Application of Bell’s theorem to computer simulation

[gill1109]

Suppose A and B are two functions with domain [0, 2 pi) x [0, 1] and range {-1, +1}, programmed in some programming language such as R, Python, Matlab or Mathematica. Suppose we use the programming language’s built in pseudo-random number generator to create numbers u_1, ..., u_N, simulating independent realisations of a random number uniformly distributed on [0, 1]; take N to be 1 million. Suppose we compute the average of the N numbers A(a, u_n) * B(b, u_n), for some specified values of a and b.

Can one find programs which compute some A and B with the just specified domain and range, such that the simulation experiment just described is very likely to very well approximate -cos(a - b), for all pairs of angles a, b corresponding to half degree steps between 0 and 360 degrees?

This question comes from a recent discussion in another thread.

Michel wrote:

Joy is 100% right. There is no such thing as a mathematical theorem with loopholes. Richard is wrong, the so-called "loopholes" in Bell's theorem are directly related to oversights in Bell's analysis. In other words, the "loopholes" disprove the theorem. Why else will Richard (and Larsson) try to create a new theorem if the original one was valid? Notwithstanding the fact that Fine first identified the time synchronization flaw in Bell's theorem. It is not simply a problem with practical application.

I replied:

Joy is not 100% right. There can be a loophole in an attempt to apply a mathematical theorem to a practical matter. Larsson and my theorem is a different theorem to the “Bell theorem seen as a math theorem”. The conditions are different, and the conclusion is different. The theorem of Lasrsson and myself, for instance, applies to the computer simulation “epr-simple”. Bell’s theorem doesn’t. See https://arxiv.org/abs/1507.00106

[FrediFizzx]

And Gill's theory says it can't be done. The answer is yes since we already did it and shot Gill's theory to pieces. And the mean deviation from -a.b is -0.00113294 which is about 0.11 percent. Pretty close. If a theorem has loopholes then it is not a theorem but just a theory.
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[minkwe]

Here is a more accurate history of how this thread started:

Now, what do you think of the comparison of Bell’s mathematical results with computer experiments?

Same concerns. Just because it is done in a computer doesn't change anything. The only problem is that it is possible to do things in a computer that do not correspond to what happens in the real world thus leading to more confusion when things like that are done. Two examples:

- resetting random number seeds. There is no analog in the real world. While useful in some cases to get reproducible results, it can result in many delusions when trying to compare with real-world experiments
- access to complete information. There is no such thing in the real world.

I have a particular kind of experiment in mind. Suppose I dream up some functions A and B, taking values in {-1, 1}, which are functions of (1) a direction represented by an angle in the interval [0, 2 pi] and (2) of a number “u” in the interval [0, 1]. I write programs, in Python, say, which compute A and B for any given values of the two arguments. I now run a computer simulation in which I simply pick two angles “a” and “b” and use Python’s built in pseudo random number generator to generate a long sequence of N draws “u_i” of random numbers between 0 and 1. My simulation then averages the N numbers A(a, u_i) * B(b, u_i). Take N equal to say 1 million.

I am not taking about QM or about experiments in laboratories with lasers and photodetectors. I’m talking about simple Python computer programs run on an ordinary PC.

Do you think it is likely that the result could be close to - cos(a - b), whatever a and b I chose? What would you expect to be the result?

Yes it is likely. I'm surprised you asked me this when I already have two computer programs that do just that. You see, you are missing an important detail in your "mind" experiment. The domain of A(.)B(.) is only defined for regions where the domains of both functions are defined. In pure mathematical terms, the functions in EPR-simple and EPR-clocked are just functions that are undefined for non-trivial regions of the 2-dimensional domain of (a,u). And when you multiply the two together, you get an even more restricted domain but you do get close to -cos(a,b). Have some imagination, Richard.

Sorry Michel, you are now saying that you can get the negative cosine when you do something different from what I described. I told you the domains of the functions A and B. You ignored what I said. You moreover added an ad hoc procedure to deal with the situation that (a, u_i) or (b, u_i) is not in the domain of A or B, respectively.

