THE PROBABILISTIC ORIGIN OF BELL'S INEQUALITY
Atominstitut der Osterreichischen Universitdten
The concept of local realism entails certain restrictions concerning the possible occurrence
of correlated events. Although these restrictions are inherent in classical physics they have
never been noticed until Bell has shown in 1964 that in general correlations in quantum
mechanics can not be interpreted in a classical way. We demonstrate how a local realistic
way of thinking about measurement results necessarily leads to limitations with regard to the
possible appearance of correlated events. These limitations, which are equivalent to Bell's
inequality can be easily formulated as an immediate consequence of our discussion.
Local realism denotes a certain way of thinking about the origin of experimental results which
can be specified by the concepts of locality and reality as defined in the EPR paper . For a
system consisting of two spatially separated parts (e.g. in a singlet state) locality means that,
"since at the time of measurement the two systems no longer interact, no real change can take
place in the second system in consequence of anything that may be done to the first system." As a
criterion for reality EPR give a reasonable proposition which reads as follows: "If, without in any
way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the
value of a physical quantity, then there exists an element of physical reality corresponding to this
Generalizing the EPR reality criterion in such a way that with regard to the singlet state
(Bohm's version) the result of any spin-measurement has to be considered as predetermined, Bell
has shown in 1964  that local realism as defined above is "incompatible with the statistical
predictions of quantum mechanics."
By applying the concepts of local realism to a three particle system, D.Greenberger,M.Horne
and A.Zeilinger  (see also ) have shown that in this case a clear cut contradiction can.be
deduced on the level of perfect correlations. Although this approach provides the most expressive
demonstration of the incompatibility of local realism with quantum mechanics it does not work
for two-particle systems. There the incompatibility arises just on the statistical level. Hence
an intuitive understanding of the contradiction between the idea of local realism and quantum
mechanics is difficult, if one is not aware of the origin of this contradiction.
One attempt in order to demonstrate the basic idea of Bell's proof in a more expressive way
has been made by E.P.Wigner , who derived a specific form of Bell's inequality by using only
simple settheoretical arguments. Recently another attempt has been made by Lucien Hardy 
who showed that the probability for a contradiction of the GHZ-kind can be greater than zero for
a two-particle system.
Nevertheless none of these approaches has provided a general argument based on the concepts
of locality and reality which explicitly demonstrates the origin of the discrepancy between local
realism and quantum mechanics. Thus our aim is to show the essential restrictions of local realism
by discussing the results of a general two-particle experiment using the assumptions of locality and
reality. As evident consequences the conditions for the fulfilment of these restrictions are equivalent
to Bell's inequality.
2 Predictions based on the knowledge of correlations
We consider the following experimental setup (cf. Fig. 1): A source emits the two parts of a
system in opposite directions. Measurements with the possible results +1 and —1 are performed
on each part by two observers A and B. Each of them may select one of two possible values of a
measurement parameter a and ft respectively. As a consequence four different experiments can
be made, corresponding to the four different combinations of the measurement parameters at\, a 2
and ft, ft.
FIG. 1. The experimental setup consists of a source which emits the two parts of
a system in opposite directions. Measurements with the possible results +1 and —1
are performed on each part by two observers A and B. Both A and B have a knob
which selects one of two possible values of a measurement parameter. In such a way 4
different experiments can be made (cf. table I).
We assume that all four experiments have been made. The results are listed in table I.
TABLE I. The correlations of the results of the 4 experiments are listed. They might
have been observed in actual experiments or calculated by quantum mechanics. P, is
the probability for different results in experiment i. In consequence the probability for
equal results P^ is 1 — Pf .
In experiment 1 with the parameters adjusted to aj and ft the observers A and B got different
results with probability Pf . In experiments 2, 3 and 4 the probabilities for different results are
Pj , P3 and Pf , respectively.
Knowing the correlations which have to be expected either from previous experiments or
quantum mechanical calculations, A is able to predict the possible results of B and vice versa.
This means that after A has for example performed a series of n measurements with the setting
c*i he can infer all possible results of B on the basis of his experimental data and the knowledge
of the correlations in experiments 1 and 2 by the following reasoning.
B's predictions by
changing nj signs
of his results
n^ = P,"
n 2 /n = P 2 "
A's predictions by
changing n 2 signs
of his results
B's predictions by
changing n 3 signs
of his results
n 3 /n = P 3 '
n./n = P.*
A's predictions by
changing n t signs
of his results
Predicted results of experiment 4
FIG. 2. A procedure is shown by which observer A (B) after having performed
a series of n measurements is able to predict the results of observer B (A) for the
two alternative settings of the measurement parameter /? (a). The actually measured
results of observers A and B are listed in the two boxes. By changing a corresponding
number of signs (n, = Pf • n for experiment i) of the measured results the predictions
are in agreement with the calculated or previously observed correlations listed in table
I. Nevertheless it turns out immediately that the predicted results of experiment 4
are consistent with the actual correlations of experiment 4 (cf. table I) only if the
inequality n, + n 3 + n 3 > n 4 (equivalent to Bell's inequality) is fulfilled.
