Bells Inequalities for the Ideal Case

Examination of the two-photon state vector in Eq. (10) or (12) shows that it implies nonlocality and lack of realism. It implies nonlocality since a measurement causes a collapse of, say, )+, Eq. (12a), to either x)i x)2 or ll )il )2, each possibility occurring with probability one half. Thus, detection of photon 1 to the left with polarization in the x direction ensures that photon 2 to the right behaves as a photon polarized in the x direction also. But, as we have already seen, the choice of x direction is quite arbitrary, so the polarization state measured for photon 2 is, in fact, determined by the measurement we choose to make on photon 1 at a position that may be spatially separated, in the relativistic sense, from the position at which the measurement on photon 2 is carried out. Lack of realism also follows from this argument, since it then is impossible to think of the individual photons possessing properties, in this case polarization, which exist independently of any measurements which may be made on them. 

This nonlocality and lack of realism inherent in quantum mechanics has inspired many attempts through the years to explain the results in terms of a theory that is both local and realistic. Without a specific local realistic theory it is, of course, not possible to predict a value for (a, b) to compare with the quantum mechanical value qm( b) in Eq. (14). However, in 1964 J. S. Bell showed for the first time that such theories place constraints on (o, b), or rather combinations of (a, b), for different values of a and b. 

Essentially the same inequality was derived by Clauser, Home, Shimony, and Holt and it is sometimes referred to as the Bell, Clauser, Home, Shimony, and Holt (BCHSH) inequality. Its importance lies in the fact that it represents a general restriction on the predictions of theories based on local realism. 

---