Joy Christian's colourful exploding balls experiment

[gill1109]

I wonder why there should be any difference. Both versions result in a computer file containing N directions s_k, k=1,...,N. You were worried that with the squishy balls, we might not have a singlet state. And with all that image analysis software, there will be measurement errors as well. But with my computer generated directions, we certainly do have the singlet state: zero total angular momentum; each particle separately has completely random angular momentum.

Again, nice try.

I'm not trying anything. Just wondering how you see that there would be any difference. But it doesn't matter: these are your problems, not mine. I think you should be worried. But you aren't. You think you know something which you think I don't know. I think I know something which I think you don't know.

OK. However you do it, do you agree that the experiment plus computer processing of collected data generates

(A) one computer file containing N directions? [Alice's angular momentum direction; taken to be equal and opposite to Bob's]; or

(B) one computer file containing N pairs of directions? [Alice's and Bob's, not necessarily equal and opposite because of measurement errors and experimental imperfections]; or

(C) two computer files each containing N directions? [One file for Alice, one for Bob; but in one-to-one correspondence. i.e. k'th direction for Alice and k'th direction for Bob both come from the same, k'th run or (synonym) trial.

Next question: how will the directions be represented?

(D) Cartesian coordinates of unit vectors in R^3?

(E) Spherical coordinates of unit vectors in R^3?

(F) Otherwise?

As you well know, when we represent points on a unit sphere in a computer, we have to choose a coordinate frame. In this case, we have to choose two coordinate frames, one for Alice, one for Bob. This is a problem which I think you should be concerned with. I am going to specify measurement directions for Alice and Bob in the same coordinate system as you use, and relative to the same coordinate frames.

(G) Do you have any idea how large N might be?

(H) Will it be known in advance or only after the experiment?

In view of our agreement of how to calculate the correlations, I am not too concerned about non-detections or data-rejections or cleaning or whatever, even if it is of non-local nature (ie based on your experimenters' analysis of the trajectories of both squishy balls).

But I do understand that the experiment is going to generate some number of pairs of directions. So at the end of the day, when I talk about run (or trial) k, k= 1, ..., N, I am referring to the post-selected set of succesful runs which actually generated a pair of directions which you and the experimenter together are happy about.

[Ben6993]

Hi Richard

I feel out of my depth again, but will comment anyway.

Your computer file (A):
It might be clearer (at least to me) if you referred to Alice's ball's angular momentum direction rather than to Alice's angular momentum direction. Two paired balls will always have opposite angular momentum directions irrespective of Alice and Bob. From Alice's viewpoint, her recording of the her ball's spin can be +1 (eg for an apparent clockwise spin) or -1 (for an apparent anticlockwise spin) with equal probability depending on her looking at the ball, to record a measurement, from a random viewpoint. So the same two balls could lead to any one of the full range of four result pairs: (1,1), (1,-1), (-1,1) and (-1,-1), with the particular result obtained depending on the angles which Alice and Bob use for the measurements.

I think you are doing the right thing in trying to establish full details of analysis in advance, and also your suggestion of using random data to test and debug the analysis programs in advance is a good step. Dummy data will presumably not show the 4π torsion effect, but when the experiment is actually run, and the lab data generated and analysed by the same programs, the experimental results will miraculously show the 4π torsion effect. Well, I hope they will. That result would be so miraculous that having James Randi himself on the adjudicating panel might not be a bad thing.

I think the following is just about on topic. On Saturday night at 2 a.m. I was dreaming. My wife says said that I reached out and patted her arm and said "are you OK?" and woke her up. Then I was supposed to have said something about worrying that she was behaving like a particle! A few seconds later I was back asleep. She was not sure whether to laugh or be concerned about my rationality. From my viewpoint, I do remember a little about the dream. But the details are very fuzzy. I was worried that she might be spinning like a particle. Somehow. And I do remember patting her to check that she was OK. I blame this proposed experiment for provoking my weird dream!

[gill1109]

Hi Richard

I feel out of my depth again, but will comment anyway.

