Joy Christian's colourful exploding balls experiment

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

Re: Joy Christian's colourful exploding balls experiment

Postby gill1109 » Sat Apr 19, 2014 6:32 am

I have rewritten my R script for evaluating the experimental results and determining the outcome of the bet.

http://rpubs.com/gill1109/Bet_v2

This version takes four disjoint subsamples from the N runs, each correlation is calculated separately on the data of each subsample.

Joy: is this what you want, then? It contradicts the instructions in your experimental paper, but if it makes you happy ...

Michel Fodje, John Reed, Fred Dieter ... please write translations into Mathematica, Python, Java... so that everyone can agree on the protocol.

Anyone who is able to simulate Joy's model can simulate the two data sets and run them through the evaluation script.
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Re: Joy Christian's colourful exploding balls experiment

Postby Joy Christian » Sat Apr 19, 2014 6:57 am

gill1109 wrote:Christian's experimental paper makes completely clear that every correlation is based on the whole sample. Will he revise that paper?


What a load of hogwash you have uploaded?

Here is my paper if anyone is interested in checking. The R code you have written for the bet does not at all follow what I have written in my paper---see, especially, equation (16). If your code did follow my instructions to the word, then you would have calculated only one E(a, b) in it. That would have been enough. But instead you have been insisting on the idiotic CHSH protocol, which is completely irrelevant to my experimental set up. I agreed with your insistence, but not so that you can do whatever you please. As I pointed out, the R code you have written is deeply flawed. It does not respect my condition of calculating each correlation separately, as demanded in the equation (16) of my paper. According to this equation, each correlation must be calculated on a separate set of particles. Where does it say in my paper that they must be calculated on the same set of particles? You can generate one list of vectors but you must sample without replacement so that no pair of vectors contributes to more than one correlation. Only then will it be equivalent to four separate sets, as both Michel and I have been insisting:

Joy Christian wrote:For the record, let me repeat that equation (16) of my experimental paper describes exactly how the expectation values E(a, b), E(a', b), E(a, b'), and E(a', b') are to be computed in my proposed experiment. Four separate sums are to be calculated as follows

E(a, b) = 1/N Sum_j A_j B_j ,

E(a, b') = 1/N Sum_j A_j B'_j ,

E(a', b) = 1/N Sum_j A'_j B_j ,

and

E(a', b') = 1/N Sum_j A'_j B'_j .

It is a matter of indifference whether N here is chosen to be the same or different for each of the four alternatives.
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Re: Joy Christian's colourful exploding balls experiment

Postby Heinera » Sat Apr 19, 2014 8:35 am

An alternative version of the bet could be that Richard only computes one correlation E(a,b) on the whole set, but he is free to pick any values of a and b he wants. He decides on values of a and b after he has received the data files from the experiment (the experiment doesn´t know a thing about detector settings anyway).

If the computed correlation differs from the QM correlation by more than +/- 0.2, we agree that QM correlations are not reproduced and Richard wins.

Even Michel seems to think that QM correlations could be achieved this way.
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Re: Joy Christian's colourful exploding balls experiment

Postby Joy Christian » Sat Apr 19, 2014 9:02 am

Heinera wrote:An alternative version of the bet could be that Richard only computes one correlation E(a,b) on the whole set, but he is free to pick any values of a and b he wants. He decides on values of a and b after he has received the data files from the experiment (the experiment doesn´t know a thing about detector settings anyway).

If the computed correlation differs from the QM correlation by more than +/- 0.2, we agree that QM correlations are not reproduced and Richard wins.

Even Michel seems to think that QM correlations could be achieved this way.


There will be no alternative version of the bet.
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Re: Joy Christian's colourful exploding balls experiment

Postby Heinera » Sat Apr 19, 2014 10:13 am

Joy Christian wrote:
There will be no alternative version of the bet.

Ok. Have you reached a conclusion as to what the original version of the bet looks like? Does Richard´s last version of the R script represent that?
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Re: Joy Christian's colourful exploding balls experiment

Postby Joy Christian » Sat Apr 19, 2014 10:27 am

Heinera wrote:
Joy Christian wrote:
There will be no alternative version of the bet.

Ok. Have you reached a conclusion as to what the original version of the bet looks like? Does Richard´s last version of the R script represent that?


