Are entangled particle outcomes determined by the unit circle?

[rlduncan]

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?

 

[Cthugha]

If Mermin’s data is truly typically of actual experiments, then the difference is noteworthy, and I assume statistically significant.

The results might be typical, but for sure not statistically significant. Your total conditional subset just consists of 31 events. One detection more in one way or the other already changes your result by 0.03.

This is not too surprising as Physics Today is rather aiming at a broader audience and not a 'hard' scientific journal, so this is rather a visualization. If you want to have reliable data, think of using a subset of thousands of detection events. For 31 events, everything is dominated by error bars. I think the original data of some experiments on entanglement have been published somewhere on the web. Maybe you manage to find them somewhere and can analyze that larger set of data. Unfortunately, I do not know where to find these exactly.

 

[Nugatory]

It is logical to assume that there is a second instruction set for the situation when different orientations are chosen.

If this second instruction set comes into play when different orientations are chosen, it seems that the choice of instruction set is necessarily non-local. The essence of Bell's argument and the experimental data is that any instruction set that works must depend on both settings, hence is non-local.

Perhaps I am misunderstanding your argument?

 

[rlduncan]

The results might be typical, but for sure not statistically significant. Your total conditional subset just consists of 31 events. One detection more in one way or the other already changes your result by 0.03.

This is not too surprising as Physics Today is rather aiming at a broader audience and not a 'hard' scientific journal, so this is rather a visualization. If you want to have reliable data, think of using a subset of thousands of detection events. For 31 events, everything is dominated by error bars. I think the original data of some experiments on entanglement have been published somewhere on the web. Maybe you manage to find them somewhere and can analyze that larger set of data. Unfortunately, I do not know where to find these exactly.

Thanks for the comments. I certainly agree with you about n=45 trials. Number of trials was not the issue. This was fully understood before deciding to post. However, 100s of n=45 trials and Mermin selects a set that he clearly states as typical (his words not mine), but contradicts the main objective of writing the article. And, hard to imagine that after 28 years and thousands of readings of his paper by professionals that not one notices or informs David Mermin.

Concerning the actual data, may be someone could provide a link. However, I am sure analyzing it will be extremely difficult to do, given that few authors give enough information to accurately assess the relevant data.

Note. If the data presented is typical of the 8000 measured photon pairs, how can it not be statistically significant?

 

[rlduncan]

If this second instruction set comes into play when different orientations are chosen, it seems that the choice of instruction set is necessarily non-local. The essence of Bell's argument and the experimental data is that any instruction set that works must depend on both settings, hence is non-local.

Perhaps I am misunderstanding your argument?

The instructions are an inherent property of the trig functions and the unit circle, the principles of trigonometry dictate the possible outcomes and there are only 6 not 8 permutations. This was one key point that I was trying to make. Nearly all previous discussions including Mermin insisted on 8 permutations which exaggerate the differences between quantum theory and local hidden variable theory offered in the post.

Place two points on the unit circle 180 degrees apart, certain to have opposite signs. Placing 3 points on the unit circle at 120 angles; certain to have one different sign which leads to 6 permutations. Alice and Bob are both using the same instructions or better yet hidden variables, a local realistic model.

The second point, 0,120,240 angles may not be the best choices of orientations to distinguish between quantum theory predictions versus hidden variable theories for the differences are small. I tried to show this by analyzing the Aspect’s experimental data as characterized by Mermin and then compare the results with the probabilities of quantum theory versus the unit circle. Some problems became immediately apparent. Does Mermin’s data need re-assessing? May be, and perhaps the inconsistency needs to be publicized.

 

[Cthugha]

This was fully understood before deciding to post. However, 100s of n=45 trials and Mermin selects a set that he clearly states as typical (his words not mine), but contradicts the main objective of writing the article. And, hard to imagine that after 28 years and thousands of readings of his paper by professionals that not one notices or informs David Mermin.

I disagree. In what way does it contradict the main objective of writing the article? I did not calculate the standard error of that value, but it should be around 0.8 to 0.12. That error is way too large to allow any conclusion. Any small subset of just 45 trials will carry such a large error. This is a mass-compatible physics article for the general public, so it is ok just for demonstrating how things are done. It would not be an acceptable basis for a publication in a solid physics journal, though.

Note. If the data presented is typical of the 8000 measured photon pairs, how can it not be statistically significant?

See above. The trial size is just too small. If you throw a coin thirty times and it ends up on heads 18 times, that is also still a typical outcome, but you cannot claim that the coin is unfair or biased as the trial size is too small.

 

[rlduncan]

See above. The trial size is just too small. If you throw a coin thirty times and it ends up on heads 18 times, that is also still a typical outcome, but you cannot claim that the coin is unfair or biased as the trial size is too small.

I understand your point. However, toss a coin a 1000 times and suppose you get 480 heads and 520 tails which gives the 0.48 probability. For practical reasons or convenience (limited space) you decide to use 50 tosses to show that coin tossing is random or 0.50. Now a data set typical of the large number of tosses would be 24 heads and 26 tails. Why would you instead condense the coin toss experiment by reporting 30 heads/20 tails which is 0.60? It would not be typical of the large number of tosses. As I interpret Mermin’s paper the data was typical of the large number of runs. Here is the quote from page 7: “Typical data from a large number of runs are shown in figure 3.” May be you interpret this differently than I do.

 

[DrChinese]

A. 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.

B.

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

C. 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)

... 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?

A. Obviously, the traditional QM viewpoint is not the same as what you describe. The agreement at identical settings is just a function of the cos^2 relationship and nothing more.

B. I see your point here, seems fine as a hypothesis.

C. If the results actually came this close between a local realistic candidate and the predictions of QM, then you would be correct. But actual results are always i) exclusive of the LR candidate; and ii) within the range of QM predictions by many standard deviations.

Nonetheless, I am always glad to see someone digging this deep into the nuts and bolts!

 

[San K]

rlduncan, another way to look at it is --

if entanglement was not real then:

- entanglement swapping would not be possible if the phenomena was local/LHV

- delayed choice quantum entanglement experiments would not give the results they currently provide

- you could not have interference between two photons without them ever meeting at the same point in time-spaceall such experiments have loopholes but i gather that they are closed or close to being closed

Dr. Chinese’s Challenge: 0,120,240 Data Set

that's an abbreviation; the full form of the challenge is:

Dr. Chineses' China challenge in Chinese while eating Chinese...

 

[lugita15]

rlduncan, I think you're reading too much into the word typical. I think Mermin just meant "this is an example of the sort of data you can get". I doubt he checked how close to the average it is.

 

[rlduncan]

A. Obviously, the traditional QM viewpoint is not the same as what you describe. The agreement at identical settings is just a function of the cos^2 relationship and nothing more.

B. I see your point here, seems fine as a hypothesis.

C. If the results actually came this close between a local realistic candidate and the predictions of QM, then you would be correct. But actual results are always i) exclusive of the LR candidate; and ii) within the range of QM predictions by many standard deviations.

Nonetheless, I am always glad to see someone digging this deep into the nuts and bolts!

Fair enough. Thanks for the reply.

 

[rlduncan]

rlduncan, I think you're reading too much into the word typical. I think Mermin just meant "this is an example of the sort of data you can get". I doubt he checked how close to the average it is.

Sure that is possible. He indeed emphasized the same-switch/same-color occurrence, the randomness of the data, and the other merely went unnoticed. Testing Bell’s inequality has taken top billing, but these simple data averages are equally important. However, I haven’t found any references to these statistics. May be you can assist me.
rlduncan