Your computer file (A):
It might be clearer (at least to me) if you referred to Alice's ball's angular momentum direction rather than to Alice's angular momentum direction. Two paired balls will always have opposite angular momentum directions irrespective of Alice and Bob. From Alice's viewpoint, her recording of the her ball's spin can be +1 (eg for an apparent clockwise spin) or -1 (for an apparent anticlockwise spin) with equal probability depending on her looking at the ball, to record a measurement, from a random viewpoint. So the same two balls could lead to any one of the full range of four result pairs: (1,1), (1,-1), (-1,1) and (-1,-1), with the particular result obtained depending on the angles which Alice and Bob use for the measurements.

I think you are doing the right thing in trying to establish full details of analysis in advance, and also your suggestion of using random data to test and debug the analysis programs in advance is a good step. Dummy data will presumably not show the 4π torsion effect, but when the experiment is actually run, and the lab data generated and analysed by the same programs, the experimental results will miraculously show the 4π torsion effect. Well, I hope they will. That result would be so miraculous that having James Randi himself on the adjudicating panel might not be a bad thing.

I think the following is just about on topic. On Saturday night at 2 a.m. I was dreaming. My wife says said that I reached out and patted her arm and said "are you OK?" and woke her up. Then I was supposed to have said something about worrying that she was behaving like a particle! A few seconds later I was back asleep. She was not sure whether to laugh or be concerned about my rationality. From my viewpoint, I do remember a little about the dream. But the details are very fuzzy. I was worried that she might be spinning like a particle. Somehow. And I do remember patting her to check that she was OK. I blame this proposed experiment for provoking my weird dream!

Great story! Very relevant I think to this thread.

About my first question to Joy (is he going to supply A, B, or C? [or something else again]): Yes your remark about nomenclature is well taken, indeed, I mean the angular momentum of the balls, not of the persons. You say "Two paired balls will always have opposite angular momentum directions irrespective of Alice and Bob". That may be true in an ideal world but we are doing an experiment. We are doing *the* experiment described in Joy's experimental paper. Moreover, the directions in which the two balls are spinning which I am talking about, are the directions supplied by computer analysis of video films or data from other sensors. That computer analysis is an integral part of the experiment. I won't get to see the "true" directions, but only some kind of measurements of those two directions, finite precision, encoded some way or other in one or two computer files. I just want to know what kind of a computer file is Joy going to hand over to the adjudicators, after the experiment (i.e. after the computer post-processing to recover spin directions), but before the calculation of binary spin outcomes and correlations between them. The adjudicators will do that calculation according to Joy's formulas and I am going to write out the programs which they might use. Joy and I and the adjudicators will discuss them and we only have a bet after we have all agreed on the "final calculation" (for the bet), which occurs *after* completion of the experiment.

You can discuss with Joy the sense or otherwise of the experiment!

Right now I just want to settle practical details of Joy and my bet, as far as they are important for me. And for that I want to resolve some possible ambiguities in his experiment's description.

Is he going to give me one file with one set of directions (Alice's balls' only) or one file with two sets of directions (Alice's and Bob's balls') or two files, one with each person's balls' directions?

What are those files going to look like? What kind of coordinate system...

[Joy Christian]

...do you agree that the experiment plus computer processing of collected data generates

(A) one computer file containing N directions? [Alice's angular momentum direction; taken to be equal and opposite to Bob's]; or

(B) one computer file containing N pairs of directions? [Alice's and Bob's, not necessarily equal and opposite because of measurement errors and experimental imperfections]; or

(C) two computer files each containing N directions? [One file for Alice, one for Bob; but in one-to-one correspondence. i.e. k'th direction for Alice and k'th direction for Bob both come from the same, k'th run or (synonym) trial.

The answer is (C). Two independent computer files, connected only by the numbers k.

Next question: how will the directions be represented?

(D) Cartesian coordinates of unit vectors in R^3?

(E) Spherical coordinates of unit vectors in R^3?

(F) Otherwise?

Does not make any difference. Let us just say (E).

(G) Do you have any idea how large N might be?

Yes. 10,000+. Ideally 1,000,000+.

(H) Will it be known in advance or only after the experiment?

