The bet on Christian's experiment

[gill1109]

I sent the following email on April 23, to Andrei Khrennikov, Hans de Raedt, Gregor Weihs. An identical copy was later sent to Joy.

A draft text had earlier been posted by me on Google groups. The text was also discussed on the FQXi forum. Joy made one correction, which I incorporated in the text sent to the adjudicators.

Dear Andrei, Hans, Gregor

Here is the protocol which Joy and I have agreed on.

Richard
====================================================


Christian's experiment http://arxiv.org/abs/0806.3078
(N exploding balls, and analysis of a a lot of video footage of those
explosions) generates two computer files each containing N directions
of angular momentum. The files will actually be plain text files
with the directions encoded using spherical coordinates
theta (azimuth), phi (zenith).

Let's call the directions of angular momentum in Alice's file u_k, k=1,...,N,
and in Bob's file v_k, k = 1, ..., N

If I pick measurement directions a and b,thinking now of directions
as unit vectors in R^3, then according to Christian's
experimental paper the outcomes left and right are

A_k = sign(a . u_k) and B_k = sign(b . v_k),

and the estimated (observed, sample, experimental ...) correlation is

E(a, b) = 1/N sum_k A_k B_k
= ( N(++) + N(--) - N(+-) - N(-+) ) / ( N(++) + N(--) + N(+-) + N(-+) )

in the obvious notation.

Christian predicts the theoretical (population, large N limit, ensemble)
correlation rho(a, b) = - a . b = - cos(angle between a and b)

Now we are going to look at two possible directions for Alice and two
for Bob. They are all in the equatorial plane so they can be described
just by azimuthal angles alpha = 0 and 90 for Alice and beta = 45 and
135 for Bob.

Christian's theory has

rho(0, 45) = - 0.7071...,

rho(0, 135) = + 0.7071...,

rho(90, 45) = - 0.7071...,

rho(90, 135) = - 0.7071....

and he predicts therefore

E(0, 45) = - 0.7071...,

E(0, 135) = + 0.7071...,

E(90, 45) = - 0.7071...,

E(90, 135) = - 0.7071....

Of course there may be some experimental and statistical error
(but N will be large - Joy and his experimenter determine how large).

I will win my bet if I show him that one of these four predictions is off target,
by an amount 0.2 or more (ie the absolute value of the difference between
observed and predicted is 1/5 or more).

He will give me two computer files named, for instance, "AliceDirections.txt"
and "BobDirections.txt"

I claim that I'll be able to show him that one of his four predictions has failed
by a large amount: E(alpha, beta) is off target by at least 0.2

If I succeed, I win. If not, I lose.

[gill1109]

I will certainly win this bet. I use the word *certainly* deliberately. I did not say "with large probability" and the only requirement on N is that it is a positive integer. The choice of the numbers 0.7071... and 0.2 was a very careful choice. Notice that 0.7071... - 0.2 > 0.5 (strictly larger than 0.5). One might notice that 4 times 0.5 is 2, and 4 times 0.5071... > 2. So 0.21 would not have worked.

There is certainty, because there is no random choice of setting directions, no tossing of classical coins in the present agreement. There will just be two computer files of directions of angular momentum u_k and v_k, k = 1, ..., N. Via a simple thought experiment, those numbers lead rather quickly to the infamous N x 4 spreadsheet of numbers +/-1 and to a proof of a CHSH-like inequality using simple arithmetic and simple logic.

Only a thought experiment, mind you, I promise I won't actually generate the Nx4 spreadsheet!

But it my mind it would contain, in the k'th row, the following four numbers,

Code: Select all

column names:     A                      A'                       B                      B'

=========================================================================================================

row k:          sign(a0 . u_k)           sign(a90 . u_k)          sign(b45 . v_k)         sign(b135 . v_k)


I also wrote these remarks on the FQXi forum. This led to the response

Your thought experiment is in error. It is conducted in the flatland of an N x 4 spreadsheet, which has the topology of the set R of real numbers. It has nothing whatsoever to do with the world we actually live in.

to which I responded

in the second phase of the experiment and the bet, Joy hands over two ordinary computer files of real numbers represented in flatland decimal notation. Which will be analysed by me and the adjudicators of our bet on flatland computers using flatland arithmetic. So the flatland thought experiment is actually highly relevant to the calculation of any of those four correlations. The u_k and v_k, k = 1, ..., N, are not going to magically change because I try out different combinations of measurement directions a and b. They will have already been frozen in flatland physics.

