A simple two-page proof of local realism

[gill1109]

Joy, this is your code with just one extra line: I also plot the difference between the two curves.

http://rpubs.com/gill1109/joychristian

Take a look at the last graphic. The largest deviation is now at around 50 degrees and it's of size 0.004. Significantly larger than the Monte Carlo error at N = 10^7.

But it's of no consequence. As our computers get faster we'll be able to get higher precision and larger sample sizes, and we'll discover that we need to make more tweaks... We'll always be able to find a new tweak. The process will converge to the analytic solution of Pearle (1970).

[Heinera]

OK, now we are well and truly done. I have revised my simulation again: http://rpubs.com/chenopodium/joychristian.

The initial state of the system (e_o, theta_o) is derived in http://libertesphilosophica.info/blog/w ... mplete.pdf, and the choice of the initial function f(theta_o) is

f(theta_o) = (1/2) sin^2(theta_o).

What is different in this version is the range of theta_o. This time any remaining wrinkles in the correlation are indeed well within the accuracy of the simulation.

I ran it in Mathematica with those parameters, 5 million trials. It's still off. The straighter part of the curve is not coming in now.

That is because whatever gives a good approximation in the 3-D model generalized by Richard does not neccessarily give a good approximation in the original 2-D model. This is due to the fact that the angle between two points each uniformly distributed on the sphere is not uniformly distributed. In particular, in the 2-D model the correlation is still straight as a ruler between 60 and 120 degrees, and it can be shown that the new version actually inreases the width of the interval where the correlation is linear.

[gill1109]

So Joy is adjusting by hand the distribution of Theta to "fit" the quantum mechanical result. I've made clear in my comments above that in my opinion these simulations do not correspond to a local realist model, but I've noticed something funny anyway. Instead of tweaking the distribution of Theta, as Joy is doing, and transforming by f, we can tweek the ditribution of s directly as a Beta(1/2,1/2) achieving similar results. To do that, in this code

http://rpubs.com/chenopodium/joychristian

just replace (comment or delete) the lines

theta <- runif(M, 0.129153, pi/2.38548) ## Christian and Minkwe's theta_0
s <- (sin(theta)^2)/2

by

s <- rbeta(M, 1/2, 1/2)

Why is this curious and funny? Because it is well known that in the Bayesian analysis of the binomial model, Beta(1/2, 1/2) is the "noninformative" (Jeffreys) prior distribution for the parameter of the model, and it kind of works here.

I tested your proposal, Zen. The result is

http://rpubs.com/gill1109/zen

Not so good I think, but of course we can now tweak the parameters of the beta distribution...

But the definitive solution can be deduced from Pearle (1970).

[Joy Christian]

I've made clear in my comments above that in my opinion these simulations do not correspond to a local realist model...

No, in a flatland they don't. We, however, do not live in a flatland.

...we can tweek the ditribution of s directly as a Beta(1/2,1/2) achieving similar results. To do that, in this code

http://rpubs.com/chenopodium/joychristian

just replace (comment or delete) the lines

theta <- runif(M, 0.129153, pi/2.38548) ## Christian and Minkwe's theta_0
s <- (sin(theta)^2)/2

by

s <- rbeta(M, 1/2, 1/2)

I just tried. It doesn't work. It produces exactly the wrong result we had been trying to correct:



You can find the correct result here (see the third graph): http://rpubs.com/chenopodium/joychristian.

[Joy Christian]

Here is a very instructive video, which I had been resisting to post until now: http://www.youtube.com/watch?v=BWyTxCsIXE4.

[gill1109]

Thanks, Richard. By the way, you know that Bernstein's Theorem tells us that we can represent any continuous distribution with support [0,1] by a sufficiently large mixture of betas. Therefore, The "magic" distribution can be approximated by such a mixture. It's a kind 21st century epicyle system.

Have you coded the simulation using the results from Pearle?

Not yet. That might be my project for this weekend.

The important thing is that Pearle shows that the "magic distribution" does exist. Now we know it exists, it makes sense to approximate it with a convenient system of epicycles.

Joy Christian already knows it exists, since according to his math it must follows from his S^3 theory. But so far we did not see a deduction from his theory, we only saw an "Ansatz", i.e., a guess, and the guess turned out to be not quite right, though close. Now the guess is being improved, empirically, but each new refinement is still only another Ansatz.

