A simple two-page proof of local realism

[Joy Christian]

There is no simple relation between the (AxB) and the (A, B) models.

One shouldn't expect a simple realtion between the (AxB) and the (A, B) models, because the translation between these models within S^3 involve a non-trivial quaternion multiplication. Please see eqs. (A.9.27) to (A.9.34) on pages 248 to 250 of this book-chapter: http://libertesphilosophica.info/blog/w ... hapter.pdf.

[Heinera]

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].

And that was excactly my point, except that calling the fit in the 2D-model (even with the original range) for "almost perfect" is stretching the meaning of the words.

And whether you change the range of theta_o or change the function f doesn't matter, since instead of changing the range one can always insert a corresponding transformation on theta_o in f, with identical results.

[minkwe]

For those interested I have now posted my new "clocked" simulation with 100% detection at https://github.com/minkwe/epr-clocked. I will start a new thread for that.

[Joy Christian]

For those interested I have now posted my new "clocked" simulation with 100% detection at https://github.com/minkwe/epr-clocked. I will start a new thread for that.

Excellent. Let the goalpost shifting begin!

[Heinera]

For those interested I have now posted my new "clocked" simulation with 100% detection at https://github.com/minkwe/epr-clocked. I will start a new thread for that.

Ok. But before I dig deeper into that, I have a question: Is this still related to Joy's model, or is it supposed to be someting different?

[Joy Christian]

For those interested I have now posted my new "clocked" simulation with 100% detection at https://github.com/minkwe/epr-clocked. I will start a new thread for that.

Ok. But before I dig deeper into that, I have a question: Is this still related to Joy's model, or is it supposed to be someting different?

I will let Michel answer your question, but my answer is that it is both. It is quite independent and stands on its own, but of course it is also relevant because one of the reasons for the continued resistance to my work is the deeply held belief among Bell-believers that the simulation Michel has produced is impossible to produce.

[Heinera]

For those interested I have now posted my new "clocked" simulation with 100% detection at https://github.com/minkwe/epr-clocked. I will start a new thread for that.

Ok. But before I dig deeper into that, I have a question: Is this still related to Joy's model, or is it supposed to be someting different?

I will let Michel answer your question, but my answer is that it is both. It is quite independent and stands on its own, but of course it is also relevant because one of the reasons for the continued resistance to my work is the deeply held belief among Bell-believers that the simulation Michel has produced is impossible to produce.

I don't think any Bell-believers believe that this is impossible.

[gill1109]

Ok. But before I dig deeper into that, I have a question: Is this still related to Joy's model, or is it supposed to be someting different?

I will let Michel answer your question, but my answer is that it is both. It is quite independent and stands on its own, but of course it is also relevant because one of the reasons for the continued resistance to my work is the deeply held belief among Bell-believers that the simulation Michel has produced is impossible to produce.

I don't think any Bell-believers believe that this is impossible.

I wrote a paper with Jan-Åke Larsson in 2004 explaining exactly why Michel's program is possible and what are its limits.
http://arxiv.org/abs/quant-ph/0312035
Bell's inequality and the coincidence-time loophole
Jan-Ake Larsson, Richard Gill
Europhysics Letters, vol 67, pp. 707-713 (2004)
The recent Zeilinger group experiment (Giustina et al.) was actually "spoilt" by the coincidence loophole, but fortunately a more careful analysis of their data showed that they had after all violated the local realism limit. Which Michel's program, obviously, doesn't and can't do.

Gisin (Geneva) and the guys at Boulder Colorado were just behind Giustina et al. in publishing such a fantastic experiment too, and both groups could claim a first: the second group because of a defective data-analysis of the first group (repaired a bit later, with the help of Jan-Åke). 2013 was an exciting year.

[Heinera]

Yup, and it is a very good paper. So I suggest that everyone participating in this and other threads actually read it.

[gill1109]

Yup, and it is a very good paper. So I suggest that everyone participating in this and other threads actually read it.

That gives me a nice warm feeling

[Joy Christian]

Yup, and it is a very good paper. So I suggest that everyone participating in this and other threads actually read it.

That gives me a nice warm feeling

I am pleased to see how flatlanders get all excited by this sort of experiments. How cute!

All these experiments ever do is confirm my theorem that ALL quantum correlations are local-realistic correlations among the points of a parallelized 7-sphere.

[minkwe]

For those interested I have now posted my new "clocked" simulation with 100% detection at https://github.com/minkwe/epr-clocked. I will start a new thread for that.

Ok. But before I dig deeper into that, I have a question: Is this still related to Joy's model, or is it supposed to be someting different?

Look at the equations, I'm using very similar concepts borrowed from the first simulation. Some of the bounds are different. I've explained it in the README file. So you can discuss this simulation on it's own merit. That is why I suggest we discuss it in the new thread started for that purpose. As concerns Gill's paper with Larson, I believe it is flawed and I have explained the reasons why.

[gill1109]

As concerns Gill's paper with Larson, I believe it is flawed and I have explained the reasons why.

However, your simulation results are exactly in line with the predictions of Larsson and Gill (2004).

[minkwe]

As concerns Gill's paper with Larson, I believe it is flawed and I have explained the reasons why.

However, your simulation results are exactly in line with the predictions of Larsson and Gill (2004).

Could we continue this discussion in the other thread, I'm dissecting your paper there and showing exactly why it is flawed. Of course a flawed result can not apply to my simulation.

[gill1109]

Could we continue this discussion in the other thread, I'm dissecting your paper there and showing exactly why it is flawed. Of course a flawed result can not apply to my simulation.

You show why you think it is flawed. You are not dissecting our paper but some caricature of it. A straw man, Joy would say.

