2}:
[A_{2}, B_{2}]_{−} ≠ O.
Observables of different particles are compatible. If
all observables are dichotomic (i.e. have only two values), their
two values being +1 and −1, then the Bell inequality can be expressed
in terms of the expectation values <A_{1}A_{2}>, etc. of the
correlation observables A_{1}A_{2},
etc. according to
<A_{1}A_{2}> −
<A_{1}B_{2}>
≤ 2 +
<B_{1}A_{2}> +
<B_{1}B_{2}>,
which is also called the ClauserHorneShimonyHolt (CHSH) inequality.
In the Aspect measurements the particles are actually
chosen to be photons. The observables correspond to measurements
of photon polarization in different directions perpendicular to
the direction of motion of the photons. The experiments in an unambiguous way show
experimental violation of the Bell inequality.

The BCHS inequality
A more general inequality than the CHSH inequality is
the BellClauserHorneShimony (BCHS) inequality. This
inequality is expressed in terms of probabilities
rather than expectation values. Therefore, it is
independent of the (eigen)values of the observables (compare). There is no restriction to
dichotomic ones, nor to the values +1 and −1. Denoting the
(eigen)values of observable A_{1} by
a_{1i}, etc., the BCHS inequality is given by
−1 ≤
p(a_{1i},b_{2m}) +
p(b_{1j},a_{2k}) +
p(b_{1j},b_{2m}) −
p(a_{1i},a_{2k}) −
p(b_{1j}) − p(b_{2m})
≤ 0,
in which the univariate probabilities (having a single digit as
argument) are marginals of the bivariate ones. Under the
restrictions, given above, the CHSH inequality can
straightforwardly be derived from the BCHS inequality (e.g.
Publ. 15).
 `Bell inequality' as an expression of `classicality'
The Bell inequality holds true in any classical statistical theory satisfying Kolmogorov's
statistical axioms (e.g. Publ. 15). This is in agreement with
the idea that `violation of the Bell inequality' marks the typically nonclassical features
of quantum mechanics expressed by the `existence of incompatible observables' and
the notion of `entanglement'.
On the other hand, it should be noted that `satisfaction of the Bell
inequality' does not imply that `everything would be classical':
in quantum mechanics the inequality is often satisfied even if
some of the observables do not commute (it, actually, requires
some care to find a set of observables for which the Bell
inequality is not satisfied). By the same token, the Bell inequalities may be
satisfied if the state exhibits a certain measure of entanglement
(e.g. Werner states^{0}).
On the other hand, `violation of the Bell inequality' can be seen as a sign of `nonclassicality'.
 Assumptions allowing derivation of the Bell inequality within quantum mechanics
As is well known, the Bell inequality can be violated by quantum mechanics. Therefore,
next to the quantum mechanical formalism additional assumptions will have to be invoked
in order that the Bell inequality be derivable within quantum mechanics. Let us consider some of these:
 i) Existence of a `quadrivariate probability distribution'
In order to analyze this question I start from a
general theorem, derived independently by Fine and by Rastall^{73}.
Theorem: A sufficient condition for derivability of the BCHS
(and, hence, the Bell) inequality is the existence of a
`quadrivariate probability distribution'^{62}
p(a_{1i},b_{1j},a_{2k},b_{2m})
from which the bivariate probabilities can be derived as marginals.
As a consequence of
Gleason's theorem^{0}
(within the domain of quantum mechanics specifying a `probability distribution' as a `linear functional of the
density operator'), for the `quadrivariate probability distribution' we should assume here that
p(a_{1i},b_{1j},a_{2k},b_{2m}) =
Tr ρ M_{ijkm},
in which M_{ijkm}
are elements of a quadrivariate POVM {M_{ijkm}}.
The importance of this theorem seems to be underestimated, probably because within the
standard formalism the existence of such a
`quadrivariate probability distribution' is impossible if incompatible standard observables are involved
(compare). However, within the generalized formalism
such restraint is unwarranted (compare).
 Criticism and appraisal of
`existence of quadrivariate probability distributions'
The problematic character of `joint probability distributions of incompatible observables' within standard
quantum mechanics is wellknown. For this reason it is not surprising that it has been thought that
such probability distributions do not exist at all, for instance, because
the requirement of `asymptotic stability
of the quadrivariate relative frequencies' might not be satisfied if
incompatible (standard) observables are involved.
If `quadrivariate probability distributions' would not exist, then the above theorem would be inapplicable.