I don't see sampling without replacement in the latest code by Richard. But I am not a programmer. In fact, I know nothing about programming.
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Re: Joy Christian's colourful exploding balls experiment

Postby Heinera » Sat Apr 19, 2014 10:55 am

Joy Christian wrote:
Heinera wrote:
Joy Christian wrote:
There will be no alternative version of the bet.

Ok. Have you reached a conclusion as to what the original version of the bet looks like? Does Richard´s last version of the R script represent that?


I don't see sampling without replacement in the latest code by Richard. But I am not a programmer. In fact, I know nothing about programming.

There is certainly sampling without replacement in Richard ´s code. Any pair of vectors is used only once, and then thrown away and never used again.

Second, but unrelated question: How can one go through a PhD program in physics and not be exposed to some computer programming? Just curious, because from my own experience, that would be impossible for a math PhD.
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Re: Joy Christian's colourful exploding balls experiment

Postby Joy Christian » Sat Apr 19, 2014 11:04 am

Heinera wrote:How can one go through a PhD program in physics and not be exposed to some computer programming? Just curious, because from my own experience, that would be impossible for a math PhD.


It is not necessary to know programing to practice either physics or mathematics. Roger Penrose, Stephen Hawking, Edward Witten, Michael Atiyah, Grigori Perelman, and Andrew Wiles seem to have done just fine without knowing any programing.
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Re: Joy Christian's colourful exploding balls experiment

Postby Heinera » Sat Apr 19, 2014 11:45 am

Joy Christian wrote:
Heinera wrote:How can one go through a PhD program in physics and not be exposed to some computer programming? Just curious, because from my own experience, that would be impossible for a math PhD.


It is not necessary to know programing to practice either physics or mathematics. Roger Penrose, Stephen Hawking, Edward Witten, Michael Atiyah, Grigori Perelman, and Andrew Wiles seem to have done just fine without knowing any programing.

Trust me: All of those six gentlemen know how to program.
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Re: Joy Christian's colourful exploding balls experiment

Postby Joy Christian » Sat Apr 19, 2014 11:54 am

Heinera wrote:
Joy Christian wrote:
Heinera wrote:How can one go through a PhD program in physics and not be exposed to some computer programming? Just curious, because from my own experience, that would be impossible for a math PhD.


It is not necessary to know programing to practice either physics or mathematics. Roger Penrose, Stephen Hawking, Edward Witten, Michael Atiyah, Grigori Perelman, and Andrew Wiles seem to have done just fine without knowing any programing.

Trust me: All of those six gentlemen know how to program.


Trust me, they don't. I happen to know at least two of those gentlemen personally. An architect does not need to know plumbing to build a building.
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Re: Joy Christian's colourful exploding balls experiment

Postby Heinera » Sat Apr 19, 2014 3:26 pm

Well, well. I don´t see much point in pushing that argument any further. Anyway, you have somehow agreed to enter into a bet that is of a kind where a lack of programming knowledge is a clear disadvantage to you. I venture to guess that none of the mentioned six gentlemen where ever in the same situation; i.e, they never had to make a public monetary bet to get their theory through.

Edit: Hawking jokingly did so. But the stake was not money.
Last edited by Heinera on Sat Apr 19, 2014 3:40 pm, edited 2 times in total.
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Re: Joy Christian's colourful exploding balls experiment

Postby Joy Christian » Sat Apr 19, 2014 3:31 pm

Heinera wrote:Well, well. I don´t see much point in pushing that argument any further. Anyway, you have somehow agreed to enter into a bet that is of a kind where a lack of programming knowledge is a clear disadvantage to you. I venture to guess that none of the mentioned six gentlemen where ever in the same situation; i.e, they never had to make a public monetary bet to get their theory through.


http://www.theguardian.com/science/2014 ... t-big-bang
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Re: Joy Christian's colourful exploding balls experiment

Postby gill1109 » Mon Apr 21, 2014 12:16 am

Joy Christian wrote:
Heinera wrote:
Joy Christian wrote:
There will be no alternative version of the bet.

Ok. Have you reached a conclusion as to what the original version of the bet looks like? Does Richard´s last version of the R script represent that?


I don't see sampling without replacement in the latest code by Richard. But I am not a programmer. In fact, I know nothing about programming.


Joy, you have demanded an alternative version of the bet. You earlier agreed with my script http://rpubs.com/gill1109/Bet, which implemented to the letter, the instructions in your paper http://arxiv.org/abs/0806.3078. I asked your supporters to translate it into some other languages (we already have Excel and Perl, and John Reed had just embarked on a Mathematica version, when you withdrew. Now you want something else.

