Prof. Gill's Triangle wave versus the Cosine paper

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Prof. Gill's Triangle wave versus the Cosine paper

Postby FrediFizzx » Fri Jun 07, 2019 2:36 pm

gill1109 wrote:I put a paper on arXiv in which I show what correlation functions can be produced by simple “local hidden variables”. https://arxiv.org/abs/1312.6403. The triangle wave versus the cosine (how to optimally approximate EPR-B Correlations by classical systems)

Richard D. Gill
(Submitted on 22 Dec 2013 (v1), last revised 27 Dec 2013 (this version, v2))
The famous singlet correlations of a composite quantum system consisting of two spatially separated components exhibit notable features of two kinds. The first kind are striking certainty relations: perfect correlation and perfect anti-correlation in certain settings. The second kind are a number of symmetries, in particular, invariance under rotation, as well as invariance under exchange of components, parity, or chirality. In this note I investigate the class of correlation functions that can be generated by classical composite physical systems when we restrict attention to systems which reproduce the certainty relations exactly, and for which the rotational invariance of the correlation function is the manifestation of rotational invariance of the underlying classical physics. I call such correlation functions classical EPR-B correlations. It turns out that the other three (binary) symmetries can then be obtained for free: they are exhibited by the correlation function, and can be imposed on the underlying physics by adding an underlying randomisation level. We end up with a simple probabilistic description of all possible classical EPR-B correlations in terms of a ``spinning coloured disk'' model, and a research programme: describe these functions in a concise analytic way.

I’m a bit fed up that no one seems to bother to read it. :?

I guess you mean that no one has commented on it yet. It is a bit hard to follow. I'm trying to study it.
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby gill1109 » Fri Jun 07, 2019 10:44 pm

Thanks Fred for starting this new thread. Yes, that's what I meant, I guess it is hard to follow... :?

In https://arxiv.org/abs/1312.6403, "The triangle wave versus the cosine", I was inspired by Steve Gull's four-over-head-projector-transparencies proof sketch http://www.mrao.cam.ac.uk/~steve/maxent2009/images/bell.pdf

Richard wrote:An intriguing possible direction involving indeed the Fourier transform is suggested by some lecture slides by Steve Gull, going back 35 years now. On his “MaxEnt 2009” web-page http://www.mrao.cam.ac.uk/~steve/maxent2009/, under the heading "Quantum Acausality and Bell’s Theorem", Steve writes
Steve Gull wrote:Many years ago (about 1984), I used to give a Mathematical Physics course to the Part II students. I illustrated the quantum paradox covered by Bell’s theorem by showing that you can’t program two independently running computers to mimic the results of spin measurements on two spin-1/2 particles in a singlet state. I believe this demonstration is actually better than Bell’s original argument.

This is a lovely proof of Bell’s theorem using the fact that the Fourier transform of the correlation function ρ has to equal the expected squared absolute value of the Fourier transform of the random function A. The actual correlation function only has three non-zero Fourier coefficients. However the Fourier transform of any realisation of A must have infinitely many non-zero coefficients, since otherwise it could not have any jumps. Since their absolute values get squared before averaging, there is no way that all but three can vanish.


In https://arxiv.org/abs/quant-ph/0301059, "Time, Finite Statistics, and Bell's Fifth Position", I wrote
Richard wrote:Jaynes (1989) does not make any attempt to specify what he understands by a local model, and expresses great surprise at very new results of Steve Gull, presented at the same conference as Jaynes’ own paper, in which a computer network metaphor is introduced and where it is shown that the singlet correlations cannot be simulated on such a network. Reference: Jaynes, E. T. (1989). Clearing up mysteries—the original goal. In J. Skilling (Ed.), Maximum Entropy and Bayesian Methods, Dordrecht, pp. 1–27. Kluwer.
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby gill1109 » Fri Jun 07, 2019 11:20 pm

I have the eBook (Jaynes 1989) if anyone would like me to look anything up for them. In Amsterdam (CWI Library) there's a hard copy which also has some pages of discussion. But I can't find them in the eBook. I'll have to make a pilgrimage to Amsterdam...

The book's publisher has changed ...
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby Joy Christian » Fri Jun 07, 2019 11:29 pm

Steve Gull wrote:Many years ago (about 1984), I used to give a Mathematical Physics course to the Part II students. I illustrated the quantum paradox covered by Bell’s theorem by showing that you can’t program two independently running computers to mimic the results of spin measurements on two spin-1/2 particles in a singlet state. I believe this demonstration is actually better than Bell’s original argument.

Gull's is not a "proof" of anything, let alone of a misnomer called "Bell's theorem." It is more like what John Weeler would call "an idea of an idea"; and what I would call "wishful thinking."

What is more, it has nothing to do with how the Bell-test experiments are done. To understand how silly "Bell's theorem" is, please read my short paper: https://arxiv.org/abs/1704.02876.

And whatever Gull's argument is, it is necessarily wrong, because a local-realist model of the singlet correlations based on quaternions already exists: https://arxiv.org/pdf/1103.1879.pdf.