The number of trials will depend on the experimental design and the budget. A team of specialist postdocs will be hired to do the experiment (applications welcome).

[gill1109]

Thanks!

I will now write an R script implementing your formulas for the correlations and CHSH quantity, based on your description of the data, and using my already declared pairs of settings for Alice and Bob. It's the easiest way for me to describe the algorithm. We can do the same calculations with any computer language - the adjudicators are going to run this part, anyway. If you want advice about whether there is any funny business in my R code or not, Chantal would surely be able to advise you very professionally. She could even prepare a Java version.

[Heinera]

It has always puzzled me that Joy Christian obviously never did any numerical experimentation on this last analysis part of the experiment, to check the feasibility of his ideas. After all, the input to this stage is just a list of (unit) vectors lambda (and a corresponding list of -lambda). How fancy can that get? Obviously, any permutation of this list will give the same computed correlations, so the only thing that is left to statistically distinguish between two lists is the distribution of the vectors. So why doesn't he just try this out, generating lists with 10 000 elements, trying a lot of different distributions? That's the good thing with the script Richard is writing now; Joy can actually try this approach. If he finds a list that nearly reproduces the four quantum correlations, there is no need to do the experiment; he will get the Nobel prize anyway. And if he experiences that no matter what list he throws into the program, the CHSH expression never exceeds 2, maybe he'll withdraw from the experiment all by himself.

[gill1109]

It has always puzzled me that Joy Christian obviously never did any numerical experimentation on this last analysis part of the experiment, to check the feasibility of his ideas. After all, the input to this stage is just a list of (unit) vectors lambda (and a corresponding list of -lambda). How fancy can that get? Obviously, any permutation of this list will give the same computed correlations, so the only thing that is left to statistically distinguish between two lists is the distribution of the vectors. So why doesn't he just try this out, generating lists with 10 000 elements, trying a lot of different distributions? That's the good thing with the script Richard is writing now; Joy can actually try this approach. If he finds a list that nearly reproduces the four quantum correlations, there is no need to do the experiment; he will get the Nobel prize anyway. And if he experiences that no matter what list he throws into the program, the CHSH expression never exceeds 2, maybe he'll withdraw from the experiment all by himself.

Sssshhhhh!

[Joy Christian]

It has always puzzled me that Joy Christian obviously never did any numerical experimentation on this last analysis part of the experiment, to check the feasibility of his ideas. After all, the input to this stage is just a list of (unit) vectors lambda (and a corresponding list of -lambda). How fancy can that get? Obviously, any permutation of this list will give the same computed correlations, so the only thing that is left to statistically distinguish between two lists is the distribution of the vectors. So why doesn't he just try this out, generating lists with 10 000 elements, trying a lot of different distributions? That's the good thing with the script Richard is writing now; Joy can actually try this approach. If he finds a list that nearly reproduces the four quantum correlations, there is no need to do the experiment; he will get the Nobel prize anyway. And if he experiences that no matter what list he throws into the program, the CHSH expression never exceeds 2, maybe he'll withdraw from the experiment all by himself.

Sssshhhhh!

Wisecracks.

[gill1109]

Here is a draft R script for the final stage of the bet.

It reads two data files. They should come out of the experiment.

To test my program, I read in two files from internet, which I created myself. Since I did not do a real experiment (whether quantum or with exploding balls) it does not give a good result. In fact the two sets of directions were completely independent of one another,
uniformly distributed.

They each contain just 100 directions, represented by spherical coordinates theta and phi, in mathematicans' notation ... the opposite from physicists, wikipedia tells me https://en.wikipedia.org/wiki/Spherical_coordinates. But what's in a name? The first coordinate is the azimuthal angle, the second is the polar angle (aka zenith angle). Both measured in radians. In the code, I call them "theta" and "phi", in that order. Ah, those good old Arabic mathematicians! (Who maybe got it from the earlier Indian civilizations ... the great chain of being. Wonderful. Where would we be without India, Arabia? (With a little help from Greece). Answer: not even in the Middle Ages).

