Bell's theorem/Probability

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Defining the variables (X, Y, Z) of a Bell's theorem experiment
A Bell's theorem experiment establishes that nature violates Bell's inequality.

Bell's theorem is  a proof by contradiction, or something that seems to "prove" something that is not true, presumably because the theorem is based on false assumption(s). We begin our proof by assuming the existence of variables that arguably don't exist. A simple configuration that is a predecessor to the Bell's theorem experiment is shown to the right. A single atom emits a pair of photons whose polarization is always measured to be at perpendicular angles.

The "local" photon passes if and only if the "remote" photon passes, as shown in cases (a) and (b).

Therefore, it is reasonable to assume that the "local" photon's value of X can be measured by observing the "remote" photon.

This is depicted in case (c), where we have attributed the value X=1 to the local photon even though no measurement has been made on it. By permitting these two de facto polarization measurements on the local photon, data is generated in the laboratory that will be shown in the next section to be "impossible".


 * See also WikiJournal of Science/ A card game for Bell's theorem and its loopholes

A simple Bell's theorem configuration
A simple experimental test of Bell's theorem for photons uses two sets of three polarizing filters, each set with the three orientations shown in the figure. All three lines of polarization are 60 degrees apart, in a symmetry deliberately chosen to simplify the analysis.

The measurement consists of placing the polarizer in front of the photon and detecting whether it was blocked or passed by the filter as it is detected. Since there are two entangled photons, the experiment can measure two variables. There are, in fact, a total of 6 variables if you include the remote photon. However, great simplification results from focusing our attention on the "local" photon. By properly rotating the orientation of the "remote" filter, we can use that measurement to ascertain the value of corresponding variable (X,Y,or Z) of the "local" photon.


 * Key ideas:
 * We know by observation that the measurement on the "remote" photon always predicts the polarization of the "local" photon
 * This permits us to effectively measure two of the three variables associated with the "local" photon. These variables (X,Y,Z) specify whether or not the "local" photon would pass a filter at the specified orientation.
 * Although measurements on the "remote" and "local" photons allow a de facto measurement of two of the three variables, they do not permit us to measure all three (X,Y,Z) variables for a given photon.

Do variables exist if they are not observed?
Since Bell's theorem is violated by experiment, we must seek a false assumption that would render the theorem invalid. That assumption seems to be the existence of the variables (X,Y,Z). Since we can only measure two of them, the third might not exist. Moreover, it is not known at the time of creation which of the two measurements will be made of the entangled photons. So if X, Y, and Z don't exist until they are measured, they somehow "pop" into existence the instant a photon strikes the detector. This resource makes no claim to understanding what it means for a variable to "exist" or "not exist".

Doctor Who fans know that a Weeping Angel does not exist while being observed. In contrast, (X,Y,Z) only exist if they are observed, which suggests that hidden variables and weeping angles are complementary entities.

If you think you understand most of this you are ready for Bell's theorem/Inequality
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