***
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby gill1109 » Sat Jun 08, 2019 6:17 am

Joy Christian wrote:Steve Gull wrote
Many years ago (about 1984), I used to give a Mathematical Physics course to the Part II students. I illustrated the quantum paradox covered by Bell’s theorem by showing that you can’t program two independently running computers to mimic the results of spin measurements on two spin-1/2 particles in a singlet state. I believe this demonstration is actually better than Bell’s original argument.

Gull's is not a "proof" of anything, let alone of a misnomer called "Bell's theorem." It is more like what John Weeler would call "an idea of an idea"; and what I would call "wishful thinking."

It’s indeed an idea of an idea. The original idea of the computer challenge. The original idea that Bell’s theorem is a theorem in the theory of (classical) distributed computing. It also is the idea of a completely different kind of proof- one based on elementary facts from Fourier analysis. Not for precocious schoolboys I think, but for average bachelor students in any STEM discipline. I took it a whole lot further, as also did some others. Even Joy Christian nowadays does computer simulation to support his physics.
Joy Christian wrote:What is more, it has nothing to do with how the Bell-test experiments are done. To understand how silly "Bell's theorem" is, please read my short paper: https://arxiv.org/abs/1704.02876.

In Bell experiments, experimenters measure correlations. Gull investigates what those correlations might tell us about the actual physics going on in those experiments.
Joy Christian wrote:And whatever Gull's argument is, it is necessarily wrong, because a local-realist model of the singlet correlations based on quaternions already exists: https://arxiv.org/pdf/1103.1879.pdf.

You’re assuming that your maths is correct. I think my maths is correct. Nature tells us that the quantum correlations are for real, or at least, bloody close to real. I think everyone agrees on that, now. So we are left arguing about maths and about logic.
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby gill1109 » Sat Jun 08, 2019 7:39 am

BTW - to Fred - I do appreciate the title of this thread, but my friends call me “Richard” not “Prof Gill”, though maybe not immediately. After a few hours/days/years perhaps. You can also just take the middle way and call me “Richard Gill”. It’s a label. It’s neutral.
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby FrediFizzx » Sat Jun 08, 2019 3:59 pm

I took another look at this paper and I am still having trouble figuring out what it is that you are actually presenting. What are the math formulas for all those graphs you present? Can one of them be presented in a complete equation string?
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby gill1109 » Sat Jun 08, 2019 8:12 pm

FrediFizzx wrote:I took another look at this paper and I am still having trouble figuring out what it is that you are actually presenting. What are the math formulas for all those graphs you present? Can one of them be presented in a complete equation string?

The whole thing is drawn by this little R script:

Code: Select all
oneplot <- function() {
    if (nswitch%%2 == 0)  times <- c(times,1)
    timesplus <- c(times,times+1,times+2,times+3)
    count <- function(t,d) {
        sum(timesplus > t & timesplus <= t+d)
    }
    points <- seq(from=0,to=1,by=0.01)
    numbers <- outer(data,points,Vectorize(count))
    corr <- 2*(apply(numbers%%2,2,sum)/1000)-1
    correlation <- c(corr,corr[100:1])
    difference <- pi*(c(points,1+points[2:101]))
    plot(difference,correlation,
        type="l",bty="n",ann=FALSE,xaxt="n",yaxt="n")
    lines(c(0,2*pi,2*pi,0,0),c(+1,+1,-1,-1,+1))
    abline(h=0)
    lines(difference,-cos(difference),col="blue")
    lines(c(0,pi,2*pi),c(-1,+1,-1),col="red")
}

nswitch <- 4
set.seed(11091951)
par(mfrow=c(3,4),oma=c(0,0,0,0),mar=c(0,0,0,0))
for( i in (1:12) ) {
    times <- sort(runif(nswitch))
    data <- 2*runif(1000)
    oneplot()
}


I'm firing up Rstudio right now to play with it again. But it's just a completely ordinary LHV model with functions A(a, lambda) and B(b, lambda) where lambda is an angle between 0 and 2 pi, a and b are angles between 0 and 2 pi, B = - A, and A is a step function taking the values +1 and -1 only. Its values on pi to 2 pi are just the values on 0 to pi, reversed. So for instance you pick four angles strictly between 0 and pi. That forms five intervals partitioning [0, pi]. "Colour" those intervals alternatingly black and white. So you get: black, white, black, white, black. Now repeat from pi to 2 pi: white, black, white, black, white. So the whole unit circle has been coloured alternatingly black and white in exactly 10 segments. And two opposite points always have opposite colours.

Now the model is: given two settings a, b (points on the unit circle), and given a point lambda uniformly at random on the unit circle, the outcomes of the two spin mesurements are the colour at point a + lambda, and the reverse of the colour at point b + lambda
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby FrediFizzx » Sat Jun 08, 2019 8:17 pm

Can you put that into a regular equation of the form A(a, lambda) = ??? , etc? I'm not going to try to decipher R code.
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby FrediFizzx » Sat Jun 08, 2019 8:48 pm

FrediFizzx wrote:Can you put that into a regular equation of the form A(a, lambda) = ??? , etc? I'm not going to try to decipher R code.

