At this moment, the 4

^{th}may at 8:47h the bet seems to be still ‘on’: Joy Christan and Richard Gill are betting whether or not a computer simulation of the exploding balls experiment can be created that can or cannot produce Bell type statistics. This bet is not the 5000 euros one for which the exploding balls experiment should actually be performed (see the earlier post on this subject).
The current bet focusses on the angle settings from Alice and Bob where the difference between classical and QM predictions are most significant: 0, 45, 90 and 135 degrees.

The exact text of the bet as stated by Richard can be found here:

Joy has responded and claimed victory by supplying this model in R: http://rpubs.com/jjc/16415

In his own words:

“Richard Gill has offered 10,000 Euros to anyone who can simulate the N directions of angular momentum vectors appearing in equation (16) of this experimental proposal of mine: http://arxiv.org/abs/0806.3078. Here I am attempting to provide such N directions. They are given by the vectors 'e' in this simulation. He has also offered further 5,000 Euros to me if my proposed experiment is realized successfully. I am hopeful that that will happen someday. The details of these challenges by Richard Gill can be found here: http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=46. While this is by no means a perfect simulation of my model, it does meet all the stringent conditions set out by Richard Gill for his challenge.

Since after the explosion the angular momentum vectors 'e' moving along the z direction will be confined to the x-y plane, a 2D simulation is good enough for my proposed experiment. ”

Gill disputes whether Joy's simulation meets the conditions of the challenge.

The agreement now is to have a jury give a verdict (see http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=52&start=10#p1935).

For the non-programmers, and for myself to get acquainted with in the R syntax, I explain this code line by line below.

The agreement now is to have a jury give a verdict (see http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=52&start=10#p1935).

For the non-programmers, and for myself to get acquainted with in the R syntax, I explain this code line by line below.

**Joy’s R code explained**

`set.seed(9875)`

angles <- seq(from = 0, to = 360, by = 1) * 2 * pi/360

K <- length(angles)

length(): gives the number of elements in the array, and stores it in a single value K.

length(): gives the number of elements in the array, and stores it in a single value K.

`corrs <- numeric(K) ## Container for correlations`

declares an
array structure named ‘corrs’ which can contain K numbers (?)

`M <- 10^5 ## Sample size. Next try 10^6, or even 10^7`

The number
10 to the power of 5 = 1000000 is stored in M.

`s <- runif(M, 0, pi)`

`t <- runif(M, 0, pi)`

`x <- cos(s)`

` `

`y <- 1.2 * (-1 + (2/(sqrt(1 + (3 * t/pi)))))`

` e <- rbind(x, y) ## 2 x M matrix; M columns of e represent the`

`## x and y coordinates of points on a circle; y -> -y => e -> -e.`

`for (i in 1:(K - 1)) {`

`…`

`}`

` alpha <- angles[i]`

` `

`a <- c(cos(alpha), sin(alpha)) ## Measurement vector 'a'`

`for (j in 1:(K - 1)) {`

` beta <- angles[j]`

` b <- c(cos(beta), sin(beta)) ## Measurement vector 'b'`

` ca <- colSums(e * a) ## Inner products of cols of 'e' with 'a'`

` cb <- colSums(e * b) ## Inner products of cols of 'e' with 'b'`

x

_{1}x_{2}x_{3}…x_{M }y_{1}y_{2}y_{3}…y_{M}
which is
multiplies with a (a

_{1}, a_{2})
giving

x

_{1}a_{1}x_{2}a_{1}x_{3}a_{1}… x_{M}a_{1 }y_{1}a_{2}y_{2}a_{2}y_{3}a_{2}… y_{M}a_{2}
colSums()
then sums the numbers per column, giving

x

_{1}a_{1+}y_{1}a_{2}x_{2}a_{1+}y_{2}a_{2}x_{3}a_{1+}y_{3}a_{2}…x_{M}a_{1 +}y_{M}a_{2}
which is
stored in ca. The same calculation is done with ‘b’ giving cb.

` N <- length(ca)`

` corrs[i] <- sum(sign(-ca) * sign(cb))/N`

So each
number in ca is first multiplied by -1 because of the ‘–ca’, and the sign
function results in a list with 1’s and -1’s. These are multiplied by the
elements in sign(cb).The resulting list of 1’s and -1’s is summed up and
divided by N. The calculated number is stored in the i-th position of corrs.

` `

` Ns[i] <- N`

When the
code in the loops is completed it continues with the statements below

`corrs[K] <- corrs[1]`

`Ns[K] <- Ns[1]`

The rest of
the code is for printing the results.

An extended
description of R can be found here: http://cran.r-project.org/doc/manuals/R-intro.html**Some concluding remarks**

- Joy's model iterates over Alice’s and Bobs angles. This is an efficient mechanism to get results for all the settings. The same results should be obtained using random (integer) angles between 0 and 360 degrees, but one might need a slightly larger 'M' to get nice results for all the angles.
- The conditions for the Bell type simulation seems to be met: All the particles are used in the result set (which excludes the detection loophole) and the measurement for Alice does not use Bob's measurement and vice versa.

Thank you, Albert Jan. This is very useful. As you know, I am not a programmer myself, so your explanations are very useful for me.

ReplyDeleteUnfortunately, this code only shows what happens when one particular choice of beta is made. Only alpha varies. Now run the code with a different beta. That would be very illuminating.

ReplyDeleteBy the way, the verdict of the jury was that the submitted data set did not win the challenge. Which is also easy for anyone else to check for themselves.

ReplyDeleteSorry: there are two plots: first with beta fixed, them with alpha fixed. But what we don't see are the results for the four combinations of alpha, beta needed for CHSH. The point is, there is a whole correlation surface. Christian shows you what the surface looks like, above two perpendicular straight lines. Here it looks good. He doesn't show you what it looks like anywhere else...

ReplyDeleteI have modified Christian's script so that it now prints out the four correlations for the four combinations of angles in a CHSH experiment. Two of the correlations are just what we want, but the other two are badly off target.

ReplyDeletehttp://rpubs.com/gill1109/joy

Albert Jan wrote above "Joy has responded and claimed victory by supplying this model in R: http://rpubs.com/jjc/16415". What he doesn't mention is that this was only the first of a whole sequence of attempts. They all failed. You can see half a dozen other attempts at http://rpubs.com/jjc

ReplyDeleteSeveral of them use the detection loophole trick. Different subsets of pairs of particles are selected for the correlations at different pairs of angles. One of them uses a most extraordinary and novel trick, namely to supply two sets of data, one for one pair of angles, the other for another pair of angles.

Of course the jury had to judge on the basis of the "final submission", I forget which one that was, but it doesn't matter, since it is hard to come up with a counter-example to a true theorem. A true theorem of arithmetic. The arithmetic of an N x 4 spreadsheet of numbers +/-1.

Actually Fred Diether got angry with his Master's failures and made his own, different, submission at one point. Unfortunately, it failed too.

I am really greatful to Joy and Fred and others for their obstinate denial of simple mathematical truth, because otherwise I would never have come up with the new (finitary) "spreadsheet" proof of Bell's theorem. I acknowledge Joy in my paper in Statistical Science http://arxiv.org/abs/1207.5103

It should be coming out any day now.