Absolutely not, I did no such thing. My functions take and just like you specified in your "computer experiment". Produce outcomes just like you specified. Now you want to change the rules.

Any astute reader will notice that the new "restatement" dramatically changes the way the question is asked by explicitly trying to exclude the solution I previously provided. The astute reader will also note that no such restriction of the domain of the functions is present in any of the works of Bell himself. In fact, Bell's own toy example used a the function which will not satisfy Richards's conditions.

Therefore any pretense that this has anything to do with Bell is blown wide open. It's a waste of time engaging with it.

[gill1109]

Bell's own toy example used a the function which will not satisfy Richards's conditions.

In Bell’s example, there would be zero probability that the sign function would ever deliver the outcome “0”.
Modify Bell’s example by replacing the outcome “0” by “+1”. The example generates exactly the same statistical outcomes.

So answering my question does actually also cover Bell’s example.

I’m waiting for your answer, or otherwise requests for further clarification.

Incidentally, later researchers have noticed that Bell’s theorem remains true if one interprets A(a, u) not as the outcome itself, but as the expectation value of the +/-1 valued outcome. The idea is that independent (possibly biased) coin tosses take place in each detection apparatus, after the hidden variables from the source have arrived there, and those are the actual outcomes.

So one can just as well take the range to be [-1, +1], if you prefer to think about it that way.

Alternatively use three uniforms for each trial: one for the source, one for Alice’s detector, one for Bob’s detector.

You can also pull three independent uniforms out of one: expand in binary, then make three streams of bits out of one.

[FrediFizzx]

Same concerns. Just because it is done in a computer doesn't change anything. The only problem is that it is possible to do things in a computer that do not correspond to what happens in the real world thus leading to more confusion when things like that are done. Two examples:

- resetting random number seeds. There is no analog in the real world. While useful in some cases to get reproducible results, it can result in many delusions when trying to compare with real-world experiments
- access to complete information. There is no such thing in the real world.

Another example: On a computer you can record trial numbers so no problem matching events up. Kind of takes out timing factors. And yes, having random seeds are a joke since you want to get approximately the same result between runs with different initial parameters. A more scientific way would be to average the results of say 10 runs. Especially for CHSH results.
.

[gill1109]

Same concerns. Just because it is done in a computer doesn't change anything. The only problem is that it is possible to do things in a computer that do not correspond to what happens in the real world thus leading to more confusion when things like that are done. Two examples:

- resetting random number seeds. There is no analog in the real world. While useful in some cases to get reproducible results, it can result in many delusions when trying to compare with real-world experiments
- access to complete information. There is no such thing in the real world.

Another example: On a computer you can record trial numbers so no problem matching events up. Kind of takes out timing factors. And yes, having random seeds are a joke since you want to get approximately the same result between runs with different initial parameters. A more scientific way would be to average the results of say 10 runs. Especially for CHSH results.
.

Setting the random seed is critically necessary when testing programs.
In real world state-of-the-art experiments you match predefined time-slots. The coincidence loophole has only historical interest. Thanks to use of heralding events, and to improvements in detector efficiency, the detection loophole has only historical interest. Thanks to martingale tests, the memory loophole has only historical interest. Thanks to randomised settings, the finite statistics loophole has only historical interest.

The experimental loopholes have all been banished. Only the metaphysical loopholes remain (superdeterminism, retrocausality). Doing computer simulations of the experiments of the last century can be fun, but it is not very relevant to today’s Bell experiments.

Lots of people misunderstood various feature of the Bell (1964) paper. By 1980 these were pretty well cleared up, though even today, amateurs start off reading the 1964 paper and if they are clever and original, start developing their own ideas. Great!

[minkwe]

So answering my question does actually also cover Bell’s example.
I’m waiting for your answer, or otherwise requests for further clarification.

Incidentally, later researchers have noticed that Bell’s theorem remains true if one interprets A(a, u) not as the outcome itself, but as the expectation value of the +/-1 valued outcome. The idea is that independent (possibly biased) coin tosses take place in each detection apparatus, after the hidden variables from the source have arrived there, and those are the actual outcomes.

