I would like to reopen a discussion on assertions made by David Mermin such as: “There is no conceivable way to assign such instruction sets to the particles from one run to the next that can account for the fact that in all runs taken together, without regard to how the switches are set, the same colors flash half the time.” Taken from his 1985 article: Is the moon there when nobody looks? Reality and the quantum theory.(adsbygoogle = window.adsbygoogle || []).push({});

He describes a simple gedanken demonstration with a source, and two detectors each with three switches. He then proceeds to explain how the data can be explained by quantum theory, that is, cos^{2}120=1/4. And, that the data violates the Bell’s inequality and there can be no instruction sets. But, and striking (at least to me), he never made the connection between the gedanken model’s 3 switches and the cosine function. Without any mention of angles prior to this; it seem to appear out of nowhere.

I suggest a model that uses angles, the unit circle, where the trig functions are defined. Place three points on the circumference of the unit circle 120 degrees apart and label them in any order Bob, Chris, and Alice. This guarantees they choose different angles. Each are free to choose an angle and their choices are independent of each other. For simplicity, in each trial Bob’s and Chris’s choices are compared to Alice’s. Now rotate all three points simultaneously by 60 degree increments and record the sequence of plus and minus given by the cosine of the angle for each. Example: B-, C-, A+ = Bob’s sign is negative, Chris’s is negative, and Alice’s is positive. You will generate 8 triplets. The unit circle cannot assign + + + or - - - triplets for these angles. I know what you are thinking. How can you generate 8 triplets when there are only 6 distinct sequences remaining? Two of them repeat as follows: Bob chooses 240 and gets B-, Chris chooses 120 gets C-, and Alice is left with 0 and gets A+. But Bob’s and Chris’s choices of angles can be reversed. Two ways to get the same sequence: - - +, this only occurs whenever Bob and Chris get the same signs as shown below for 1) & 2) or 7) & 8).

240, 120, 0...........Bob/Alice....Chris/Alice.......(D = different signs, S = same signs)

1) B-, C-, A+...............D....................D

2) C-, B-, A+...............D....................D

0, 120, 240

3) B+, C-, A-...............D....................S

4) C+, B-, A-...............S....................D

180, 300, 60

5) B-, C+, A+..............D....................S

6) C-, B+, A+..............S....................D

60, 300, 180

7) B+, C+, A-..............D....................D

8) C+, B+, A-..............D....................D

Reordering the 8 triplets as BCA yields: (- - +), (- - +), (+ - -), (- + -), (- + +), (+ - +), (+ + -), (+ + -)

P(S) = 4/16 = ¼ same as quantum theory!

It is unlikely that quantum theory would have contradicted the unit circle. The fact that an analysis of the unit circle demonstrates the quantum prediction is indeed satisfying, but what are the implications? The hidden variable is the unit circle and the properties exist at all times, yet the above data (according to Mermin) violates Bell’s inequality. Does this discovery advance the debate over classical reality vs quantum reality, local vs nonlocal interactions, or Bell Theorem? Or, does it muddle the debate even more? Was Einstein both right and wrong? No pun intended.

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# Unit Circle and Quantum Predictions [hidden variable model agrees with QM?]

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