Note, however, that the theorem is valid not only for probabilities, but, more generally, also
for relative frequencies at each single value of N. Hence,
it is sufficient that a `quadrivariate relative frequency' exist for each value of N;
`asymptotic stability' is not required for application of the theorem.
Moreover, `quadrivariate probability distributions
of incompatible observables' do exist if the restriction to the `standard formalism' is
relinquished (compare). Hence, applicability of the theorem
is secured at least within the generalized formalism.

ii) Compatibility of all (standard) observables that are involved
Note that the above theorem is satisfied
if the four standard observables A_{1}, B_{1}, A_{2},
and B_{2} that are involved in the Bell inequality are `all mutually compatible'.
Then a `quadrivariate probability distribution' exists, viz. (compare)
p(a_{1i},b_{1j},a_{2i},b_{2j}) =
Tr ρE_{1i}F_{1j}
E_{2i}F_{2j},
in which {E_{ni}} and {F_{nj}} (n = 1,2) are the (mutually commuting)
spectral representations of the four mutually compatible standard observables
A_{n} and B_{n}, respectively.
For this `quadrivariate probability distribution'
the Bell inequality is satisfied for any density operator
ρ.
It is evident from the
`derivability of the Bell inequality from compatibility of all observables that are involved' that
`incompatibility of observables' is a necessary condition lest the Bell inequality be violated:
there is no violation of the Bell inequality without incompatibility.
Experimental violation of the Bell inequality by
certain quantum mechanical (standard) observables in EPRBell experiments
implies that, at least for these observables, neither the
`possessed values principle' can hold, nor can there exist a
`joint probability distribution of values of these observables'.
This result is hardly surprising, since
incompatibility is wellknown to be at the basis of both
the `invalidity of the possessed values principle', and of the
`nonexistence of joint probability distributions of standard observables'.
 iii) Existence of `quadruples of measurement results'
In order that `quadrivariate relative frequencies' (as well as quadrivariate probability distributions
p(a_{1i},b_{1j},a_{2k},b_{2m})) exist there is one important
requirement that should be satisfied,
viz. the four measurement results a_{1i}, b_{1j}, a_{2k}, and b_{2m}
should be attributable to a common quantum event,
so as to be liable `to be combined into the quadruple
(a_{1i},b_{1j},a_{2k},b_{2m}) as the argument of a relative frequency or probability'.
Since `quadrivariate probability distributions' cannot exist unless such quadruples exist,
we have here a more fundamental possibility to explain the nonexistence of
`quadrivariate probability distributions' in case of the Aspect measurements. Indeed,
how can the existence of such quadruples be justified if the four measurement results
a_{1i}, b_{1j}, a_{2k}, and b_{2m}
are not obtained coincidentally in one single experiment but rather
by means of four different measurements (cf. figure 10), each measurement probing its own quantum events
(e.g. Publ. 41)?
I am aware of three ways in which `existence of quadruples of measurement results'
has been implemented also if incompatible observables are considered:
iiia) application of the `possessed values principle';
iiib) application of `counterfactual definiteness';
iiic) generalized measurements.

iiia) Quadruples and the `possessed values principle'
The Bell inequality can be derived if every element of an
ensemble has a welldefined value for the four observables that
are involved. Hence, the inequality could be derived within quantum mechanics if the
possessed values principle were satisfied (e.g. section 9.4.1 of
Publ. 52). From the experimentally found possibility of
`violating the Bell inequality if incompatible observables are involved'
it follows that the `possessed values principle' cannot be generally valid. Hence, it seems that such derivations,
which are characteristic of an `objectivisticrealist
interpretation of quantum mechanical observables', are as obsolete as is that interpretation
(compare).
On the other hand, in a `contextualisticrealist interpretation of quantum mechanics'
the `nonexistence of quadruples for the standard Aspect measurements' can in a natural way be explained
by the fact that in these experiments different contexts are involved. This points into the direction
of `objectivity as a cause of derivability of the Bell inequality' (compare).
Within quantum mechanics `quadruples of measurement results are to be expected only in a `joint measurement
of four observables'. Within the `standard formalism' these observables must necessarily be mutually compatible.
Hence, within the domain of application of the standard formalism
`derivability of the Bell inequality' is not to be expected if incompatible observables are involved.

