Bell Tests With No "non-detections"
Consider, for example, David Bohm's thought-experiment (Bohm, 1951), in which a molecule breaks into two atoms with opposite spins. Assume this spin can be represented by a real vector, pointing in any direction. It will be the "hidden variable" in our model. Taking it to be a unit vector, all possible values of the hidden variable are represented by all points on the surface of a unit sphere.
Suppose the spin is to be measured in the direction a. Then the natural assumption, given that all atoms are detected, is that all atoms the projection of whose spin in the direction a is positive will be detected as spin up (coded as +1) while all whose projection is negative will be detected as spin down (coded as −1). The surface of the sphere will be divided into two regions, one for +1, one for −1, separated by a great circle in the plane perpendicular to a. Assuming for convenience that a is horizontal, corresponding to the angle a with respect to some suitable reference direction, the dividing circle will be in a vertical plane. So far we have modelled side A of our experiment.
Now to model side B. Assume that b too is horizontal, corresponding to the angle b. There will be second great circle drawn on the same sphere, to one side of which we have +1, the other −1 for particle B. The circle will be again be in a vertical plane.
The two circles divide the surface of the sphere into four regions. The type of "coincidence" (++, −−, +− or −+) observed for any given pair of particles is determined by the region within which their hidden variable falls. Assuming the source to be "rotationally invariant" (to produce all possible states λ with equal probability), the probability of a given type of coincidence will clearly be proportional to the corresponding area, and these areas will vary linearly with the angle between a and b. (To see this, think of an orange and its segments. The area of peel corresponding to a number n of segments is roughly proportional to n. More accurately, it is proportional to the angle subtended at the centre.)
The formula (1) above has not been used explicitly — it is hardly relevant when, as here, the situation is fully deterministic. The problem could be reformulated in terms of the functions in the formula, with ρ constant and the probability functions step functions. The principle behind (1) has in fact been used, but purely intuitively.
Thus the local hidden variable prediction for the probability of coincidence is proportional to the angle (b − a) between the detector settings. The quantum correlation is defined to be the expectation value of the product of the individual outcomes, and this is
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- (2) E = P++ + P−− − P+− − P−+
where P++ is the probability of a '+' outcome on both sides, P+− that of a + on side A, a '−' on side B, etc..
Since each individual term varies linearly with the difference (b − a), so does their sum.
The result is shown in fig. 1.
Read more about this topic: Local Hidden Variable Theory
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