This is your last chance. I think that what you now want, contrarily to your own paper http://arxiv.org/abs/0806.3078, the following: http://rpubs.com/gill1109/Bet_v2

You say you don't see any sample without replacement here.

The line
Code: Select all
Sample <- sample(c(1, 2, 3, 4), N, replace = TRUE)

produces a vector of length N consisting of a random sample with replacement from the four numbers 1, 2, 3, and 4. Thus the numbers "1", ..., "4" indicate membership of four disjoint subsamples. Then four sets of lines like
Code: Select all
A <- sign(cos(AliceTheta[Sample == 1] - Alpha))
B <- - sign(cos(BobTheta[Sample == 1] - Beta))

(and similarly with 2, 3 and 4) pulls out the lines of the original data file corresponding to samples 1, 2, 3 and 4 and calculate the outcomes of the measurements in directions Alpha and Beta (and thereafter, the sample correlation), exactly as in http://arxiv.org/abs/0806.3078, only now using four sub-samples instead of each time the whole sample. (Why the result will differ by more than something in the order of 1 / sqrt N, i.e., by more that regular sampling variation, is a total mystery to me).

Please get some of your supporters to translate this code into Mathematica and Python so we can test all versions on various test data sets and all agree that they do the same thing. Your supporters will thereby be able to assure you that they do exactly what you want them to do.
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Re: Joy Christian's colourful exploding balls experiment

Postby gill1109 » Mon Apr 21, 2014 12:30 am

Joy Christian wrote:Now consider a large ensemble of such balls, identical in every respect except for the relative locations of the two lumps (affixed randomly on the inner surface of each shell). The balls are then placed over a heater—one at a time—at the center of an EPR-Bohm type setup [6], with the common plane of their shells held perpendicular to the horizontal direction of the setup. Although initially at rest, a slight increase in temperature of each ball will eventually eject its two shells towards the observation stations, situated at a chosen distance in the mutually opposite directions. Instead of selecting the directions a and b for observing spin components, however, one or more contact-less rotational motion sensors—capable of determining the precise direction of rotation—are placed near each of the two stations, interfaced with a computer. These sensors will determine the exact direction of the angular momentum lambda_j (or −lambda_j) for each shell, without disturbing them otherwise, at a designated distance from the center. The interfaced computers can then record this data, in the form of a 3D map of all such directions. Once the actual directions of the angular momenta for a large ensemble of shells on both sides are fully recorded, the two computers are instructed to randomly choose the reference directions, a for one station and b for the other station—from within their already existing 3D maps of data—and then calculate the corresponding dynamical variables sign (lambda_j · a) and sign (−lambda_j · b). This “delayed choice” of a and b will guarantee that the conditions of parameter independence and outcome independence are strictly respected within the experiment [2]. It will ensure, for example, that the local outcome sign (lambda_j · a) remains independent not only of the remote parameter b, but also of the remote outcome sign (−lambda_j · b). If in any doubt, the two computers can be located at a sufficiently large distance from each other to ensure local causality while selecting a and b. The correlation function for the bomb fragments can then be calculated using the formula

E(a, b) = 1/N sum_{j =1}^N {sign (lambda_j · a)} {sign(−lambda_j · b)}, (16)

where N is the number of trials.


Apparently written by Joy Christian (Department of Physics, University of Oxford), http://arxiv.org/pdf/0806.3078v2.pdf, page 4. Am I missing a later version?

The only change I asked from Joy, and he accepted, was that the bet which we made is determined by just four of these correlations, denoted E(0, 45), E(0, 135), E(90, 45), E(90, 135). The correspondence between the angles 0, 45, 90, 135 degrees and directions a, b is that the four directions a, b are in the same equatorial plane. The angles alpha and beta are the longitude of four points on the equator. Latitude = 0 degrees.

Joy's theory predicts three correlations equal to - 0.7 and one equal to + 0.7. Bell's theory predicts three correlations equal to - 0.5 and one correlation equal to + 0.5. Supposing that the sign pattern comes out as both expect, I have proposed that the bet is settled on whether the average value of the four absolute correlations is larger or smaller than 0.6. The exact value 0.6 would mean that the adjudicators have to toss a coin to choose a winner.

