# Question about Nick Herbert's Bell proof

bruce2g
I've just finished reading Nick Herbert's book 'Quantum Reality,' and I was a bit puzzled by his intuitive proof of Bell's theorem.

The part that puzzles me is the graphs on pp. 223 and 224, Fig 12.4 and 12.5. These show the 'matches per four marks' as a function of 'calcite difference.' They say that when the calcite difference is 90 degrees, the matches are zero.

It seems to me that when the calcite difference is 90 degrees, the correlation is zero, but then the matches should be 2 out of 4, since there's a 50-50 chance of a match when the correlation is zero. It seems to me that the calcite difference would need to be 180 degrees to achieve 0 matches.

Am I missing something, or did Herbert confuse 'correlation = zero' with 'matches = zero?'

Bruce

Mentor
Herbert has it right. At 0 degrees the system exhibits perfect correlation (= 1); at 90 degrees, perfect anti-correlation (= -1). No randomness at all at those angles.

bruce2g
Doc Al said:
Herbert has it right. At 0 degrees the system exhibits perfect correlation (= 1); at 90 degrees, perfect anti-correlation (= -1). No randomness at all at those angles.

Thanks. The problem stems from the fact that Herbert defines PC (polarization correlation) as "the fraction of matches" between the two calcites (p. 217). So actually, PC is a probablility between zero and 1, and it's not a statistical correlation (-1 to 1).

Since I have a statistics background, I got a little confused when he said (again on p. 217) "At zero degrees, PC = 1; at ninety degrees, PC = 0." As you noted, the actual (statistical) correlation at 90 degrees is -1.

My guess is that someone like Aspect or Bell started calling the coincidence count rate a "correlation" several decades ago, and it stuck; so the word "correlation" when used in the phrase "polarization correlation" has a different quantitative meaning than it has in normal statistics.

Just for the record, if 'r' is the statistical correlation and 'p' is the probability of a match (the 'polarization correlation'),
r = E(XY) = (1)p + (-1)(1-p) = 2p - 1, and
p = (r+1)/2
(based on the fact that XY = 1 when they match and -1 when they don't).

Other than this little confusion, Herbert's compact proof of Bell's theorem is terrific!

Gold Member
bruce2g said:
Since I have a statistics background, I got a little confused when he said (again on p. 217) "At zero degrees, PC = 1; at ninety degrees, PC = 0." As you noted, the actual (statistical) correlation at 90 degrees is -1.

You are exactly correct. The statistical view is different than how "correlation" is used with Bell tests. There are some places where it is actually presented as you describe (-1 to 1), but the majority have the range going from 0 to 1. That is because the results then nicely match the cos^2 theta function that is the driver for the quantum mechanical predictions.

DrChinese said:
There are some places where it is actually presented as you describe (-1 to 1), but the majority have the range going from 0 to 1. That is because the results then nicely match the cos^2 theta function that is the driver for the quantum mechanical predictions.
I have a question. Is not the range [-1 to 0.0 to + 1] completely different than [0 to +1] ? If so, which is the correct range to use for QM predictions, or does it not matter ?