iiib) Quadruples and `counterfactual definiteness'
As discussed here `counterfactual definiteness'
does not address the objectivisticrealist idea of `quantum mechanical measurement results
as objective properties of the microscopic object', but it is referring to
`contextualisticrealist' or even `empiricist' ideas in which measurement results are
either `properties of the microscopic object, influenced by the measurement' or
`postmeasurement properties of the measuring instrument'. Hence, the `possessed
values principle' is not thought to be applicable.
Nevertheless, it might be possible to construe
for each of the four EPRBell experiments of figure 10
quadruples (a_{1i},b_{1j},a_{2k},b_{2m})
by adding to the two `measurement results
found in a measurement (e.g. a_{1i}, a_{2k})' two other results (viz. b_{1j}, b_{2m})
`that would have been found if the other two observables had been measured instead of the actually measured
ones'. Once again, it follows from the ensuing derivability of the Bell inequality that such a `counterfactual definiteness'
cannot obtain if the Bell inequality is violated.
Note that the idea of `counterfactual definiteness'
is contradicting the idea of Heisenberg disturbance,
and, therefore would not even be warranted if reality can be attributed to the latter idea.
Since `Heisenberg disturbance' is corroborated by applications of the generalized formalism
(compare), it follows that `counterfactual definiteness'
can now be considered obsolete too.
 iiic) Quadruples and `generalized measurements'
If a measurement has four pointers, and hence is yielding for every quantum event a
`quadruple of measurement results',
then the Bell inequality is satisfied as a result of the theorem. This is the case, for instance,
in a `joint measurement of
four compatible standard observables', for which the existence of the quadrivariate probability distribution
p(a_{1i},b_{1j},a_{2k},b_{2m}) =
Tr ρE_{1i}F_{1j}E_{2k}F_{2m}
is warranted by (standard) quantum mechanics.
However, within the domain of application of the generalized formalism
it is possible to consider joint measurements in which incompatible observables are involved. An example
is given in figure 11. From the `experimentally obtained quadruples' it is possible to derive
relative frequencies that in the asymptotic (large N) limit can be compared with the
quadrivariate probabilities predicted by `generalized quantum mechanics'.
Presently, measurements of the type of figure 5
are performed by an increasing number of groups (compare chapter 8 of
Publ. 52), corroborating the generalized theory.
As a result of the existence of `quadruples of measurement results', in the `generalized Aspect measurement of
figure 11' the Bell inequality must be satisfied.
Note that in the joint probabilities of these experiments `measurement results'
have to be taken in the empiricist sense, implying that here `derivability of
the Bell inequality' is neither a consequence of the `possessed values principle', nor of `counterfactual
definiteness'.
 The assumption of `locality'
 Early developments
The assumption of `locality' as a `condition allowing a derivation of the Bell inequality'
has a long history. The idea of `nonlocality of the quantum world', allegedly following from
`experimental violation of the Bell inequality'
has its origin both within quantum mechanics and within hidden variables (subquantum) theory.

First, within quantum mechanics there was Einstein's tradeoff between
`(Copenhagen) completeness' and `locality',
barring `locality' if quantum mechanics would have to be complete (and vice versa).
Indeed, `nonlocality' seemed to be
a necessary consequence of the Copenhagen way of interpreting in the EPR experiment the state vector
as a `description of an individual object (rather than an ensemble)',
allegedly necessitating (nonlocal) strong von Neumann projection
(also).

Second, within hidden variables theory: although Bell did not endorse the Copenhagen completeness thesis,
and, hence, could have relied on Einstein's assumption as regards
`locality of an ensemble interpretation
of the state vector', he was so impressed by Bohm's nonlocal hidden variables theory
that with Bohm he became firmly convinced that the quantum world must be nonlocal notwithstanding
`experimental corroboration of the principle of local commutativity'.
For these reasons he did not try to prove that
`generic hidden variables theories' would be impossible, but he just set out to prove the `impossibility of
local hidden variables theories'.
Bell's "success" in this endeavour convinced him that the assumption of `locality of hidden variables theory' is crucial
for being able to prove the Bell inequality to be necessarily satisfied. For the majority of physicists,
by the 1960s still endorsing the `Copenhagen interpretation' (but presumably mostly in the realist sense
of quantum mechanics textbooks, compare), the `apparent support of
EPRnonlocality
by Bell's result' has probably been reason enough to accept the idea that `reality behind
the quantum mechanical phenomena must be nonlocal'.