In my new R code http://rpubs.com/gill1109/Bet_v2 I randomly assign each of the N runs to just one of the four pairs of settings (0, 45), (0, 135), (90, 45), (90, 135). In my opinion we would get identical results if this allocation was done in distributed fashion: one script chooses between alpha = 0 and 90 for each of the N runs, another script chooses between 45 and 135 for each of the N runs. One could then also calculate sign( a. lambda_k) and sign( b . -lambda_k ) by two separate computer programs, generating traditional CHSH style data sets: N runs, giving for Alice N pairs of a setting (0 or 90) and an outcome (+/- 1) , and for Bob N pairs of a setting (45 or 135) and an outcome (+/- 1). Then finally one further computer script computes the four correlations E(alpha, beta) for each of the four disjoint subsets of the runs, defined by the possible pairs of setting values.

Joy's own computer experts are welcome to write the Mathematia or Python versions of the scripts which do this job. I will test that they do the same as my script. Once we are all agreed, we can offer them to the adjudicationg committee. The Vaxjo conference is coming soon.

Conference site:
http://lnu.se/subjects/mathematics/conferences/quantum-theory-from-problems-to-advances---qtpa/qtpa?l=en

Conference poster:
http://lnu.se/polopoly_fs/1.97904!QTPA%20affisch%20140609.pdf

The adjudicators will all be there. I plan to announce our bet. If it has been called off, I will announce that it is called off. If we are still haggling about whether all N runs are used for all four correlations, or whether four disjoint random samples will be used, I will announce that fact.

Maybe Joy should come too.
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Re: Joy Christian's colourful exploding balls experiment

Postby Joy Christian » Mon Apr 21, 2014 2:36 am

On the FQXi blog Richard Gill wrote: “CHSH inequality is a dead issue. Bell's so-called theorem is a dead issue.”

Yes! Finally something I can agree with.

He continues:

“The real question is whether experiment (and simulation) reproduces the theoretical correlation surface which can be derived both by QM and by Christian-LHV:

rho(alpha, beta) = - cos(alpha - beta).”

Yes, again.

He continues:

“Interestingly, the experiment described in the paper http://arxiv.org/abs/0806.3078 (author J. Christian) is very different from all conventional experiments. The paper describes how N runs are performed, resulting in N sets of video film; then, for k = 1, ... , N,two directions u_k and - u_k are computed by analysing the video footage of the k'th run....”

So far so good, but alas he continues:

“... and after that (back home, so to speak), settings alpha and beta are repeatedly chosen at random and A_k(alpha) is calcuated, A_k(alpha) = sign(a . u_k) where a is the direction corresponding to alpha. Similarly B_k(beta) = sign(b . - u_k).”

Wrong!

There is nothing random about alpha and beta. They are settings chosen only once! They are fixed angles or vectors. What is random is the “hidden variable”, u_k in his notation.

Does he now see his error?

Does he now see his error in what he wrote next?

“http://arxiv.org/abs/0806.3078 makes perfectly clear that the same set of N video films of N exploding balls is used to compute two single sets of directions: the directions u_k and - u_k. After that, according to the formulas in the paper, experimental correlations E(alpha, beta) are determined on the basis of the complete set of N runs at a fine by random cloud of points alpha, beta.”

Is he being sneaky again, or is this a genuine oversight on his part?
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Re: Joy Christian's colourful exploding balls experiment

Postby gill1109 » Mon Apr 21, 2014 3:13 am

I think there are two J. Christian's. One who posts on arXiv, one who posts in this forum.

No wonder it is difficult getting him / them to agree to the terms of a bet.

Perhaps the J. Christian who writes here can get his friends to study this new script:

http://rpubs.com/gill1109/Bet_v2

Give us translations into Mathematica and Python. Is it OK for the bet?

Christian has now moved the goal-posts once, dramatically. Will he do it again?