[gill1109]

I tried, it is a bit better, but not much.

http://rpubs.com/gill1109/zen

Joy and Richard,

Would you please replace

s <- rbeta(M, 1/2, 1/2)

with

s <- rbeta(M, 1.29, 4.56)

to check if there is a better fit?

Thank you very much.

[Joy Christian]

I tried, it is a bit better, but not much.

http://rpubs.com/gill1109/zen

So I am better than R. Cool.

I tried the library(MASS) thing, but it does not give me a result in closed form.

[minkwe]

Interesting discussions. To improve the number of samples at each point, I suggest you reduce the angle resolution to something like 22.5. That way you will get ~23 times more samples at each angle for the same N. This should allow to investigate the discrepancy further and test various distributions.

Secondly, I suggest to try:

beta(pi, 1)/pi

(http://en.wikipedia.org/wiki/Beta_distribution, I don't know if it is the same as the version in R, not the beta-prime distribution )

Also, the problem might not be the 1/2 sin(t)^2 distribution but the integral. The magnitude of A and B functions may be incomplete and need to be scaled. Since sign(-0.1) = sign(-1.0) it might not matter for the individual outcomes of (+1 or -1) but may for the initial conditions. Maybe we are struggling because we are focusing on the wrong side of the expression |C| < 1/2 sin(t)^2. Maybe we should be looking for a scale factor for |C|.

[Joy Christian]

Interesting discussions... Maybe we should be looking for a scale factor for |C|.

My worry in looking for a scale factor for |C| is that that might mess up the perfect anti-correlation requirement at equal settings. It is not good enough to have 98% perfect anti-correlation at equal settings. That violates what is predicted by QM (I know---operationally that might not be a problem, but logically it is a problem).

[minkwe]

Interesting discussions... Maybe we should be looking for a scale factor for |C|.

My worry in looking for a scale factor for |C| is that that might mess up the perfect anti-correlation requirement at equal settings. It is not good enough to have 98% perfect anti-correlation at equal settings. That violates what is predicted by QM (I know---operationally that might not be a problem, but logically it is a problem).

I don't think it would because the anti-correlation is based on the sign of C only not it's magnitude.

[Joy Christian]

Interesting discussions... Maybe we should be looking for a scale factor for |C|.

My worry in looking for a scale factor for |C| is that that might mess up the perfect anti-correlation requirement at equal settings. It is not good enough to have 98% perfect anti-correlation at equal settings. That violates what is predicted by QM (I know---operationally that might not be a problem, but logically it is a problem).

I don't think it would because the anti-correlation is based on the sign of C only not it's magnitude.

Yes, you have a point.

Have you looked at the latest version of my simulation yet? http://rpubs.com/chenopodium/joychristian

It does not seem to work as well in Mathematica as in R. I wonder whether it would work in Python.

[minkwe]

Have you looked at the latest version of my simulation yet? http://rpubs.com/chenopodium/joychristian

It does not seem to work as well in Mathematica as in R. I wonder whether it would work in Python.

Yes I have. I also tried those parameters in the python version and it did not work. As Fred suggested, there must be an inherent difference between the 2D and 3D distributions.

[gill1109]

We have two basic simulation approaches. The original Christian-Roth simulation of A times B, not of A and B separately. Originally programmed in Java by Chantal, later ports to JavaScript (Daniel Sabsay) and Mathematica (John Reed) and R (me).

Then we have the Minkwe (Michel Fodje; Python) approach in which A and B are simulated separately. It is a true event-based simulation. No theoretical short cuts. This has been later ported to R (me; Chantal) and generalized /extended in various ways.

The first, AB approach, is definitely different from the second, (A, B) approach. The second approach is mathematically isomorphic to (a generalization) of Caroline Thompson's model / Marshall et al / Pearle. From Pearle we know that it can be tuned to give *exactly* the cosine.

[Heinera]

But even with the second (A, B) approach, the 2D vs the 3D generalization are sufficiently different so that the the same input formula will give different correlations, as previous posters have discovered. This is why Fred and minkwe are now getting worse results; they run the 2D model (in Mathematica and Python, respectively), while Christian fine tuned this with Richard 3D generalization.

