Bells Inequality

The progress undergone in recent years toward a solution of the problem raised by Einstein 1927 has become possible because of two developments (1) A few experimental techniques, in particular, the methods for measuring very short times with good accuracy, have permitted in recent years the execution of several experiments which in their essence are practical versions of the thought experiment proposed and discussed in 1935 by Einstein, Podolski, and Rosen.39 (2) A procedure of analysis of their results has been made possible by the work of Bell40 who has derived in the frame of local realism a relation (the Bell inequality) obeyed by local realistic theories but violated by quantum mechanics. 

Although not all the results of the experiments carried on until now are consistent with one another, most of them not only violate the Bell inequality but are also in good agreement with the prediction of quantum mechanics. The experimental errors are still rather large but in a few years this matter will be definitely settled by the results of further, more accurate experiments. It is rewarding that the final word even in this philosophical debate started in 1927, when quantum mechanics was in its infancy, will be definitely settled by accurate results of experiments designed and carried on with the most refined techniques of half a century later. 

The Bell states have a unique feature that all the information on polarization properties is completely contained in the (polarization-) correlations between the separate photons, while the individual particle does not have any polarization prior to measurement. In other words, all the information is distributed among two particles, and none of the individual systems carries any information. This is the essence of entanglement. At the same time, these (polarization-) correlations are stronger than allowed classically, since they violate bounds imposed by local realistic theories via the Bell-inequality [Bell 1964] or they lead to a maximal contradiction between such theories and quantum mechanics as signified by the Greenberger-Horne-Zeilinger theorem... 

Experiments with photons showed that the Bell inequalities are not satisfied. This means that either there is instantaneous communication even between extremely distant particles entangled stated ), or that the particles do not have some definite properties before the measurement is performed (more details about this are given at the end of this chapter). 

A Test fora Common Sense The Bell Inequality... 

Fig. 1.15. Bell inequalities. A bar launching gun adopts stochastic positions (of equal probability) when rotating about the axis. Each time, the full magazine of bars is loaded. The slits also may be rotated about the axis. The bars arrive at slits A and B. Some will go through and be detected behind the slits.

Now there are four differences between Detector A and Detector B (shown in bold). In Experiment IV, the number of differences could not be larger (Bell inequality). In our case, it could be four or fewer. When would it be fewer That would happen when accidentally the bold figures (i.e., the differences of Experiments II and IE with respect to those of Experiment I) coincide. In this case, this would be counted as a difference in Experiments U and IE, while in Experiment IV, it would not be counted as a difference. 

It would be interesting to perform a real experiment similar to Bell s to confirm the Bell inequality, as discussed in the next section. This opens the door for deciding in a physical experiment whether ... 

Bell s experiments II-IV have been carried out. Common sense indicates that, even if the two photons in a pair have random polarizations (perfectly correlated though always the same, like the bars), they still have some polarizations i.e.. it may be unknown but definite (as in the case of the bars i.e., what E,P, and R believed happens). Hence, the results of the photon experiments would have to fulfil the Bell inequality. However, the photon experiments have shown that the Bell inequality is violated, but the results remain in accordance with the prediction of quantum mechanics. 

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