I have also posted a R html notebook with the script so everyone can see a "test run" for themselves: http://rpubs.com/gill1109/Bet

Code: Select all

## I use mathematician's notation:
## theta is the azimuthal angle, phi is the polar angle, measured in radians.
## Since my measurement directions all lie in equatorial plane I just extract "theta"

AliceDirections <-
    read.table("http://www.math.leidenuniv.nl/~gill/AliceDirections.txt")
   
names(AliceDirections) <- c("theta", "phi")

head(AliceDirections)             ## N pairs theta, phi (N rows, 2 columns)

NAlice <- nrow(AliceDirections)

NAlice

AliceTheta <- AliceDirections$theta         # Alice's azimuthal angles

head(AliceTheta)

BobDirections <-
    read.table("http://www.math.leidenuniv.nl/~gill/BobDirections.txt")
   
names(BobDirections) <- c("theta", "phi")

head(BobDirections)

NBob <- nrow(BobDirections)

NBob

if (NAlice != NBob) print("Error: particle numbers don't match") else
    print("Go ahead!")

BobTheta <- BobDirections$theta            # Bob's azimuthal angles

head(BobTheta)

## First pair of measurement directions

Alpha <- 0 * pi / 180
Beta <- 45 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E11 <- mean(A * B)

## Second pair of measurement directions

Alpha <- 0 * pi / 180
Beta <- 135 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E12 <- mean(A * B)

## Third pair of measurement directions

Alpha <- 90 * pi / 180
Beta <- 45 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E21 <- mean(A * B)

## Fourth pair of measurement directions

Alpha <- 90 * pi / 180
Beta <- 135 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E22 <- mean(A * B)

CHSH <- E12 - E11 - E21 - E22

CHSH

if (CHSH > 2.4) print("Congratulations, Joy") else
    print("Congratulations, Richard")


If there is something "illegal" about the two data-sets, the program will crash, which of course means that I have won! But obviously Joy and the experimenter can check that this doesn't happen, before submitting the data to the adjudicators.

[gill1109]

It has always puzzled me that Joy Christian obviously never did any numerical experimentation on this last analysis part of the experiment, to check the feasibility of his ideas. After all, the input to this stage is just a list of (unit) vectors lambda (and a corresponding list of -lambda). How fancy can that get? Obviously, any permutation of this list will give the same computed correlations, so the only thing that is left to statistically distinguish between two lists is the distribution of the vectors. So why doesn't he just try this out, generating lists with 10 000 elements, trying a lot of different distributions? That's the good thing with the script Richard is writing now; Joy can actually try this approach. If he finds a list that nearly reproduces the four quantum correlations, there is no need to do the experiment; he will get the Nobel prize anyway. And if he experiences that no matter what list he throws into the program, the CHSH expression never exceeds 2, maybe he'll withdraw from the experiment all by himself.

Sssshhhhh!

Wisecracks.

We do science mainly because it is great fun, right?

[Joy Christian]

Here is a draft R script for the final stage of the bet.

It reads two data files. They should come out of the experiment.

To test my program, I read in two files from internet, which I created myself. Since I did not do a real experiment (whether quantum or with exploding balls) it does not give a good result. In fact the two sets of directions were completely independent of one another,
uniformly distributed.

They each contain just 100 directions, represented by spherical coordinates theta and phi, in mathematicans' notation ... the opposite from physicists, wikipedia tells me https://en.wikipedia.org/wiki/Spherical_coordinates. But what's in a name? The first coordinate is the azimuthal angle, the second is the polar angle. Both measured in radians. In the code, I call them "theta" and "phi", in that order.

I have also posted a R html notebook with the script so everyone can see a "test run" for themselves: http://rpubs.com/gill1109/Bet

Code: Select all

## I use mathematician's notation:
## theta is the azimuthal angle, phi is the polar angle, measured in radians.
## Since my measurement directions all lie in equatorial plane I just extract "theta"

AliceDirections <-
    read.table("http://www.math.leidenuniv.nl/~gill/AliceDirections.txt")
   
names(AliceDirections) <- c("theta", "phi")

head(AliceDirections)             ## N pairs theta, phi (N rows, 2 columns)

NAlice <- nrow(AliceDirections)

NAlice

AliceTheta <- AliceDirections$theta         # Alice's azimuthal angles

head(AliceTheta)

BobDirections <-
    read.table("http://www.math.leidenuniv.nl/~gill/BobDirections.txt")
   
names(BobDirections) <- c("theta", "phi")

head(BobDirections)

NBob <- nrow(BobDirections)

NBob

if (NAlice != NBob) print("Error: particle numbers don't match") else
    print("Go ahead!")