Even today `corroboration of violation of the Bell inequality
by the standard Aspect measurements
(cf. figure 10)' is widely interpreted as `experimental evidence of violation of locality',
although it is realized that this `nonlocality' is unobservable
by means of quantum mechanical measurements. For this reason
it has been conjectured that the `nonlocality believed to obtain in EPRBell experiments'
is another type of `nonlocality' than the one
expressed by parameter dependence. It is
referred to as `hidden nonlocality', `inseparability' (because of its alleged relation to `entanglement',
compare),
or even by fancy names like `passionatadistance (as opposed to actionatadistance)', or `peaceful coexistence'.
It is widely believed that
`there is no violation of the Bell inequality without EPRnonlocality'.
Note, however, that this belief is unwarranted, because,
as argued here and
here, `derivability of the Bell inequality'
may have quite a different reason than `nonlocality' (e.g. Publ. 43).
 Later developments
 Within the standard formalism
It was realized only later that, under certain assumptions (like iiia) or iiib)),
the `Bell inequality' can be derived within quantum mechanics without reliance on subquantum theories,
even though `possible experimental violation of the Bell inequality' makes these assumptions questionable. Indeed,
assumptions iiia) and iiib) are inspired by the
classical paradigm (`quantum mechanical measurement results'
being treated as if they are independent
of the way they are measured), and, for this reason, do not seem to be
very attractive.
On the other hand, the `theorem on the derivability of the Bell inequality from the
existence of a quadrivariate probability distribution' is useful also here. However, within the standard
formalism it is applicable only if all observables that are involved are mutually compatible
(compare).
Yet, we can learn from it something very important, viz. that `violation of the Bell inequality'
necessarily involves incompatible observables (compare).
This implies that, in agreement with the principle of local commutativity,
quantum mechanics does not suggest that `violation of the Bell inequality by
standard Aspect measurements
(in which two compatible observables are measured jointly)' would be caused by any `nonlocal disturbance'.
On the contrary, since
`incompatibility of observables' can only obtain if observables are measured in the same region,
`violation of the Bell inequality' is a local^{13} affair.
Since the `standard formalism of quantum mechanics' is not able to deal with `joint probability distributions of
incompatible observables', this is as far as conclusions can be drawn from the standard formalism.
It is not equipped to deal with `possible causes of violation of the Bell inequality as a result of
joint measurement of incompatible standard observables
(compare)'.
 Within the generalized formalism
The generalized formalism
has more resources in this respect, and is able
to deal with `quadrivariate probability distributions of incompatible observables'
(compare).
By applying the `generalized formalism' to a generalized Aspect measurement
it is possible to study simultaneously in both arms of the interferometer
`mutual (Heisenberg) disturbance in a joint measurement
of incompatible standard observables (as expressed by the Martens inequality)'.
It is then not only seen that
`unobserved observables may have values' (compare), but even how these values change if
the measurement arrangement is changed. It is then important to note that
`Heisenberg disturbance of measurement results in one arm of the interferometer'
is dependent only on `what changes are made to the arrangement in that very arm'.
In the standard Aspect measurements `violation of the Bell inequality'
is not caused by `disturbance of the measurement results in one arm of the interferometer
by a measurement carried out in the other arm', but it can be seen as a consequence
of the `dependence on the measurement arrangement of local
Heisenberg disturbances' in
`joint nonideal measurements of incompatible standard observables'
carried out simultaneously in both arms of the interferometer.
Hence, if prejudices are abandoned that were caused by restricting oneself to the
`standard formalism of quantum mechanics',
`violation of the Bell inequality' may be seen to have a local explanation, viz.
dependence of `Heisenberg disturbance' on the `measurement arrangement that is present
at the position of each measurement'. The quadrivariate probability
measured in the generalized Aspect measurement has been obtained under the influence of
a cumulative effect of `disturbances separately realized
in both arms of the interferometer'.
Only by considering `measurements within the domain of application of the
generalized formalism is it possible to establish `Heisenberg disturbance'
as an alternative explanation of `violation of the Bell inequality', making obsolete explanations
based on an alleged, unobservable `nonlocality'.
 Methodological remark on `nonlocality versus
(local) Heisenberg disturbance'
`Nonlocality of the quantum world' is not based on any experimental evidence,
the latter corroborating
`parameter independence of (bilocally performed) EPRBell experiments'. Contrary to an oftenheard assertion,
`violation of the Bell inequality by the standard Aspect measurements' is not
`experimental evidence of nonlocality', because, contrary to Bell's conviction, locality is not a crucial
assumption in derivations of the Bell inequality (compare, also
Publ. 43).
