Dr. Chinese’s Challenge: 0,120,240 Data Set Data Features (Quantum Theory): P(B|A) = 1; P(B|Aʹ) = .25; P(B) = .50 Eq. (1) “A” means “same setting”, “Aʹ” means “different setting”, and “B” means “different outcome”, “Bʹ” means “same outcome”. Here is a quote from David Mermin’s paper (Is the moon there when nobody looks? Reality and the quantum theory): “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.” A Geometric Explanation Any explanation for the experimental data set for a two-valued variable must take into account that the source prepares the spin singlet state particle with opposite spins, which are then sent to different detectors. The fact that P(B|A) = 1 is guaranteed when identical settings are chosen is an instruction! Indeed a hidden instruction when observing any single run or trial. As such, when the orientations are at the same setting the outcomes S(+,+) and S(-,-) are forbidden. What about the second feature, P(B|Aʹ) = 1/4? Are there forbidden outcomes? It is logical to assume that there is a second instruction set for the situation when different orientations are chosen. Lacking a balancing instruction means P(B) = 1/2 is no longer certain! As an example, in a modified coining tossing experiment in which selecting the same coin gives opposite outcomes, then P(B) = 2/3. And, this doesn’t match the quantum theory prediction of random outcomes for entangled particles. If it exists, the second hidden instruction set originates from the unit circle and basic principles of trigonometry. First, opposite spins between the two entangled particles are sustained by rotating θ + π degrees between the two particles A and B placed on the unit circle and B(θ + π) = -A(θ), where A and B are the outputs. Let A(θ) equal the sign designated by cosine of the angle. For example, A(0) is + (because the cosine of zero is positive) and B(π) is -. Second, consider three points on the unit circle at angles 0, 120, 240 degrees. At the reference angle A(0) = +. Rotating counter clock-wise to the second quadrant assigns B(120) = -, and rotating to the third quadrant assigns B(240) = -. This gives the + - - which is one of the possible permutations. However, and this is very significant, the outcomes +++ or --- are forbidden at these angles. It is geometrically impossible for any three points placed on the unit circle at 120 degrees angles to simultaneously have the same signs. Thus, the hidden second instruction set. These permutations are unquestionably allowed for traditional two-valued variable experiments such as in coin tossing. In actual entanglement experiments, polarizers are randomly rotated at selected orientations just as above and surely the physics of these rotations must obey the fundamental laws of trigonometry. Therefore, the only allowable permutations by the unit circle for Alice and Bob are: 1) GGR-RRG 2) GRG-RGR 3) GRR-RGG 4) RGG-GRR 5) RGR-GRG 6) RRG-GGR. Unit Circle Data Set AB(θ,θ)------------AB(θ,θ+120)-----AB(θ,θ+240) GR----------------- GR----------------- GG GR----------------- GG----------------- GR GR----------------- GG----------------- GG RG----------------- RR----------------- RR RG----------------- RR----------------- RG RG----------------- RG----------------- RR Thus the unit circle outcomes dictate the following theoretical probabilities: P(B|A) = 6/6 = 1; P(B|Aʹ) = 4/12 = 1/3 = .33; P(B) = 10/18 = .56 Eq. (2) Shown below are the calculated probabilities for the data outcomes as presented in the article by Mermin. P(B|A) = 14/14 = 1; P(B|Aʹ) = 10/31 = .32; P(B) = 24/45 = .53 Eq. (3) Note. For simplicity, Mermin analyzed the data for outcomes of the same color. Mermin clearly stipulates that the data presented is a fragment of the actual data collected by Aspect and others. However, he also states that the presented data is typical from a large number of runs. On page 6 Mermin notes that Figure 5 shows that the pattern of colors is completely random (I assume buttressed by the P(B) = .53). Compare probabilities from Eq. 2) and Eq. 3), which shows excellent agreement. The P(B) = .53 cannot be used to confirm either the quantum theory (.50) or unit circle (.56) prediction for probability of getting a different outcome. More troublesome, from Eq. 3) the experimental P(B|Aʹ) = .32 and does not match the quantum theory prediction of Eq. (1) of P(B|Aʹ) = .25! But does match the unit circle’s P(B) = .33. If Mermin’s data is truly typically of actual experiments, then the difference is noteworthy, and I assume statistically significant. When the polarizations are measured at 120 degree angles do these experiments in reality confirm the quantum theory predictions or the unit circle properties/expectations?