BobTheta <- BobDirections$theta            # Bob's azimuthal angles

head(BobTheta)

## First pair of measurement directions

Alpha <- 0 * pi / 180
Beta <- 45 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E11 <- mean(A * B)

## Second pair of measurement directions

Alpha <- 0 * pi / 180
Beta <- 135 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E12 <- mean(A * B)

## Third pair of measurement directions

Alpha <- 90 * pi / 180
Beta <- 45 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E21 <- mean(A * B)

## Fourth pair of measurement directions

Alpha <- 90 * pi / 180
Beta <- 135 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E22 <- mean(A * B)

CHSH <- E12 - E11 - E21 - E22

CHSH

if (CHSH > 2.4) print("Congratulations, Joy") else
    print("Congratulations, Richard")


If there is something "illegal" about the two data-sets, the program will crash, which of course means that I have won! But obviously Joy and the experimenter can check that this doesn't happen, before submitting the data to the adjudicators.

What does this suppose to tell me? I would understand it better if I see a plot for E(a, b), producing a linear correlation like this: /\. Then we are in business.

[gill1109]

What does this suppose to tell me? I would understand it better if I see a plot for E(a, b), producing a linear correlation like this: /\. Then we are in business.

Ask your favourite R-programmer to get that plot out of this. Ask them to explain to you whether or not this code does what you want it to do. If you have a better understanding of Mathematica, Python or Java, ask your favourite friendly computer-savvy friend to translate my code to another language. Michel, Chantal, John Reed ... they could all do this job in a flash. Fred is learning how to run R from Mathematica. He can test this code.

I can write the math formulas for this code ... but then I would just reproduce the formulas from your experimental paper!

Don't trust me; find friends whom you trust, who can check that this works as required. Have them translate the script into some other computer languages. I will check the translations (just a question of checking that same input gives same output). Later we'll ask the adjudicators. We can offer them several equivalent computer scripts, as well as your paper as mathematical "original".

To do tests which give the triangle wave correlation, you would need to create different data sets, and of course, you need to calculate all the correlations, not just four.

In the data set in my example it is not true that Bob's directions are the opposite of Alice's. Moreover, my test data-sets only have N = 100. Very noisy.

To explain my code a bit: the observed directions have spherical coordinates (theta, phi). Project to the equatorial plane, and they have polar coodinate (r, theta). r is irrelevant. The measurement directions are written in the code, you can see them, we agreed on them. They are to be thought of as directions in the equatorial plane.

You recognise the expression: sign(cos(theta - alpha))? theta is the "hidden variable" of Alice's spin, and alpha is Alice's measurement direction.

[gill1109]

What does this suppose to tell me? I would understand it better if I see a plot for E(a, b), producing a linear correlation like this: /\. Then we are in business.

PS I can add some lines of code, add some instructions to draw some graphs, and replace the data-sets with new ones, so that you see the triangle wave, /\ . Piece of cake. But why would you trust the other piece of code, which I wrote to compute CHSH?

Never trust an Englishman! You have to find some trustworthy friends who can help you out here. Or learn R.

[gill1109]

My God, I am such a kind person.

(My friends always tell me that I am much too nice).

Anyway: I relented, and just for you Joy, because this is all such fun, I added a few lines to the script, showing the triangle wave, with N = 1000.