Often EPRnonlocality is assumed to be a subquantum issue, being different from
`negation of the locality expressed by the quantum mechanical
principle of local commutativity'.
It, then, could be doubted whether a `local explanation on the basis of
(generalized) quantum mechanics' can be seen as an alternative to
Bell's explanation by `nonlocality of the underlying subquantum theory'.
It could be surmised that two different mechanisms are at stake,
both tending to violate the Bell inequality.
However, in my view this idea is unattractive for two reasons:
i) the dubious role played by the assumption of `locality' in derivations of the `Bell inequality'
both within quantum mechanics and within subquantum theory
(compare).
In my view the determining factor allowing `derivation of the Bell inequality' is not `locality'
but rather objectivity (within standard quantum mechanics represented by the
possessed values principle,
within subquantum theory by the assumption of quasiobjectivity);
ii) the methodological principle that the cause of a phenomenon,
described by two different theories at different levels of sophistication,
should be ontologically the same in the two theories. Thus, it does not make sense to explain at the macroscopic level
the rigidity of a billiard ball
by some `macroscopic (nonlocal) influence', whereas at the microscopic level it is explained by
`atomic interactions acting at a microscopic scale',
unless the macroscopic influence can be reduced to the atomic interactions. By the same token
it is sound physical methodology at the levels of description of quantum mechanics and
subquantum theory to try to explain `experimental violation of the Bell inequality' by one and the same cause.
Even though the existence of `nonlocal interactions at the subquantum level, unobservable at the quantum mechanical level
of observation' cannot be logically excluded, I therefore prefer trying to understand at the subquantum level
the `hard evidence provided by the (generalized) quantum mechanical formalism as applied to
joint nonideal measurement of incompatible standard observables'
on the basis of ideas provided by that formalism.
Some ideas pointing into the direction where to look for the abovementioned `reconciliation of quantum mechanics and
subquantum theory' are briefly discussed here. At present it does not
seem useful, however, to invest too much effort in such an endeavour because we do not have any experimental
clue as to the details of such a subquantum theory, which, moreover, would be really fruitful only if its domain
of application would exceed the domain of application of quantum mechanics.
 Nonentanglement (separability)
The `Bell inequality' is satisfied for any nonentangled (or separable) state
described by a density operator of the type
∑_{n} p_{n}ρ_{1n}
ρ_{2n}. This follows easily from the
existence of a quadrivariate probability distribution reproducing the
bivariate probabilities of the standard Aspect measurements for these states, viz.
p(a_{1i},b_{1j},a_{2k},b_{2m}) =
∑_{n} p_{n}p_{1ni}q_{1nj}p_{2nk}q_{2nm},
in which p_{1ni} = Tr_{1}ρ_{1n}E_{1i},
q_{1nj} = Tr_{1}ρ_{1n}F_{1j},
etc.^{79}
Hence, within quantum mechanics
there is no violation of the Bell inequality without entanglement.
Until Werner proved that the Bell inequality is also satisfied by certain
inseparable states^{0}
it could be thought that, in agreement with the dichotomy
between quantum and classical mechanics, the `Bell inequality' would be drawing a sharp line
between entangled and separable states.
It is obvious by now^{78} that the situation is not that simple.
Study of `entanglement'
(important for applications in `quantum computation' and `quantum information') has become an elaborate subdiscipline,
attempts at developing `alternative entanglement criteria (replacing, or at least supplementing,
the Bell inequality)' being an important branch. It is felt that, like `incompatibility' is for `measurement',
`entanglement' is an additional resource for `preparation', offering new possibilities of experimentation
in the abovementioned fields.
 `Nonlocality' versus `entanglement'
Comparing the (valid) relation between the `violation of the Bell inequality' and `entanglement', given above,
with the (dubious) analogous assertion with respect to `nonlocality' may
explain the equally dubious relation often assumed between `entanglement' and `(hidden) nonlocality'
(manifest by the costum of equating `entanglement' with `inseparability', compare).
`Correlations entailing violation of the Bell inequality' are often referred to as `nonlocal correlations'
(thus suggesting that `correlations entailing satisfaction of the Bell inequality' would be `local correlations'),
even though `all correlations obtained in bilocal (EPRBell) measurements'
are nonlocal (even though the principle of local commutativity is satisfied).