[FrediFizzx]

Bzzzzt! Wrong! Try again. There is no such thing as A' station and B' station. There is only an A station and a B station.

[gill1109]

Bzzzzt! Wrong! Try again. There is no such thing as A' station and B' station. There is only an A station and a B station.

I do not say there is an A station and an A' station, and a B station and a B' station !

There are no "stations" at all in the last stage of the experiment+bet

There are two computer files, each on a different USB stick, the files are text files
called AliceDirections.txt and BobDirections.txt.

They contain directions u_k and v_k, k = 1, ..., N encoded in spherical coordinates theta, phi,
and represented in ordinary decimal notation to say 10 significant figures.
Two floating point real numbers, e.g. separated by a few spaces, per line.
N lines of pairs of real numbers in both files.

These four times N numbers are all going to be calculated (but not all at the same time)

sign(a0 . u_k), sign(a90 . u_k), sign(b45 . v_k), sign(b135 . v_k),

k = 1, ..., N,

and these four correlations are all going to be calculated (but not all at the same time)

E(0, 45) = 1/N sum_k sign(a0 . u_k) sign(b45 . v_k) =?= - 0.7071...
E(0, 135) = 1/N sum_k sign(a0 . u_k) sign(b4135 . v_k) =?= + 0.7071...
E(90, 45) = 1/N sum_k sign(a90 . u_k) sign(b45 . v_k) =?= - 0.7071...
E(90, 135) = 1/N sum_k sign(a90 . u_k) sign(b135 . v_k) =?= - 0.7071...

I am not going to put these numbers into one spreadsheet. I am not going to call them A, A', B, B'.

I have four different computer programs which will each read the same two data files AliceDirections.txt and BobDirections.txt, and then each compute just one of the four correlations according to Joy's formulas, as agreed in our bet.

If you like I can show you the programs.

I believe that I will observe that one of those correlations is off by 0.2 or more, and if so, I'll report which one it is to the adjudicators. The adjudicators can then check. They also have a copy of the two files on the two USB sticks. They don't have to use my programs, they can use any computer language they like. Everyone will be able to see what calculation is done and what comes out of the calculation. The two data files AliceDirections.txt and BobDirections.txt will be posted on internet so everyone can check that my claim is right (if I'm right), or that Joy's claim is right (if he's right). They can do the calculations in a spreadsheet, by a perl script, with R, Python or with Mathematica, just as they like.

Now the problem for Joy is that by some simple algebra it is easy to check that

- E(0, 45) + E(0, 135) - E(90, 45) - E(90, 135) <= 2

which means that it is not possible for all four of - E(0, 45), + E(0, 135), - E(90, 45), and - E(90, 135), to be strictly larger than 0.5 at the same time. At least one of those four must be less than or equal to 0.5 hence less that 0.7071... - 0.2 = 0.5011 ...

ie at least one of the four correlations will be off Joy's target by at least 0.2

The only assumption I'm making is that N >= 1.

[Joy Christian]

The only assumption I'm making is that N >= 1.

This is not the only assumption you are making. You do not know what other assumptions you are making even after I have been hammering them on the heads of all Bell's believers for the past seven years. In any case, the correct prediction for the CHSH style bound for my proposed experiment is given by equation (9.100) of this paper. If you read this paper, or read this page of my blog, then it may finally dawn on you that when you do all your calculations as you describe above you will find

E(0, 45) = 1/N sum_k sign(a0 . u_k) sign(b45 . v_k) = - 0.7071...

E(0, 135) = 1/N sum_k sign(a0 . u_k) sign(b4135 . v_k) = + 0.7071...

E(90, 45) = 1/N sum_k sign(a90 . u_k) sign(b45 . v_k) = - 0.7071...

E(90, 135) = 1/N sum_k sign(a90 . u_k) sign(b135 . v_k) = - 0.7071...

You will then be forced to part with your 5000 euroes in humiliation. If you are a gentleman as you claim you are, then you will also apologize to me profusely.

[Joy Christian]

Over on the FQXi blog Richard Gill wrote:

Sorry Joy I thought you had rejected my offer of 10 000 Euro's if you could submit two computer files which win the bet before Wednesday June 11, 2014

That's the day I give my talk at Vaxjo.

But I can put it back on the table.

To make it more of a competition, let me say that this offer is open to absolutely anyone. But after June 11 it goes down to 5000.