This is what the other Joy Christian wrote on http://arxiv.org/abs/0806.3078:
Joy Christian wrote:Now consider a large ensemble of such balls, identical in every respect except for the relative locations of the two lumps (affixed randomly on the inner surface of each shell). The balls are then placed over a heater—one at a time—at the center of an EPR-Bohm type setup [6], with the common plane of their shells held perpendicular to the horizontal direction of the setup. Although initially at rest, a slight increase in temperature of each ball will eventually eject its two shells towards the observation stations, situated at a chosen distance in the mutually opposite directions. Instead of selecting the directions a and b for observing spin components, however, one or more contact-less rotational motion sensors—capable of determining the precise direction of rotation—are placed near each of the two stations, interfaced with a computer. These sensors will determine the exact direction of the angular momentum lambda_j (or −lambda_j) for each shell, without disturbing them otherwise, at a designated distance from the center. The interfaced computers can then record this data, in the form of a 3D map of all such directions. Once the actual directions of the angular momenta for a large ensemble of shells on both sides are fully recorded, the two computers are instructed to randomly choose the reference directions, a for one station and b for the other station—from within their already existing 3D maps of data—and then calculate the corresponding dynamical variables sign (lambda_j · a) and sign (−lambda_j · b). This “delayed choice” of a and b will guarantee that the conditions of parameter independence and outcome independence are strictly respected within the experiment [2]. It will ensure, for example, that the local outcome sign (lambda_j · a) remains independent not only of the remote parameter b, but also of the remote outcome sign (−lambda_j · b). If in any doubt, the two computers can be located at a sufficiently large distance from each other to ensure local causality while selecting a and b. The correlation function for the bomb fragments can then be calculated using the formula

E(a, b) = 1/N sum_{j =1}^N {sign (lambda_j · a)} {sign(−lambda_j · b)}, (16)

where N is the number of trials.


Apparently he actually meant, that the whole experiment has to be repeated, over and over again, for each new pair of reference directions! Moreover we are going to have to wait an awful long time till we coincidentally hit on the four pairs of reference directions which we agreed on. In the equatorial plane, 0 and 90 degrees longitude for Alice, 45 and 135 for Bob (four combinations). Seems hard to believe that this was the intention ... still if we go on long enough, eventually we will be doing the experiment with the four pairs (a, b) which are needed to resolve the bet. Seems an awful waste of resources.

Actually, reading his text closely, he says we are only allowed to compute correlations for pairs of angles "from within their already existing 3D maps of data". I always found this really weird. But I imagined that if N is large we will have observed directions for Alice close to the directions I want for Alice, and directions for Bob close to the directions I want for Bob (we have two 3D maps, right?). I imagined that since the maps *already existed* we were always talking about the same maps based on the same video footage of the same N exploding balls, for any correlations we want to calculate. Silly me.
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Re: Joy Christian's colourful exploding balls experiment

Postby Joy Christian » Mon Apr 21, 2014 3:34 am

"Randomly" is not the same word as "repeatedly".

In the context of this sentence,

"Once the actual directions of the angular momenta for a large ensemble of shells on both sides are fully recorded, the two computers are instructed to randomly choose the reference directions, a for one station and b for the other station---from within their already existing 3D maps of data",

"randomly" means "arbitrarily."

Anyone who has any understanding of Bell's local-realistic framework knows that what is random is the hidden variable (lambda in his notation), NOT the detector directions. They are chosen "randomly" in the sense that they are chosen "arbitrarily." We are not seeking a distribution of a or b. They do not tell us anything about the geometry of the physical space, because WE are choosing them, arbitrarily, or "freely." The distribution we seek in the experiment is that of the hidden variable, lambda. It is the distribution of lambda which will tell us something about the geometry and topology of the physical space. The distributions (or not) of a and b are put in by us, by hand. They will tell us only that which we have put in.
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Re: Joy Christian's colourful exploding balls experiment

Postby Joy Christian » Mon Apr 21, 2014 3:45 am

gill1109 wrote:Apparently he actually meant, that the whole experiment has to be repeated, over and over again, for each new pair of reference directions! Moreover we are going to have to wait an awful long time till we coincidentally hit on the four pairs of reference directions which we agreed on. In the equatorial plane, 0 and 90 degrees longitude for Alice, 45 and 135 for Bob (four combinations). Seems hard to believe that this was the intention ... still if we go on long enough, eventually we will be doing the experiment with the four pairs (a, b) which are needed to resolve the bet. Seems an awful waste of resources.

Actually, reading his text closely, he says we are only allowed to compute correlations for pairs of angles "from within their already existing 3D maps of data". I always found this really weird. But I imagined that if N is large we will have observed directions for Alice close to the directions I want for Alice, and directions for Bob close to the directions I want for Bob (we have two 3D maps, right?). I imagined that since the maps *already existed* we were always talking about the same maps based on the same video footage of the same N exploding balls, for any correlations we want to calculate. Silly me.


Are you deliberately being dumb just to frustrate me? We are talking about actual spin directions, say u_k. And then a reference observation direction a. The component of spin is to be measured about a. How does one do that? Well, Bell already told us how in his 1964 paper. See his equation (9). We have to compute

A(a, u_k) = sign(a.u_k).

