Quantum Correlations beyond Bell’s Inequalities

Abstract

In 1964, John Bell¹ achieved a remarkable feat. His theorem states that a very general class of local realistic theories is incompatible with quantum mechanics. This accomplishment is of an exceptional nature, because John Bell did not have to know explicitly the details of the individual theories which were excluded by his theorem. The theorem is usually stated in terms of the spin correlations either between two fermions or between two photons, where in both cases both particles enjoy a two-dimensional spin space. (See the early review by Clauser and Shimony.²) The most often used state to discuss these correlations is the two-fermion singlet state

$$ |\psi \rangle = {1 \over {\sqrt 2 }}{(| \uparrow \rangle _1}| \downarrow {\rangle _2} - |{ \downarrow _1}| \uparrow {\rangle _2}). $$

((1))

Whenever we write such a state, as a product of kets we imply the tensor product because each particle is defined in its own Hilbert space. Nearly all the experiments with spin correlations exploited the correlations between two photons in atomic cascades. The first such experiment was performed by Clauser and Freedman³ with the development culminating in the experiments of Aspect et al.⁴ In the last one of these experiments, time-varied measurement was used in such a way that the photons were switched to different polarizers while in-flight. This was intended to rule out any conspiratory type of local theory where, through some yet unknown communication channels between detectors and source, the quantum correlations are established. Yet, due to an unfortunate numerical coincidence between switching frequency and photon flight time,⁵ that experiment cannot be called conclusive. Furthermore, this experiment used periodic switching, and it is easy to see that the definitive experiment would have to use a purely random switch.