If B selects the parameter ft (experiment 1) the probability for different results has to be Pf (cf.
table I). For n -► oo this means that n x = Pf • n results have to be different. By reversing ni
signs of his results A arrives at a series of possible results of B which correspond to the known
correlations. Because there are wl ,^"l Wl )i different ways of changing n a signs of n numbers A ends
up with a list of w ,,"l n y different predictions for the possible outcomes of B's measurements. In
the same way A is able to infer the possible results B could get, if B selects ft (experiment 2) by
changing n 2 = P* • n signs of his results. In Figure 2 the results of A are listed in the box in row
A a , . The predictions he derives from these data are symbolized by vertical lines in rows Bp t and
B/jj . Each line corresponds to one way of changing ni and n 2 signs, respectively. By this means
the predicted results correspond to the known correlations.
Now let's assume that observer B actually selects the parameter ft and performs a series of n
measurements. In figure 2 his results are listed in the box in row B^ . Of course they correspond
to one of the predictions by A.
Not knowing what A has done observer B himself predicts all possible results A could get if
he selects the parameter a t (experiment 1) or a 2 (experiment 3) (cf. table I) by considering all
possible ways of changing ni or n 3 signs of his results. Again the actual results of A correspond
to one of the predictions by B as it is shown in figure 2.
3 Bell's inequality
In the previous section we have shown how it is possible for A to predict all results B could
obtain and vice versa. In the following we are going to apply the locality assumption that "no real
change can take place in the second system in consequence of anything that may be done to the
first system" . Moreover we assume in agreement with realistic approaches that "unperformed
experiments have results"  or in other words that predicted results have the status of potential
If we now ask what A could have measured if he had selected the parameter a 2 (experiment 3)
instead of ai (experiment 1), we just have to take into consideration the predictions by observer
B to find the answer. Based on his actual results and the known correlation in experiment 3 (cf.
table I) observer B has predicted all results A could have got if he had chosen a 2 (cf. figure 2). As
a consequence of the locality assumption the results of B, which are the basis of his predictions,
are independent of anything that may be done by A. Because of this independence all of B's
predictions have the status of potential reality, which means that if A had selected a 2 he actually
would have got one of the results predicted by B.
In the same way we find the answer to the question what B could have measured if he had
selected the parameter ft (experiment 2) instead of ft (experiment 1) by considering the predic-
tions by observer A (cf. figure 2). It is important to notice that because of the locality assumption
we can make independent use of the predictions by B and A to answer the question what A and
B could have measured if they had selected a 2 and ft, respectively.
Since we know all possible results A and B could have got if they had chosen a 2 and ft,
respectively (experiment 4), we may now try to find out if these results are consistent with the
known correlation of experiment 4 Pf (cf. table I). For this purpose we take one of the results B
could have got if he had selected ft (row B& in figure 2), change n 2 signs to get the actual results
of A (box in row A 0l ), change n^ signs to get the actual results of B (box in B^) and change n 3
signs to end up with one of the results A could have got if he had selected a 2 (row A Q2 in figure
2). Of course we could also do the same thing the other way round but anyway the results A
could have got are connected to the results B could have got by the following transformation rule
(cf. figure 2): Reverse n 2 signs in the first, nj signs in the second and n 3 signs in the last step
or the other way round. Doing this the maximum number of signs one can change is simply
n x + n a + n 3 . This result of local realistic reasoning is consistent with the observed correlation
in experiment 4 if and only if n, + n a + n 3 > n 4 = Pj • n . If this condition is violated, then
not a single pair of the predicted results of experiment 4 (cf. figure 2) is correlated in agreement
with experience because there is no pair with more than n x + n 2 + n 3 different signs.
It follows immediately that this condition is equivalent to Bell's inequality:
ni + n 2 + n 3 > n 4
t = 1,2,3,4
Pf + P* + ?f > ?f
Pf + P= = l 1 = 1,2,3,4
1 - Pf + 1 - Pj + 1 - Pj > 1 - P^
E, = P= - Pf
i = 1,2,3,4
E! + E 2 + E 3 - E 4 < 2
We get (3) by adding inequalities (1) and (2) and using the definition of the expectation value of
the product of the results E, in experiment t (i = 1,2,3,4).
We have shown that just by discussing the possible results of a general two-particle experiment
in a local realistic way one is directly led to a condition for the consistency between quantum
mechanics and the concept of local realism.
The crucial point in the argumentation is on the one hand the assumption that A's and B's
data are determined locally, which means that A's (B's) results are completely independent of
the measurement parameter selected by B (A). On the other hand by assuming that unperformed
experiments have results A's and B's predictions can be combined in order to get a prediction of
experiment 4 (unperformed). It turns out that this kind of counterfactual reasoning is inconsistent
with the results one obtains by actually making experiment 4.
I would like to thank Prof. Anton Zeilinger, Dr.Johann Summhammer, Dr.Marek Zukowski and
Clemens Ulrich for useful discussions on this and related topics.
This work has been supported by the Austrian Fonds zur Forderung der wissenschaftlichen
Forschung, grant No. P8781-PHY.
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