As well as the money you'll get my public apologies to Joy Christian, my withdrawal of quite a few papers by myself, and a lot of publicity. First come first served.

My bet with Joy remains standing. Of course once someone else has submitted winning computer files, I will publish them, and Joy will be able to collect on his bet with me, as well.

To which I replied:

So potentially I could collect 15,000 euros from you. Now that I call a generous offer.

Well, I will settle for 10,000 euros and call you a gentleman.

So, to heat up the competition, let me declare that I am working on the problem full time, with a two-pronged strategy. I rather not reveal full details at this stage.

[gill1109]

There is no humiliation in science. It is a collective enterprise of humanity.

If Joy wins I will be amazed, temporarily short of cash, and delighted to witness the birth of a new age of science. Let the competition heat up! Are there ego's in science? Our ego's push our efforts to push the envelope. They take us to the cutting (=bleeding) edge. On the other hand there should be no question of shame or humiliation in being proven wrong.

[Joy Christian]

Dear All,

Here is my claim to collect 10,000 Euros from Richard Gill: http://rpubs.com/jjc/16415.

Please read the preamble of the simulation for further details.

Joy Christian

[gill1109]

Dear All,

Here is my claim to collect 10,000 Euros from Richard Gill: http://rpubs.com/jjc/16415.

Please read the preamble of the simulation for further details.

Joy Christian

To claim the 10 000 Euro's you must adapt the script so that it outputs two files each containing the same number (say N) of directions: "AliceDirections.txt", and "BobDirections.txt". It's fine by me if the directions are all in the plane, hence points on S^1, hence two files of N real numbers between 0 and 360 degrees, rather than the earlier agreed spherical coordinates theta, phi (the zenith angle phi is indeed superfluous, in view of the restricted set of measurement directions involved in the bet, and the definition of the measurement functions).

[Joy Christian]

Dear All,

Here is my claim to collect 10,000 Euros from Richard Gill: http://rpubs.com/jjc/16415.

Please read the preamble of the simulation for further details.

Joy Christian

To claim the 10 000 Euro's you must adapt the script so that it outputs two files each containing the same number (say N) of directions: "AliceDirections.txt", and "BobDirections.txt". It's fine by me if the directions are all in the plane, hence points on S^1, hence two files of N real numbers between 0 and 360 degrees, rather than the earlier agreed spherical coordinates theta, phi (the zenith angle phi is indeed superfluous, in view of the restricted set of measurement directions involved in the bet, and the definition of the measurement functions).

OK. Here are the two files:

AliceDirections.txt: vector e_k := ( cost(s_k), -1+(2.08/(sqrt(1+(3 x t_k/pi))), 0), where s_k is randomly picked from [0, pi] and t_k is randomly picked from [0, pi].

BobDirections.txt: vector -e_k := ( cost(s_k), +1-(2.08/(sqrt(1+(3 x t_k/pi))), 0), where s_k is randomly picked from [0, pi] and t_k is randomely picked from [0, pi].

Here k is an integer from [1, 2, 3, ..., 10^7].

So there you have it. These are the two files you are asking for, with the explicit simulation provided here: http://rpubs.com/jjc/16415

[gill1109]

OK. Here are the two files:

AliceDirections.txt: vector e_k := ( cost(s_k), -1+(2.08/(sqrt(1+(3 x t_k/pi))), 0), where s_k is randomly picked from [0, pi] and t_k is randomly picked from [0, pi].

BobDirections.txt: vector -e_k := ( cost(s_k), +1-(2.08/(sqrt(1+(3 x t_k/pi))), 0), where s_k is randomly picked from [0, pi] and t_k is randomely picked from [0, pi].

Here k is an integer from [1, 2, 3, ..., 10^7].

So there you have it. These are the two files you are asking for, with the explicit simulation provided here: http://rpubs.com/jjc/16415

Please have one of your friends make the two files for you. If I make them, and if they then fail the test, then you will blame me, right? (Algebraically challenged, third rate statistician, not even a mathematician, probably less than fourth rate programmer!). Michel is a splendid programmer, believe me! Chantal too!

BTW so you are going to give me x, y and z coordinates instead of spherical coordinates? It's not a problem, but I need to know in advance. You first give me two files, and I then tell you a pair of directions such that E(a, b) is 0.2 or more off your prediction. a and b chosen, as agreed, by me, from the usual set of four "CHSH" pairs. I will calculate E(a, b) according to the recipe which we agreed together (the recipe in your experimental paper, indeed). Everyone will be able to check who's right.