We are doing just Bell's own local model. if you are confused about Bell's own equation (9) then you shouldn't be in this business.
Last edited by Joy Christian on Mon Apr 21, 2014 4:20 am, edited 1 time in total.
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Re: Joy Christian's colourful exploding balls experiment

Postby gill1109 » Mon Apr 21, 2014 4:02 am

Joy Christian wrote:Are you deliberately being dumb just to frustrate me? We are talking about actual spin directions, say u_k. And then a reference observation direction a. The component of spin is to measured about a. How does one do that? Well, Bell already told us how in his 1964 paper. See his equation (9). We have to compute A(a, u_k) = sign(a.u_k). We are doing just Bell's own local model. if you are confused about Bell's own equation (9) then you shouldn't be in this business.

We are not "doing just Bell's own local model". We are following Joy Christian's instructions concerning Joy Christian's experiment. There is a Joy Christian speaking today in this forum and there is a Joy Christian who authored a paper http://arxiv.org/abs/0806.3078 on arXiv. If they disagree, we seem to be in difficulties.

But great, seems we are back with my original "variant 1" script http://rpubs.com/gill1109/Bet. Suits me fine. Joy Christian today does agree, after all, with Joy Christian of 2008.

John Reed already converted http://rpubs.com/gill1109/Bet to Mathematica so your Mathematica-savvy supporters can check it is OK. But you complained - Michel Fodje got you really scared - and then you demanded "variant 2". So I wrote out a script for variant 2 http://rpubs.com/gill1109/Bet_v2, , and I also asked John Reed to convert it to Mathematica, but now you are telling me this is a waste of time.

Your choice. Variant 1 or Variant 2. It always seemed to me that http://arxiv.org/abs/0806.3078 calls for Variant 1. But maybe I'm just being dumb, after all, I am only a statistician (even though I was taught by Stephen Hawking, among others).
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Re: Joy Christian's colourful exploding balls experiment

Postby gill1109 » Mon Apr 21, 2014 4:42 am

Mathematica version of http://rpubs.com/gill1109/Bet (variant 1).
Code: Select all
(* Richard Gill's bet with Joy Christian  *)
(* Translated into \
Mathematica by John Reed  *)

aliceDirections =
  ReadList["http://www.math.leidenuniv.nl/~gill/AliceDirections.txt", \
{Real, Real}];

aliceTheta =
  Table[aliceDirections[[i, 1]], {i, Length[aliceDirections]}];

nAlice = Length[aliceTheta]


bobDirections =
  ReadList["http://www.math.leidenuniv.nl/~gill/BobDirections.txt", \
{Real, Real}];

bobTheta = Table[bobDirections[[i, 1]], {i, Length[bobDirections]}];

nBob = Length[bobTheta]


nAlice


If[nAlice != nBob, Print["Error: particle numbers don't match"],
 Print["Go Ahead"]]


mean[a_, b_, alpha_, beta_] :=
 Mean[-Sign[Cos[a - alpha]] Sign[Cos[b - beta]]] // N

(* First pair of measurement directions  *)
alpha = 0 Degree;
beta = 45 Degree;
E11 = mean[aliceTheta, bobTheta, alpha, beta];

(* Second pair of measurement directions  *)

alpha = 0 Degree;
beta = 135 Degree;
E12 = mean[aliceTheta, bobTheta, alpha, beta];

(* Third pair of measurement directions  *)
alpha = 90 Degree;
beta = 45 Degree;
E21 = mean[aliceTheta, bobTheta, alpha, beta];

(* Fourth pair of measurement directions  *)
alpha = 90 Degree;
beta = 135 Degree;
E22 = mean[aliceTheta, bobTheta, alpha, beta];

CHSH := E12 - E11 - E21 - E22

CHSH


If[CHSH > 2.4, Print["Congratulations, Joy"],
 Print["Congratulations, Richard"]]

Congratulations, Richard

(* Second experiment  *)


AliceTheta = RandomReal[{0, 360 Degree // N}, 1000];
BobTheta = -AliceTheta;

correlation = ConstantArray[0, 37];

a = Sign[Cos[AliceTheta]];

Do[
 delta = (i - 1) 10 Degree;
 b = -Sign[Cos[BobTheta - delta]];
 correlation[[i]] = Mean[a b], {i, 37}]

ListPlot[correlation, AxesLabel -> {Degrees, Correlation},
 DataRange -> {0, 360}]

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