So one can just as well take the range to be [-1, +1], if you prefer to think about it that way.

Alternatively use three uniforms for each trial: one for the source, one for Alice’s detector, one for Bob’s detector.

You can also pull three independent uniforms out of one: expand in binary, then make three streams of bits out of one.

Sorry, don't hold your breath. Your tactics are not worth my effort. You claim to be "applying" Bell's theorem to computer simulations but you fashion the rules to exclude a solution I already provided to you. This is not the first time. I remember when you re-defined the term "clocked". You introduce restrictions that Bell himself never thought of. So I'll leave you to play in that beautiful sandbox you've created alone. Or invite some of your pals and Buddhists and high school students to celebrate how everything is connected through a mystical non-local "force". You should read Bell's Omni interview. He was also talking about Buddhists for some reason.

[gill1109]

So answering my question does actually also cover Bell’s example.
I’m waiting for your answer, or otherwise requests for further clarification.

Incidentally, later researchers have noticed that Bell’s theorem remains true if one interprets A(a, u) not as the outcome itself, but as the expectation value of the +/-1 valued outcome. The idea is that independent (possibly biased) coin tosses take place in each detection apparatus, after the hidden variables from the source have arrived there, and those are the actual outcomes.

So one can just as well take the range to be [-1, +1], if you prefer to think about it that way.

Alternatively use three uniforms for each trial: one for the source, one for Alice’s detector, one for Bob’s detector.

You can also pull three independent uniforms out of one: expand in binary, then make three streams of bits out of one.

Sorry, don't hold your breath. Your tactics are not worth my effort. You claim to be "applying" Bell's theorem to computer simulations but you fashion the rules to exclude a solution I already provided to you. This is not the first time. I remember when you re-defined the term "clocked". You introduce restrictions that Bell himself never thought of. So I'll leave you to play in that beautiful sandbox you've created alone. Or invite some of your pals and Buddhists and high school students to celebrate how everything is connected through a mystical non-local "force". You should read Bell's Omni interview. He was also talking about Buddhists for some reason.

You're wrong, Michel.

I used the word "clocked" and when I used it, I explained what I meant by it. You seem to be good at missing crucial phrases and sentences in fairly complex text and mathematics. I'm sorry. Fortunately, other people can read and understand, and maybe they can explain to you using better words.

Bell himself in "Bertlmann's socks" introduced the restrictions which I put into my theorems and which experimenters put into their experiments. He explained it very well, I think, but some people think this paper is too long. Then they miss the crucial part.

You can't please them all!

You wrote two Bell simulations.
I proved theorems which apply to those two simulations.
Check out:
https://arxiv.org/abs/1507.00106
Event based simulation of an EPR-B experiment by local hidden variables: epr-simple and epr-clocked
https://www.preprints.org/manuscript/202001.0045/v3

Joy and Fred have written several simulations.
I proved theorems which apply to a number of those simulations.
Check out:
https://arxiv.org/abs/1505.04431
Pearle's Hidden-Variable Model Revisited
Entropy 2020, 22(1), 1; https://doi.org/10.3390/e22010001

Bell's original theorem, seen as a piece of pure mathematics, applies to the simulation program which I presented in this thread.
It explains why, indeed, the Bell-CHSH bound is not exceeded (bar statistical fluctuations)
Check out:
https://arxiv.org/abs/quant-ph/0110137
Accardi contra Bell (cum mundi): The Impossible Coupling
https://arxiv.org/abs/quant-ph/0301059
Time, Finite Statistics, and Bell's Fifth Position

The results in these papers were further developed by several researchers and used (and cited) by the experimenters of the four 2015 "loophole-free" experiments.

[minkwe]

You wrote two Bell simulations.
I proved theorems which apply to those two simulations.

They don't, the simulations demonstrate that. The simulations also demonstrate that Bell was short-sighted.