That `entanglement' cannot be equated with `hidden nonlocality'
is evident from the fact that, contrary to `hidden nonlocality', `entanglement' has observational consequences
within the domain of quantum mechanics.
Indeed, independently of whether the state is entangled or not entangled,
the quantum state can be experimentally determined by means of
quantum tomography^{0},
in which `bilocal measurements of the Aspect type' are carried out (for the case of two spin1/2 objects this
can even be done within
the `standard formalism'^{77}; for higherdimensional spaces
`bilocal measurements of generalized observables' are necessary for complete
determination, e.g. Publ. 47).
Also from a preparative point of view it seems improbable that `entanglement' has any relation to
`nonlocality'. Indeed, in the Aspect measurements
the entangled state is there already when the particles are close together.
In case they previously were not entangled, entanglement may very well be brought about
by means of local interactions in the process of pair formation.
 The Bell inequality in the
generalized formalism
By generalizing the quantum mechanical
formalism so as to encompass experiments that can be interpreted
as `joint measurements of incompatible observables', it is possible
to derive the Bell inequality also within an empiricist
interpretation.
 A generalized Aspect measurement
An example is the following generalized
Aspect measurement, which is a variation of Aspect's
switching experiment. In this latter experiment the
experimental arrangements of the four polarization observables
were set up simultaneously, be it in different places, so as to
avoid mutual disturbance: by means of a switching mechanism
photon n (n = 1,2) is directed either toward the measurement
arrangement for A_{n} or B_{n}.
Hence, depending on the switches, one of the experiments of figure
10 is performed.
figure 11
In the generalized Aspect measurement considered here the photon switches
are replaced by partly transparent mirrors (see figure 11),
having transparencies γ_{n},
0 ≤ γ_{n} ≤ 1, n = 1,2. So, in a sense the Aspect switching experiment is a
special case of the generalized Aspect measurement, the values of
γ_{n}
being restricted to 0 or 1, and being switched (pseudo)randomly between these values.
In the generalized experiment the values of γ_{n}
are kept stationary. In this experiment there are four
detectors (pointers), viz. D_{1}, D′_{1}, D_{2}, and D′_{2}.
Since it is possible to determine for each
detector and for each individual photon pair whether it has
detected a photon or not, it is possible to construe a
`quadrivariate probability distribution'
p(a_{1i},b_{1j},a_{2k},b_{2m}) of
the responses of the four detectors (each response being, for instance,
+1 or −1 as the answer to the `question whether a photon is detected'
is `yes' or `no'; note, however, that the choice of the (eigen)values is
unimportant: they can just as well be taken as + or −, as was done
in the discussion of the
joint nonideal measurement of incompatible polarization observables).
From the existence of the `quadruples of the measurement results' the Bell
inequality can be derived.

The POVM of the generalized Aspect measurement can
be derived from the POVM of the joint nonideal measurement
of two polarization observables (which, actually, coincides with
the present arrangement as far as measurement of one photon is
concerned, compare figures 5 and
11). Thus we get as a POVM
R_{ijkm} = R_{(1)ij} R_{(2)km},
in which R_{(1)ij} and R_{(2)km} are the POVMs of the measurements of photons 1 and 2,
respectively, given by (for n=1,2):
R_{(n)++} = O, R_{(n)+−} =
γ_{n}E_{(n)+},
R_{(n)−+} = (1 − γ_{n})E′_{(n)+},
R_{(n)−−} = 1 − γ_{n}E_{(n)+}
− (1 − γ_{n})E′_{(n)+}.
By taking the product of the two POVMs of the measurement procedures it is expressed that these
measurement procedures, in agreement with the principle of local commutativity,
do not influence each other.
The `quadrivariate probability distribution'
of the generalized Aspect measurement is
p_{(γ1,γ2)}(a_{1i},b_{1j},a_{2k},b_{2m}) =
Tr ρ R_{(1)ij} R_{(2)km}.
Taking for granted, analogously to what is usually done in the
standard theory, that relative frequencies have a limit for large numbers,
the existence of a `quadrivariate probability distribution' is a consequence of the
existence of `quadruples of measurement results for each
individual photon pair'. Derivability of a Bell inequality from
the existence of such a probability distribution is evidence of
the existence of a classical statistical Kolmogorovian model for
measurement results jointly obtained within a single measurement
setup (compare).