IOW, show the measurement functions as regular equations then show the average of the product of the measurement functions. Then do the code. It will be more understandable that way.
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby gill1109 » Sun Jun 09, 2019 1:17 am

FrediFizzx wrote:
FrediFizzx wrote:Can you put that into a regular equation of the form A(a, lambda) = ??? , etc? I'm not going to try to decipher R code.

IOW, show the measurement functions as regular equations then show the average of the product of the measurement functions. Then do the code. It will be more understandable that way.

Take an even number and angles . Colour the segments "black", "white", ..., "black". If there is just one segment , and it is coloured black. Colour in complementary way: are coloured "white", "black", ..., "white". The colour assigned to end-points does not matter. We have now coloured the entire unit circle with our two colours "white" and "black" except that we didn't determine the colours of the finitely many points on the boundaries between intervals of fixed colour. Each point is opposite a point of the opposite colour, so the total length of white segments and the total length of black segments are equal. Now give the circle a random rotation chosen uniformly between 0 and . Define according to whether the colour of the randomly rotated circle at point is black or white. Define .. A and B aren't defined at finitely many angles, but we don't have to worry about this. The probability is zero that A is undefined at any particular given point.
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby FrediFizzx » Sun Jun 09, 2019 1:48 am

gill1109 wrote:
FrediFizzx wrote:
FrediFizzx wrote:Can you put that into a regular equation of the form A(a, lambda) = ??? , etc? I'm not going to try to decipher R code.

IOW, show the measurement functions as regular equations then show the average of the product of the measurement functions. Then do the code. It will be more understandable that way.

Take an even number and angles . Colour the segments "black", "white", ..., "black". If there is just one segment , and it is coloured black. Colour in complementary way: are coloured "white", "black", ..., "white". The colour assigned to end-points does not matter. We have now coloured the entire unit circle with our two colours "white" and "black" except that we didn't determine the colours of the finitely many points on the boundaries between intervals of fixed colour. Each point is opposite a point of the opposite colour, so the total length of white segments and the total length of black segments are equal. Now give the circle a random rotation chosen uniformly between 0 and . Define according to whether the colour of the randomly rotated circle at point is black or white. Define .. A and B aren't defined at finitely many angles, but we don't have to worry about this. The probability is zero that A is undefined at any particular given point.

Well, you need to put that in a regular measurement function equation so one can do the product calculation. If you can't do that, then people are going to have a hard time understanding it. You asked for help with this paper. That is my suggestion to make it better. Do the math. It may be hard, but necessary. Connect everything you said there and put them into the two measurement function equations. You should be able to just reverse engineer the R code for it.
.
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby gill1109 » Sun Jun 09, 2019 7:45 am

It is trivial to write out a formula with computer algebra. And trivial to draw the curve, after that. But not interesting I think. Instead I computed many such correlation functions by simulation and drew pictures of them. I will write an R shiny interactive web page. My description tells you exactly how to do it yourself.

The pictures show that the triangle wave is misleading, atypical.
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby FrediFizzx » Sun Jun 09, 2019 12:20 pm

Sorry, but I am not going to bother "to do it myself." Too busy with other things. It is up to the writer of a paper to make the paper as understandable as possible for the reader.
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby gill1109 » Sun Jun 09, 2019 9:53 pm

FrediFizzx wrote:Sorry, but I am not going to bother "to do it myself." Too busy with other things. It is up to the writer of a paper to make the paper as understandable as possible for the reader.

Sure. The paper isn't finished. I'm looking for a co-author who will help me make it more accessible, and no doubt, hopefully indeed, add their own ideas. No hurry. Obviously, it would have to be someone who kind of "gets the idea" based on reading the current version, despite its inadequacies.

It's not my first priority at the moment, and life is short... I have to prepare a quantum talk this week at Växjö with some new and exciting results, and also prepare a forensic science / nonparametric Bayes talk next week in Leiden, different topic, but also with new and exciting results. Then there are a lot of other ongoing projects which need attention. I will probably leave this particular item to my scientific heirs. They can publish it in an "in Memoriam" to me.
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby FrediFizzx » Sun Jun 09, 2019 10:19 pm

Well, good luck with your talks. And good luck with finding a co-author. I'm still scratching my head a bit with what significance it really has. I will look at it some more when I have time.
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Re: Prof. Gill's Triangle wave versus the Cosine paper

Postby gill1109 » Tue Jun 11, 2019 7:44 am

BTW I have put a draft of my Växjö slides (talk: tomorrow, end of afternoon) on slideshare:

https://www.slideshare.net/gill1109/yet-another-statistical-analysis-of-the-data-of-the-loophole-free-experiments-of-2015

Actually the slides contain about three sub-talks. I have to prune out quite a few pages, but also add quite a few as well. If anyone is interested in talking about them I put start a new thread, when I can put definitive slides are on-line, probably tomorrow evening after the conference dinner.
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