Local Hidden Variable Theory - Local Hidden Variables and The Bell Tests

Local Hidden Variables and The Bell Tests

The principle of "locality" enables the assumption to be made in Bell test experiments that the probability of a coincidence can be written in factorised form:

(1)

where is the probability of detection of particle with hidden variable by detector, set in direction, and similarly is the probability at detector, set in direction, for particle, sharing the same value of . The source is assumed to produce particles in the state with probability .

Using (1), various Bell inequalities can be derived, giving restrictions on the possible behaviour of local hidden variable models.

When John Bell originally derived his inequality, it was in relation to pairs of indivisible spin-1/2 particles, every one of those emitted being detected. In these circumstances it is found that local realist assumptions lead to a straight line prediction for the relationship between quantum correlation and the angle between the settings of the two detectors. It was soon realised, however, that real experiments were not feasible with spin-1/2 particles. They were conducted instead using photons. The local hidden variable prediction for these is not a straight line but a sine curve, similar to the quantum mechanical prediction but of only half the "visibility".

The difference between the two predictions is due to the different functions and involved. By assuming different functions, a great variety of other realist predictions can be derived, some very close to the quantum-mechanical one. The choice of function, however, is not arbitrary. In optical experiments using polarisation, for instance, the natural assumption is that it is a cosine-squared function, corresponding to adherence to Malus' Law.

Bell's theorem assumes that measurements are made at random, and not in principle determined by the universe at large. If this assumption were to be incorrect, as proposed in superdeterminism, conclusions drawn from Bell's theorem may be invalidated. Such arguments are generally called loophole theories.