 Relation to the `standard Aspect measurements'
By taking (γ_{1},γ_{2}) =
(1,1), (1,0), (0,1), or (0,0) the generalized Aspect measurement reduces to one of the
four `standard Aspect measurements depicted in figure 10'. Hence, even if only two
pointers are present in each of these latter experiments, `quadrivariate probability distributions' do exist.
`Quadruples of measurement results' can be seen to exist also in the limiting cases. Indeed, we find
γ_{n} = 1: 
R_{(n)++} = R_{(n)−+} = O,
R_{(n)+−} = E_{(n)+}, R_{(n)−−} = E_{(n)−},


γ_{n} = 0: 
R_{(n)++} = R_{(n)+−} = O,
R_{(n)−+} = E′_{(n)+}, R_{(n)−−} = E′_{(n)−}.


Hence, to the `observables that are not measured
because there does not come any photon in their direction'
should be attributed the value `no photon detected' (note that it is conventional whether this value
is represented either by 0, −1, or by any other digit).
This implies that in each of the four
`standard Aspect measurements' to all of the four standard observables a welldefined measurement result
can be attributed:^{74} "unmeasured" observables may have values!
It is easily seen that each of the `standard Aspect measurements'
has its own `quadrivariate probability distribution
p_{(γ1,γ2)}(a_{1i},b_{1j},a_{2k},b_{2m})'.
In agreement with the existence of quadruples of measurement results
the `Bell inequality' is satisfied by the `generalized Aspect measurement', and, hence, separately by each
of the four `standard Aspect measurements' (in which the missing measurement results are obtained
according to prescription given above).
 `Bell inequality' and `Heisenberg disturbance'
In order to understand why the four `standard Aspect measurements' do not satisfy the `Bell inequality'
if their directly observed results are combined, it is useful to realize that the four `quadrivariate
probability distributions',
p_{(1,1)}(a_{1i},b_{1j},a_{2k},b_{2m}),
p_{(1,0)}(a_{1i},b_{1j},a_{2k},b_{2m}),
p_{(0,1)}(a_{1i},b_{1j},a_{2k},b_{2m}),
p_{(0,0)}(a_{1i},b_{1j},a_{2k},b_{2m}),
are different from each other. This difference can be understood on the basis of
Heisenberg's disturbance theory of measurement
(taken in a `determinative sense' rather than in `Heisenberg's preparative sense'),
to the effect that in each arm of the interferometer `Heisenberg disturbance' obtains
in the way discussed here, independently of what happens in the other arm
of the interferometer.
It is easily verified that the `mutual disturbance of observables A_{1} and
B_{1}' is different for different values of γ_{1}
(and analogously for photon 2). This is expressed by the Martens inequality,
applied independently in each arm of the interferometer.

In order to derive a Bell inequality for the combined `standard Aspect measurements' it would be necessary to
have one single `quadrivariate probability distribution'
from which the bivariate ones of the four measurements could be derived as marginals.
However, if such a `quadrivariate probability distribution' would exist, the Bell inequalities could not be violated.
Hence, experimental evidence is in disagreement with the
existence of this special `quadrivariate probability distribution'.
The same holds true for the existence of `quadruples of
measurement results applicable in all of the four experiments', which also would imply satisfaction
of the `Bell inequality' by the combined results of the four `standard Aspect measurements'.
Due to `Heisenberg disturbance in a joint
nonideal measurement of incompatible observables' the existence of
such quantities is not to be expected if incompatible
observables are involved. Hence, from the point of view of
quantum mechanics we do not have any reason to expect
that the `standard Aspect measurements' should satisfy the `Bell
inequality' if their measurement results
are combined in one single expression (like e.g. the CHSH inequality).
On the contrary, the different ways `Heisenberg disturbance' is
affecting the `bivariate probabilities in an arm of the interferometer
when either A_{n} or B_{n} (n = 1 or 2)
is measured ideally', makes it utterly plausible that, even if it is assumed
that it is possible to reproduce the same physical initial condition
in each of the four `standard Aspect measurements' (compare Publ. 41)
this does not entail the existence of `unique quadruples of measurement results
that could be used as the argument of a quadrivariate probability distribution',
and, consequently, could warrant `satisfaction of the Bell inequality'.

`Violation of the Bell inequality' is a
consequence of the incompatibility of certain of the
observables. Therefore it can be accounted for by Heisenberg
disturbances taking place locally in each of the arms of
the interferometer. We do not have any reason to assume that
there is any influence of the `measurement in one arm of the
interferometer' on the `measurement results obtained in the other
arm'. In this respect there is no difference between the `standard Aspect measurements'
and the `generalized one', in all cases the POVM of the Aspect measurement being a
direct product of the POVMs of the measurements performed in each of the arms.