Bell's original theorem, seen as a piece of pure mathematics, applies to the simulation program which I presented in this thread.
It explains why, indeed, the Bell-CHSH bound is not exceeded (bar statistical fluctuations)

You still do not get it do you? Bell's inequalities and the CHSH are an absolute bound. No amount of statistical fluctuation can violate it. Violating the inequalities is a mathematical impossibility. This is what you'll never get. You may want to sneak in some other shifted goal post and claim it is Bell's theorem but sorry I'm not buying it.

https://arxiv.org/abs/quant-ph/0110137
Accardi contra Bell (cum mundi): The Impossible Coupling
https://arxiv.org/abs/quant-ph/0301059
Time, Finite Statistics, and Bell's Fifth Position

The results in these papers were further developed by several researchers and used (and cited) by the experimenters of the four 2015 "loophole-free" experiments.

To the extent that they listened to you, they have been misled by you. They think by using pre-determined time windows they are avoiding the problem of photon-identification. They are not.

The simple fact is that the probability of a particle being matched with it's pair is nonseparable and no amount of time window trickery before, during or after the experiment can avoid it. You are too smart not to know this but yet you keep insisting that pre-agreed time windows avoid the problem.

It doesn't matter when the time-windows are decided. All you need to do is look at the number of matched pairs for each pair of settings in all the experiments done to date to confirm that the probability of being matched is dependent on both settings. This crucial point is conveniently ignored in all the data analyses.

Unless particles carry an index with them which they announce on arrival at the station, any hopes of avoiding this issue is dead. This is one of the reasons why you never understood my "epr-clocked" simulations or even "epr-simple" --- divorced from reality. Sometimes, I wonder if you have ever carried out an actual physics experiment in real life before.

[gill1109]

Michel, as Bell explained, one does not have to match photons. One should forget about theoretical words like "photon".

I see you still haven't actually read my paper on your simulation models. Clearly, you don't want to.

I last did physics experiments myself at high-school, but I have since worked many times with experimental physicists on real experiments, and many times analysed data from real experiments.

Which experiments do you believe show that "the probability of being matched is dependent on both settings"? In the 2015 experiments one does not even attempt to match photons belonging to the same emitted pair.

[FrediFizzx]

@gill1109 Doesn't matter. You are finished. Bell's theory is dead. Gill's theory is dead. Time to get real, get over it and move onnnnnnnnnnnnnnn!
.

[gill1109]

You wrote two Bell simulations. I proved theorems which apply to those two simulations.

They don't, the simulations demonstrate that. The simulations also demonstrate that Bell was short-sighted.

Bell's original theorem, seen as a piece of pure mathematics, applies to the simulation program which I presented in this thread. It explains why, indeed, the Bell-CHSH bound is not exceeded (bar statistical fluctuations)

You still do not get it do you? Bell's inequalities and the CHSH are an absolute bound. No amount of statistical fluctuation can violate it. Violating the inequalities is a mathematical impossibility. This is what you'll never get. You may want to sneak in some other shifted goal post and claim it is Bell's theorem but sorry I'm not buying it.

There are two points here. Let me focus on Michel’s key assertions.

They don't, the simulations demonstrate that. The simulations also demonstrate that Bell was short-sighted.

*My* analysis of *your* simulations in the paper by me about your simulation models (which it seems you never read) show that your simulations do exactly what is predicted by the theorems of Larsson (detection loophole) and of Larsson & Gill (coincidence loophole).

Bell's inequalities and the CHSH are an absolute bound. No amount of statistical fluctuation can violate it.

Not true! See my simulations http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=492
Latest version: https://rpubs.com/gill1109/CHSH_urn2

This includes a simulation of Bell’s own model, the one giving the triangle wave instead of the negative cosine. But for a CHSH experiment: pairs of settings chosen by fair coin tosses. In this situation the hidden variable, it turns out, can be reduced to a variable with only eight outcomes, each having equal probability. The sample size is small: 1000. The statistical variation in “S” is big, meaning that violations of CHSH are common.

[minkwe]

Bell's inequalities and the CHSH are an absolute bound. No amount of statistical fluctuation can violate it.

Not true! See my simulations http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=492
Latest version: https://rpubs.com/gill1109/CHSH_urn2

You will never understand this point. Bell's inequalities and the CHSH are an absolute bound. No amount of statistical fluctuation can violate it by even the smallest of margins. This is the meaning of Bell's mathematical proof of those inequalities. Claiming that statistical fluctuations can violate it is also the same as claiming that Bell made a mistake in deriving the inequalities. He didn't His mistake was in the application of the inequalities to physics not in the derivation of the inequalities.

