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The Subtleties of Signal Matching
with Fallible Detectors

This is a careful derivation of the probability of detecting a matching of two binary signals when the signal detectors are not completely accurate. The fallibility of the detectors introduces the possibility of false matches. But detectors which are always wrong are just as good as detectors that are always right in detecting true matches.

Let the binary signals be denoted as 0 and 1. Let the signals arriving at detectors A and B have probabilities given by p11, p10, p01, and p00. Let PA and PB be the probabilities that the detectors A and B give correct readings.

The probability that a match is recorded, R, is then

R = p11[PAPB + (1-PA)(1-PB)]
+ p00[PAPB + (1-PA)(1-PB)]
+ p01[(1-PA)PB + PA(1-PB)]
+ p10[PA(1-PB) + (1-PA)PB]

These reduce to

R = [p11 + p00][PAPB + (1-PA)(1-PB)]
+ [p01 + p10] [(1-PA)PB + PA(1-PB)]

Let [p11 + p00], the probability of a true match, be denoted as M. Then [p01 + p10], the probability of a true mismatch is (1-M). Therefore

R = M[PAPB + (1-PA)(1-PB)]
(1-M)[(1-PA)PB + PA(1-PB)]

When terms in the brackets are expanded the above equation becomes

R = M[1 - (PA+PB) + 2PAPB]
+ (1-M)[(PA+PB) - 2PAPB]
which can be further reduced to
R = [(PA+PB) - 2PAPB] + M[1 - 2(PA+PB) + 4PAPB]

The first bracketed term is the probability of a false match; the second is the probability of detecting a true match. If PA=PB=P the above formula reduces to

R = [2P - 2P²] + M[1 -4P + 4P²]
or, equivalently
R = 2P(1-P) + M[1 - 4P(1-P)]

Errors in the detectors make a difference only if one errs and the other does not. That is where the terms involving P(1-P) arise. An important point is that even if M=0, R>0 if P>0. The worst case is not P=0 but P=½. In that case R=½.

In the testing of Bell's Theorem the placing of the detectors at an angle is, in a sense, just a way of introducing fallibility in the detectors. In the literature there is the statement that the correlation should be 1/2 if the quantum mechanic analysis is correct but different from 1/2 if it is wrong and that the experimental estimate of the correlation is 1/2. The above analysis shows that a simple classical analysis gives 1/2 as the predicted value.

(To be continued.)

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