This implies that the measurement results
for photons 1 and 2 are statistically independent if the
initial states of the photons are uncorrelated (i.e. if ρ_{12} =
ρ_{1}ρ_{2}).
Any correlation between the measurement results of the two photons must be a
consequence of a preparation of the twophoton object,
such that ρ_{12} ≠ ρ_{1}ρ_{2}.
No correlation, either local or "nonlocal", is established
by the measurement procedure.

Bell inequality and interpretations of quantum mechanics
Which significance is attributed to `(violation of) the Bell inequality' is dependent on which
particular interpretation of the quantum mechanical formalism which is being entertained. Barring the
objectivisticrealist
interpretation of quantum mechanical observables (which is ruled
out because it would entail the Bell inequality to be satisfied
for all quantum mechanical observables), we must consider the
contextualisticrealist and the
empiricist interpretations.

`Bell inequality' within a `contextualisticrealist interpretation'
Contrary to the `empiricist interpretation' the `contextualisticrealist
interpretation' allows that in the EPR experiment of figure 8
a value is attributed to an observable of particle 2, even when that particle is not
interacting with a measuring instrument. `Nonlocality' is suggested
by Bohr's application of the correspondence principle, implying that the
contextual meaning of a quantum mechanical observable of
particle 2 must be accounted for by referring to the measurement
arrangement for particle 1. By overlooking the
fundamental difference between EPR and EPRBell experiments
this idea of `nonlocal contextuality' has proliferated to (allegedly) apply to EPRBell experiments too.
However, there is no reason to believe that this makes sense.
Even if the `distant measurement arrangement of particle 1' would
contribute to the `context of particle 2', then the contribution of
the `measurement arrangement for particle 2 itself' would certainly
outweigh this by far. Because of the experimental corroboration of
`parameter independence' it seems reasonable to suppose that the
influence of the far measuring instrument is negligible. Then,
in EPRBell experiments the two particles each have
their own context. This `local contextuality'
can be implemented on the basis of Heisenberg's disturbance theory of measurement,
by taking into account a local influence of the measuring
instrument for observable A_{1} on the
incompatible observable B_{1}, and vice versa (in an EPRBell experiment
analogously for A_{2} and B_{2}).
Heisenberg disturbance can explain the nonexistence of `quadruples
(a_{1i},b_{1j},a_{2k},b_{2m}) which are valid
in all of the four standard Aspect measurements', the nonexistence of similar
`quadrivariate probability distributions', and the `inapplicability of
counterfactual definiteness'.
EPRBell experiments do not seem to
frustrate a `local contextualisticrealist interpretation
of the quantum mechanical formalism'. However, this latter
interpretation is hard to maintain in the face of the preparative features of
EPR experiments (as opposed to EPRBell experiments). This makes the `empiricist interpretation'
preferable to the `contextualisticrealist one'.

`Bell inequality' within the `empiricist interpretation'
In the `empiricist interpretation' there
is no danger of confusing EPR and EPRBell experiments. The `Bell
inequality' could only be derived if it were possible to
dispose of either `quadruples of
measurement results
(a_{1i},b_{1j},a_{2k},b_{2m})', or
a `joint probability distribution of the four observables', valid in
all of the four standard Aspect experiments. If incompatible observables are involved,
then, as a result of `mutual exclusiveness of measurement arrangements', there is no way to construe either of these
within the `standard formalism of quantum mechanics'.
The Bell inequality simply is not derivable within the standard formalism.
However, the `generalized formalism of quantum mechanics',
being a direct consequence
of the empiricist idea to associate a quantum mechanical
observable with the `pointer positions of a measuring instrument',
has more resources in this respect. As seen both from the general theory
and from the example of the generalized Aspect measurement,
`joint probability distributions of incompatible observables' turn out to be viable concepts,
describing measurement results of real measurements, including the standard Aspect ones.
The possibilities provided by the `extension of the domain of quantum mechanics offered by
the generalized formalism' to apply the `Bell inequality' to `measurements of the Aspect type'
give additional insight into the mechanisms at work in such measurements. It turns out that,
far from having to introduce `spooky nonlocality', the experiments can be understood on the
basis of good old `Heisenberg disturbance' (generalized from a contextualisticrealist
to an empiricist sense).