The data is not read from files but generated at random on the spot.

http://rpubs.com/gill1109/Bet

Extended script:

Code: Select all

## I use mathematician's notation: theta is the azimuthal angle, phi is the polar
## (aka zenith) angle; both measured in radians.
## Reference: https://en.wikipedia.org/wiki/Spherical_coordinates
## Since my measurement directions all lie in equatorial plane I just extract "theta"


AliceDirections <-
    read.table("http://www.math.leidenuniv.nl/~gill/AliceDirections.txt")
   
names(AliceDirections) <- c("theta", "phi")

head(AliceDirections) ## N pairs theta, phi (N rows, 2 columns)

NAlice <- nrow(AliceDirections)

NAlice

AliceTheta <- AliceDirections$theta # Alice's azimuthal angles

head(AliceTheta)

BobDirections <-
    read.table("http://www.math.leidenuniv.nl/~gill/BobDirections.txt")
   
names(BobDirections) <- c("theta", "phi")

head(BobDirections)

NBob <- nrow(BobDirections)

NBob

if (NAlice != NBob) print("Error: particle numbers don't match") else
    print("Go ahead!")

BobTheta <- BobDirections$theta  # Bob's azimuthal angles

head(BobTheta)

## First pair of measurement directions

Alpha <- 0 * pi / 180
Beta <- 45 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E11 <- mean(A * B)

## Second pair of measurement directions

Alpha <- 0 * pi / 180
Beta <- 135 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E12 <- mean(A * B)

## Third pair of measurement directions

Alpha <- 90 * pi / 180
Beta <- 45 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E21 <- mean(A * B)

## Fourth pair of measurement directions

Alpha <- 90 * pi / 180
Beta <- 135 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E22 <- mean(A * B)

CHSH <- E12 - E11 - E21 - E22

CHSH

if (CHSH > 2.4) print("Congratulations, Joy") else
    print("Congratulations, Richard")

## Another experiment

AliceTheta <- runif(1000, 0, 360) * pi / 180
BobTheta <- - AliceTheta

Delta <- seq(from = 0, to = 360, by = 10) * pi / 180
Correlation <- numeric(length(Delta))
A <- sign(cos(AliceTheta))
i <- 0
for (delta in Delta) {
    i <- i+1
    B <- - sign(cos(BobTheta - delta))
    Correlation[i] <- mean(A * B)
}
plot(Correlation)


[Joy Christian]

My God, I am such a kind person.

(My friends always tell me that I am much too nice).

Anyway: I relented, and just for you Joy, because this is all such fun, I added a few lines to the script, showing the triangle wave, with N = 1000.

The data is not read from files but generated at random on the spot.

http://rpubs.com/gill1109/Bet

Extended script:

Code: Select all

## I use mathematician's notation: theta is the azimuthal angle, phi is the polar
## (aka zenith) angle; both measured in radians.
## Reference: https://en.wikipedia.org/wiki/Spherical_coordinates
## Since my measurement directions all lie in equatorial plane I just extract "theta"


AliceDirections <-
    read.table("http://www.math.leidenuniv.nl/~gill/AliceDirections.txt")
   
names(AliceDirections) <- c("theta", "phi")

head(AliceDirections) ## N pairs theta, phi (N rows, 2 columns)

NAlice <- nrow(AliceDirections)

NAlice

AliceTheta <- AliceDirections$theta # Alice's azimuthal angles

head(AliceTheta)

BobDirections <-
    read.table("http://www.math.leidenuniv.nl/~gill/BobDirections.txt")
   
names(BobDirections) <- c("theta", "phi")

head(BobDirections)

NBob <- nrow(BobDirections)

NBob

if (NAlice != NBob) print("Error: particle numbers don't match") else
    print("Go ahead!")

BobTheta <- BobDirections$theta  # Bob's azimuthal angles

head(BobTheta)

## First pair of measurement directions

Alpha <- 0 * pi / 180
Beta <- 45 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E11 <- mean(A * B)

## Second pair of measurement directions

Alpha <- 0 * pi / 180
Beta <- 135 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E12 <- mean(A * B)

## Third pair of measurement directions

Alpha <- 90 * pi / 180
Beta <- 45 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E21 <- mean(A * B)

## Fourth pair of measurement directions

Alpha <- 90 * pi / 180
Beta <- 135 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E22 <- mean(A * B)

CHSH <- E12 - E11 - E21 - E22

CHSH

if (CHSH > 2.4) print("Congratulations, Joy") else
    print("Congratulations, Richard")

## Another experiment

AliceTheta <- runif(1000, 0, 360) * pi / 180
BobTheta <- - AliceTheta

Delta <- seq(from = 0, to = 360, by = 10) * pi / 180
Correlation <- numeric(length(Delta))
A <- sign(cos(AliceTheta))
i <- 0
for (delta in Delta) {
    i <- i+1
    B <- - sign(cos(BobTheta - delta))
    Correlation[i] <- mean(A * B)
}
plot(Correlation)


Now we are talking. This will do.