[gill1109]

But even with the second (A, B) approach, the 2D vs the 3D generalization are sufficiently different so that the the same input formula will give different correlations, as previous posters have discovered. This is why Fred and minkwe are now getting worse results; they run the 2D model (in Mathematica and Python, respectively), while Christian fine tuned this with Richard 3D generalization.

[Joy Christian]

But even with the second (A, B) approach, the 2D vs the 3D generalization are sufficiently different so that the the same input formula will give different correlations, as previous posters have discovered. This is why Fred and minkwe are now getting worse results; they run the 2D model (in Mathematica and Python, respectively), while Christian fine tuned this with Richard 3D generalization.

This is not quite correct. As you yourself observed earlier, it is the range of theta_o that is responsible for the difference in the 2D and 3D versions of the Fodje simulation. The input formula, f(theta_o), is the same in both versions, at least in the latest attempts of the simulation.

[Heinera]

But even with the second (A, B) approach, the 2D vs the 3D generalization are sufficiently different so that the the same input formula will give different correlations, as previous posters have discovered. This is why Fred and minkwe are now getting worse results; they run the 2D model (in Mathematica and Python, respectively), while Christian fine tuned this with Richard 3D generalization.

This is not quite correct. As you yourself observed earlier, it is the range of theta_o that is responsible for the difference in the 2D and 3D versions of the Fodje simulation. The input formula, f(theta_o), is the same in both versions, at least in the latest attempts of the simulation.

I assume that both Fred and Fodje updated the range of theta_o to match your new values, ran the simulation, and got different results than you. Or did they do something else?

[gill1109]

It is the range of theta_o that is responsible for the difference in the 2D and 3D versions of the Fodje simulation. The input formula, f(theta_o), is the same in both versions, at least in the latest attempts of the simulation.

No, there was initially a very big difference (big improvement) in lifting Minkwe's (A, B) simulation from S^1 to S^2, and I'm talking here about the hidden variable e_0, not theta_0.
http://rpubs.com/gill1109/EPRB23big

The original JCS simulation by Chantal Roth, the one I call the AxB simulation, the one which Reed put in Mathematica, was always S^2 based.
http://rpubs.com/gill1109/JCS2opt

The best result yet, as far as I know, is that of
http://rpubs.com/gill1109/ChaoticUnsharpBall1

Please compare this with
http://rpubs.com/gill1109/joychristian
http://rpubs.com/chenopodium/joychristian

All of these are Minkwe - S^2 - (A, B) based; just variations on the distribution of S = sin(f(theta_0)) = cos(R).
However none of these are spot on, because none of them is quite the same as Pearle's (1970) solution, which - as one can see from Pearle's derivation - is unique. There is only one way to hit the target spot-on. It has been known for nearly 45 years.

What Fred has been doing is pushing the Reed Mathematica (AxB) simulation to its limit by refining the bin width for the angles and increasing the sample size. All that does is to reveal that it is indeed systematically off target. There is no simple relation between the (AxB) and the (A, B) models).

[Joy Christian]

But even with the second (A, B) approach, the 2D vs the 3D generalization are sufficiently different so that the the same input formula will give different correlations, as previous posters have discovered. This is why Fred and minkwe are now getting worse results; they run the 2D model (in Mathematica and Python, respectively), while Christian fine tuned this with Richard 3D generalization.

This is not quite correct. As you yourself observed earlier, it is the range of theta_o that is responsible for the difference in the 2D and 3D versions of the Fodje simulation. The input formula, f(theta_o), is the same in both versions, at least in the latest attempts of the simulation.

I assume that both Fred and Fodje updated the range of theta_o to match your new values, ran the simulation, and got different results than you. Or did they do something else?

Yes, that is what they did, as far as I know. But what I am saying is that in my, Fred's, and Michel's latest attempts we all used the same input formula, namely 1/2 sin^2(theta_o). The difference between what I saw and what they saw was due to two things: (1) my simulation is a 3D version and their simulations are 2D versions, and (2) I get an almost perfect fit for theta_o in an interval other than [0, pi/2], whereas they get an almost perfect fit for theta_o in the standard interval [0, pi/2].