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N95- 13969 



THE PROBABILISTIC ORIGIN OF BELL'S INEQUALITY 

Gunther Krenn 

Atominstitut der Osterreichischen Universitdten 

Schiittelstrafie 115 

A-1020 Wien 



Abstract 

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. 

1 Introduction 

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 [1]. 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 
physical quantity." 

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 [2] 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 [3] (see also [4]) 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 [5], who derived a specific form of Bell's inequality by using only 



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simple settheoretical arguments. Recently another attempt has been made by Lucien Hardy [6] 
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. 




Source 




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 . 



Experiment 


A 


B 


Correlation 


1 


Ot\ 


ft 


pf 


2 


Otl 


ft 


n 


3 


a 2 


ft 


n 


4 


a 2 


ft 


p? 



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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. 



l <Xi 



B 



P 



<*2 



B 



B'sdata 



0; 



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 



>u 



B's predictions by 
changing n 3 signs 
of his results 



n 3 /n = P 3 ' 



r^ 



n./n = P.* 
~ ^ 

A's predictions by 
changing n t signs 
of his results 



A's data 



1 



• 1 

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. 



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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" [1]. Moreover we assume in agreement with realistic approaches that "unperformed 
experiments have results" [7] or in other words that predicted results have the status of potential 
reality. 

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 



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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 



n 



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^ 



(1) 



(2) 



E, = P= - Pf 



i = 1,2,3,4 



E! + E 2 + E 3 - E 4 < 2 



(3) 

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). 

4 Discussion 

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. 



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5 Acknowledgments 

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. 

References 

[1] A.Einstein,B.Podolsky, and N. Rosen," Can Quantum- Mechanical Description of Physical Re- 
ality Be Considered Complete?" Phys.Rev.47,777-780(1935). 

[2] John S.Bell," On the Einstein Podolsky Rosen Paradox" Physics 1,195-200(1964) 

[3] D.M.Greenberger,M.Horne,and A.Zeilinger," Going beyond Bell's theorem", in Bell's The- 
orem,Quantum Theory,and Concepts of the Universe,edited by M.Kafatos (Kluwer Aca- 
demic,Dordrecht,The Netherlands, 1989) ,pp.73-76 

[4] D.M.Greenberger,M.Horne,A.Shimony,and A.Zeilinger, "Bell's theorem without inequalities" ,- 
Am. J.Phys.58,1 131-1 143(1990) 

[5] Eugen P.Wigner,"On Hidden Variables and Quantum Mechanical Probabilities" Am.J.Phys. 
38,1005-1009(1970) 

[6] Lucien Hardy,"A New Way to Obtain Bell Inequalities" Phys.Lett. A 161, 21-25 (1991) 
[7] Asher Peres, "Unperformed experiments have no results" Am.J.Phys. 46, 745-747 (1978) 



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