[gill1109]

Now we are talking. This will do.

Joy, you are a great guy!

That's one small step for a man, one giant leap for mankind.

PS Meanwhile John Reed has told me he did my "homework" over on the thread

A silly computer experiment ... or, the heart of the matter?
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=40

which was to rewrite a small part of my R script here, in Mathematica. So soon we will have the whole thing here, also in Mathematica version, and Fred and others can verify that it is a "true" implementation of your own (experimental paper) instructions.

Actually that was a little computer experiment which I hope that Michel Fodje would dare to actually perform, in connection with "his" thread
Bell & CHSH type inequalities and experiments
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=39
but so far he has a lot of reasons why he won't play any game which I set up for him ...

[Joy Christian]

Now we are talking. This will do.

Joy, you are a great guy!

That's one small step for a man, one giant leap for mankind.

PS Meanwhile John Reed has told me he did my "homework" over on the thread

A silly computer experiment ... or, the heart of the matter?
http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=40

which was to rewrite a small part of my R script here, in Mathematica. So soon we will have the whole thing here, also in Mathematica version, and Fred and others can verify that it is a "true" implementation of your own (experimental paper) instructions.

Very good. I look forward to seeing it.

[gill1109]

It has always puzzled me that Joy Christian obviously never did any numerical experimentation on this last analysis part of the experiment, to check the feasibility of his ideas.

Joy has the courage of his convictions. That is something to be respected.

[Ben6993]

.... about a page or two ago, Richard displayed code:

## Fourth pair of measurement directions

Alpha <- 90 * pi / 180
Beta <- 135 * pi / 180
A <- sign(cos(AliceTheta - Alpha))
B <- - sign(cos(BobTheta - Beta))
E22 <- mean(A * B)

CHSH <- E12 - E11 - E21 - E22

CHSH
if (CHSH > 2.4) print("Congratulations, Joy") else
print("Congratulations, Richard")

If I understand Richard's four sets of results: the cosine for the angle between Alice's and Bob's angles give either 0.7071 or -0.7071 {as cos 45 = 1/sqrt 2}. This is the target for each pair of settings of Alice and Bob. The sawtooth function (if I have calculated it correctly as -1 + 45/90) is either 0.5 or -0.5 in each case. So it is a matter of avoiding the 0.5 outcomes and getting as close to the 0.7071 outcomes as possible. Richard's cut off of 2.4 for the CSHS value is equivalent to an average of 0.6010 for each of the four sets of results, and is equivalent to getting as close as to 85% of the cosine correlation. But that is assuming that the CHSH uses the sum of the absolute values of the correlations in the four sets, i.e. 4 x 0.7071 = 2.8 is the maximum for the cosine curve, and 4* 0.5 = 2 is the maximum obtained for the sawtooth graph. So 2.4 gives the cutoff half way between the two. If the CHSH sometimes uses -0.7071 + 0.7071 + 0.7071 + 0.7071 then CHSH is immediately reduced to a maximum of 1.4142 for the exact cosine curve so that would give a fail even for the cosine curve, which would not seem correct.

These four sets of results give only four places on the sawtooth & cosine graph. I suspect that Richard is less likely to win with four settings than if the settings are spread out across the whole 360 degrees, but he will know better.

[Heinera]

It has always puzzled me that Joy Christian obviously never did any numerical experimentation on this last analysis part of the experiment, to check the feasibility of his ideas.

Joy has the courage of his convictions. That is something to be respected.

In the imperfect world of ours, that kind of courage is only truly respected if the convictions turn out to be correct...