[Joy Christian]

OK, I have now studied Philip Pearle's 1970 paper thoroughly and analyzed his analytical solution extensively.

His solution does not produce perfect cosine correlation in Richard Gill's simulation.

I replaced my initial function f(theta_o) with Pearle's analytical solution in Richard Gill's simulation as follows:

theta <- runif(M, 0, pi) ## Pearle's alpha from his 1970 paper
g <- (2/3)*((((theta)-sin(theta))/((sin(theta/2))^2))+(((pi-theta)-sin(theta))/((cos(theta/2))^2))) ## Pearle's solution g(alpha), see his eq. (23)
s <- g*((sin(theta/2)^2)/2)

This produces cosine correlation, but not as perfectly as with the initial function f(theta_o) of my model: http://rpubs.com/chenopodium/joychristian.

This proves that my model is fundamentally different and it has nothing whatsoever to do with data rejection or detection loophole.

But some of us already knew that.

[gill1109]

OK, I have now studied Philip Pearle's 1970 paper thoroughly and analyzed his analytical solution extensively.

His solution does not produce perfect cosine correlation in Richard Gill's simulation.

I replaced my initial function f(theta_o) with Pearle's analytical solution in Richard Gill's simulation as follows:

theta <- runif(M, 0, pi) ## Pearle's alpha from his 1970 paper
g <- (2/3)*((((theta)-sin(theta))/((sin(theta/2))^2))+(((pi-theta)-sin(theta))/((cos(theta/2))^2))) ## Pearle's solution g(alpha), see his eq. (23)
s <- g*((sin(theta/2)^2)/2)

This produces cosine correlation, but not as perfectly as with the initial function f(theta_o) of my model: http://rpubs.com/chenopodium/joychristian.

This proves that my model is fundamentally different and it has nothing whatsoever to do with data rejection or detection loophole.

But some of us already knew that.

Interesting! "Now we have a contradiction, now we can make some progress".

I think you are forgetting Pearle's equation (22) for the probability density of r and equation (4) relating beta and r. He has a sphere of random radius r < 1 in his story. One has to translate it into a story with a fixed radius 1 by projecting the smaller sphere outward to the bigger 1. As I read him, formula (23) for g is *derived* from the other choices. So it's (22) which needs to be programmed, not (23).

B****y hard paper to read!

[Joy Christian]

I think you are forgetting Pearle's equation (22) for the probability density of r and equation (4) relating beta and r. He has a sphere of random radius r < 1 in his story. One has to translate it into a story with a fixed radius 1 by projecting the smaller sphere outward to the bigger 1. As I read him, formula (23) for g is *derived* from the other choices. So it's (22) which needs to be programmed, not (23).

B****y hard paper to read!

Phil's paper is not all that difficult to understand. Apart from some details, his logic is quite straightforward and easy to follow.

I have not forgotten his eq. (22), which is an explicit expression for the probability density as a function of . But this probability density is integrated out to obtain the solution (23) he is after. It is only the fraction that appears on the LHS of eq. (5) for quantum probabilities. Once an explicit solution for the fraction is found---as he does in eq. (23)---there is no need to worry about anymore, because it has been integrated out to obtain . The only ad hoc choice he makes in deriving the fraction is that of , but even this choice is rendered irrelevant by normalization. He does note, however, that "the fraction of undetected events can be reduced somewhat by a different choice of ; the extent of this reduction is an open question." So his solution is not all that "unique" after all.

[gill1109]

Note that Pearle talks about mushroom-shaped regions cut out of the unit ball. Not about circular regions on the unit sphere.

In terms of our simulation models, thought of as chaotic ball models with random radius R of the circular caps, his model has R = pi r where r is a random element of [0,1] with probability density rho(r) given by formula (22). The translation to your S = sin(f(theta_0)) is through the formula cos(R) = S, or equivalently S = arc cos R.

So we need to draw random R from the distribution I just described here, or equivalently S = arc cos(R). That's a nice challenge since we are given a density, not a cumulative distribution function, and the c.d.f. appears to be hard to compute ... so the probability integral transform method hard to implement. Rejection sampling might be a solution.

You are right, he does not show that his solution is unique; on the contrary there appear to be many solutions. Interesting. But he does claim his solution is exact!

But anyway, what I am saying is that to simulate his model within the scheme we are using (generalizations of Michel Fodje's simulation model / Caroline Thompson's simulation model) we do not need to do anything with the function g(.). What we need to do, and the only thing we need to do, is to draw a random value "r" from the probability distribution determined by Pearle's (22).

But there is something very funny with formula (22). If the "r^2" really is there on the left hand side, the probability distribution of "r" has a huge peak at 0. There would be an incredibly low detection rate. But if the r^2 is a misprint, then there is something funny, because the right hand side doesn't integrate to 1. And it's rather funny that no-one has ever published a simulation model based on the Pearle paper. I wonder if anyone ever checked all the math. (I admit, I didn't either).

[gill1109]

Now I get it! Pearle is of course using physicist's notation from 50 years ago.

rho(r) is not a probability density.

4 pi r^2 rho(r) *is* a probability density - see formula (1).

This the density I am looking for. So we must pick "r" from the probability density equal to the right hand side of (22) times 4 pi. I think I can manage that (with the rejection method).

This R code (including numerical check) computes and plots the probability density of "r":

Code: Select all

L <- 10^6
r <- seq(from=1, to = 2*L-1, by = 2) / (2*L)
halfpir <- pi * r / 2
rho <- 4 * pi *sin(halfpir) / (3 * ( 1 + cos(halfpir) )^3 )
plot(r, rho, type = "l")
sum(rho)/length(rho)


And, joy oh joy, this density is easy to integrate between 0 and r giving us an explicit invertible formula for the cdf!