[Heinera]

Bell's inequalities and the CHSH are an absolute bound. No amount of statistical fluctuation can violate it.

Not true! See my simulations http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=492
Latest version: https://rpubs.com/gill1109/CHSH_urn2

You will never understand this point. Bell's inequalities and the CHSH are an absolute bound. No amount of statistical fluctuation can violate it by even the smallest of margins. This is the meaning of Bell's mathematical proof of those inequalities. Claiming that statistical fluctuations can violate it is also the same as claiming that Bell made a mistake in deriving the inequalities. He didn't His mistake was in the application of the inequalities to physics not in the derivation of the inequalities.

Did you even try to run his code at https://rpubs.com/gill1109/CHSH_urn2, with different settings for the seed and N? Did you find that the CHSH expression can easily get close to 4 for small N? Did you find that for large N it almost never strays far above 2?

[gill1109]

Bell's inequalities and the CHSH are an absolute bound. No amount of statistical fluctuation can violate it.

Not true! See my simulations http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=492
Latest version: https://rpubs.com/gill1109/CHSH_urn2

You will never understand this point. Bell's inequalities and the CHSH are an absolute bound. No amount of statistical fluctuation can violate it by even the smallest of margins. This is the meaning of Bell's mathematical proof of those inequalities. Claiming that statistical fluctuations can violate it is also the same as claiming that Bell made a mistake in deriving the inequalities. He didn't His mistake was in the application of the inequalities to physics not in the derivation of the inequalities.

Michel, you will never understand Heinera and my points, because you will never study my proofs of *extensions* of Bell’s results. You also refuse to think about Heinera’s urn model or look at my simulations of the urn model. These things (my martingale theory, the urn model, simulations thereof) are related. The probabilistic extensions of the CHSH inequality were designed to bring Bell’s ideas closer to applications, whether in physics or in computer science (study of simulation models). These applications have been successful! The probabilistic bounds have been further refined and are used now routinely by experimentalists. Jan-Åke Larsson and I have extended Bell’s work to allow for detection and coincidence loopholes. Experimentalists are now able to avoid those loopholes entirely.

If you can’t read the theory because you don’t know enough probability theory, look at my simulation code instead. Run it, play with it. Ask questions about it. Make your own Python code of the CHSH urn model.

My latest R version includes a simulation with a randomly filled urn, and a simulation with an optimally filled urn. You might wonder how I came up with the optimal urn, and in what sense it’s optimal.

[minkwe]

Michel, you will never understand Heinera and my points, because you will never study my proofs of *extensions* of Bell’s results. You also refuse to think about Heinera’s urn model or look at my simulations of the urn model. These things (my martingale theory, the urn model, simulations thereof) are related.

Your work is related to your work. Color me surprised. The same misconceptions carry on from beginning to end. Misconceptions about physics. It's like Bell was talking about you when he said:

Bell: Then in 1932 [mathematician] John von Neumann gave a "rigorous" mathematical
proof stating that you couldn't find a non-statistical theory that would give the same
predictions as quantum mechanics. That Von Neumann proof in itself is one that must
someday be the subject of a Ph.D. thesis for a history student. Its reception was quite
remarkable. The literature is full of respectful references to "the brilliant proof of Von
Neumann;" but I do hot believe it could have been read at that time by more than
two or three people.
Omni: Why is that?
Bell: The physicists didn't want to be bothered with the idea that maybe quantum
theory is only provisional. A horn of plenty had been spilled before them, and every
physicist could find something to apply quantum mechanics to. They were pleased
to think that this great mathematician had shown it was so. Yet the Von Neumann
if you actually come to grips with it, in your hands! There is nothing to it. It's not just flawed,
it's silly. If you look at the assumptions made, it does not hold up for a moment.
It's the work of a mathematician, and he makes assumptions that have a mathematical symmetry to them.
When you translate them into terms of physical disposition, they're nonsense.
You may quote me on that: The proof of Von Neumann is not merely false but foolish

The probabilistic extensions of the CHSH inequality were designed to bring Bell’s ideas closer to applications

Translation: Bell's ideas were divorced from reality.