[Joy Christian]

OK. Here are the two files:

AliceDirections.txt: vector e_k := ( cost(s_k), -1+(2.08/(sqrt(1+(3 x t_k/pi))), 0), where s_k is randomly picked from [0, pi] and t_k is randomly picked from [0, pi].

BobDirections.txt: vector -e_k := ( cost(s_k), +1-(2.08/(sqrt(1+(3 x t_k/pi))), 0), where s_k is randomly picked from [0, pi] and t_k is randomely picked from [0, pi].

Here k is an integer from [1, 2, 3, ..., 10^7].

So there you have it. These are the two files you are asking for, with the explicit simulation provided here: http://rpubs.com/jjc/16415

Please have one of your friends make the two files for you. If I make them, and if they then fail the test, then you will blame me, right? (Algebraically challenged, third rate statistician, not even a mathematician, probably less than fourth rate programmer!). Michel is a splendid programmer, believe me! Chantal too!

BTW so you are going to give me x, y and z coordinates instead of spherical coordinates? It's not a problem, but I need to know in advance. You first give me two files, and I then tell you a pair of directions such that E(a, b) is 0.2 or more off your prediction. a and b chosen, as agreed, by me, from the usual set of four "CHSH" pairs. I will calculate E(a, b) according to the recipe which we agreed together (the recipe in your experimental paper, indeed). Everyone will be able to check who's right.

Actually, I am not going to ask anyone. I am satisfied that the simulation speaks for itself. It does what you claimed to be impossible to do. If someone wants to help me to do what you are asking for, then I will happily accept their help. But, as I said, the simulation speaks for itself, so I don't see the point. It is pretty clear from the second plot that none of the four E(a, b) are off by more than 0.02, even with just 10^7 trials. So there is no way you are going to get your 0.2 or more deviation.

As for the choice of coordinates, I have no idea which coordinates will be convenient for the experimentalists. To me this is a trivial issue. Translation from one coordinate system to another shouldn't be an issue at all.

By the way, the accuracy of this simulation can be improved in several different ways. I will just have to play around with it a bit more to get some improvements.

[Heinera]

I have already dumped Joy's list of vectors to file. Just let me finsih dinner preparations, then I'll put them in a publicly available place.

[Joy Christian]

I have already dumped Joy's list of vectors to file. Just let me finsih dinner preparations, then I'll put them in a publicly available place.

OK. Thanks.

[Heinera]

The file is here:

JoyVector.txt

It is a dump of the 2*10^7 matrix e in Joy's program. Since he fixes seed in his program, it is the same matrix for every run.

Warning: The file is HUGE. Right click on the link and chose save; don't left click on it, because your browser will crash when trying to display it.

Alternative: It is much easier to regenerate the matrix in R (with the same seed as Joy uses) and do computations in there.

[Joy Christian]

For the benefit of those not familiar with my proposed experiment, let me spell out what this ugly looking simulation is all about. Mimicking quantum mechanical predictions, my proposed experiment makes the following four predictions for a macroscopic EPR-Bohm experiment:

E(0, 45) = - 0.7071...,

E(0, 135) = + 0.7071...,

E(90, 45) = - 0.7071...,

E(90, 135) = - 0.7071....

These are the well known "Bell test" correlations involving the four pairs of the "Bell test" angles, each to be calculated separately.

Now, because of his deep-seated belief in the Bell's so-called theorem, Richard Gill claims that one of these four correlations would be off target by at least 0.2 (i.e., the absolute value of the difference between the observed and predicted correlation would be 1/5 or more) in my proposed experiment. He has therefore offered 10,000 Euros to anyone who can simulate the hidden variables involved in my experiment which do not exhibit deviation from the correlations by more than 0.2.

In the above simulation, however, I have accomplished just that. It is easy to see from the last plot of the simulation that none of the four corrections deviate by more than 0.02 from the values predicted by my local model. So the net deviation cannot even add up to 0.2. Therefore Richard Gill now owes me 10,000 Euros.

[Heinera]

Therefore Richard Gill now owes me 10,000 Euros.

Did you check E(90, 135)?

[Joy Christian]

Therefore Richard Gill now owes me 10,000 Euros.

Did you check E(90, 135)?

Did you?

[Heinera]

Therefore Richard Gill now owes me 10,000 Euros.

Did you check E(90, 135)?

Did you?

Oh, yes.

[Joy Christian]

Did you check E(90, 135)?

Did you?

Oh, yes.

Good.