These applications have been successful!

That's the claim of a mathematician. Once you translate that into terms of physical disposition, it's nonsense.

The probabilistic bounds have been further refined and are used now routinely by experimentalists.

Translation: Bell's work was divorced from reality. What kind of "theorem" needs to be refined for experiments? Reminds me of epicycles.

Jan-Åke Larsson and I have extended Bell’s work to allow for detection and coincidence loopholes.

Translation: Bell's work had loopholes and was divorced from reality.

Experimentalists are now able to avoid those loopholes entirely.

That's the claim of a mathematician. Once you translate that into terms of physical disposition, it's nonsense.

If you can’t read the theory because you don’t know enough probability theory, look at my simulation code instead. Run it, play with it. Ask questions about it. Make your own Python code of the CHSH urn model.

I know enough probability theory to point out errors in your work that you didn't even realize were there. You know what I'm talking about. No sane person goes on the internet and picks "random nobodies who don't understand probability theory" to write articles about. Besides, you have no clue what I know and don't know. I will decide what I program or don't program, your patronizing attitude is stale already.

[gill1109]

The probabilistic extensions of the CHSH inequality were designed to bring Bell’s ideas closer to applications

Translation: Bell's ideas were divorced from reality.

These applications have been successful!

That's the claim of a mathematician. Once you translate that into terms of physical disposition, it's nonsense.

The probabilistic bounds have been further refined and are used now routinely by experimentalists.

Translation: Bell's work was divorced from reality. What kind of "theorem" needs to be refined for experiments? Reminds me of epicycles.

Jan-Åke Larsson and I have extended Bell’s work to allow for detection and coincidence loopholes.

Translation: Bell's work had loopholes and was divorced from reality.

Experimentalists are now able to avoid those loopholes entirely.

That's the claim of a mathematician. Once you translate that into terms of physical disposition, it's nonsense.

If you can’t read the theory because you don’t know enough probability theory, look at my simulation code instead. Run it, play with it. Ask questions about it. Make your own Python code of the CHSH urn model.

I know enough probability theory to point out errors in your work that you didn't even realize were there. You know what I'm talking about. No sane person goes on the internet and picks "random nobodies who don't understand probability theory" to write articles about. Besides, you have no clue what I know and don't know. I will decide what I program or don't program, your patronizing attitude is stale already.

Dear Michel,

I do know there is a great deal of recent literature by physicists, experimental and theoretical, on Bell type experiments, which you are wilfully ignoring; just as you wilfully choose not to acquaint yourself with mathematical literature on simulation models. I wrote about your simulation models because they were talked about on this forum, and because I wanted to compare them to Pearle’s model, to Caroline Thompson’s models, and others. I thought it was useful to the participants of this forum to work out an application of Larsson’s modified CHSH bound to “epr-simple” and to Larsson and my modified CHSH bound for “epr-clocked”. Bell’s original work did not take account of the detection loophole or the coincidence loophole. As you can see if you read “Bertlmann’s socks” he understood them perfectly well, and outlined an experimental disposition which would eradicate them. Experimentalists understood that well, and slowly worked toward implementation of Bell’s protocol.

I have no idea what errors in my work you are talking about. I don’t recall you finding any mistake in my probability theory. If you mean the error in my description of the amount of explanation of your code in your GitHub site, you will recall that I corrected my oversight after you stopped making threats and switched to civilized discourse.

Returning to this exchange:

The probabilistic bounds have been further refined and are used now routinely by experimentalists.

Translation: Bell's work was divorced from reality. What kind of "theorem" needs to be refined for experiments? Reminds me of epicycles.

My probabilistic bounds were applied by physicists in numerous published experiments, and later refined by physicists and used in yet more published experiments too. Published in the best physics journals. Don’t study my work. Read the papers on the Delft, Munich, Vienna and NIST experiments; the ones published by the experimentalists themselves.