Is Bell's 1971 CHSH the same as Gill's CHSH version?

[FrediFizzx]

Here is the reference to Bell (1971):

Introduction to the hidden-variable question
JS Bell
Foundations of quantum mechanics, 171-181

Preprint:
http://cds.cern.ch/record/400330/files/CM-P00058691.pdf

I guess the clue we have to what I am saying above is that Bell says, "In practice, there will be some occasions on which one or both instruments simply fail to register either way. One might then count A and/or B as zero in defining P, <A>, and <B>; (4.4) remains true and the following reasoning remains valid." Then he goes on with his CHSH derivation with out further explanation of how he goes from the first step to the second when adding in the a', b' parts and a', b parts. For me, it is only true if we make the assumption that for large enough N,

<A(a, L)> = <A(a', L)> = <B(b, L)> = <B(b', L)> = 0

Unless someone else has more insight about this, I guess I will go with that. Seems a bit odd though.

[gill1109]

Here is the reference to Bell (1971):

Introduction to the hidden-variable question
JS Bell
Foundations of quantum mechanics, 171-181

Preprint:
http://cds.cern.ch/record/400330/files/CM-P00058691.pdf

I guess the clue we have to what I am saying above is that Bell says, "In practice, there will be some occasions on which one or both instruments simply fail to register either way. One might then count A and/or B as zero in defining P, <A>, and <B>; (4.4) remains true and the following reasoning remains valid." Then he goes on with his CHSH derivation with out further explanation of how he goes from the first step to the second when adding in the a', b' parts and a', b parts. For me, it is only true if we make the assumption that for large enough N,

<A(a, L)> = <A(a', L)> = <B(b, L)> = <B(b', L)> = 0

Unless someone else has more insight about this, I guess I will go with that. Seems a bit odd though.

Fred I don't know what your problem is with Bell's derivation (which has been adopted by many, many writers since 1971), and I don't see where any assumption about mean values zero is needed or used. In fact I know for sure that no such assumption is needed anywhere. Bell is using the fact that the expectation of a conditional expectation equals the ordinary (unconditional) expectation. If you are not familiar with this kind of standard fact from probabiliy, you need to study any introductory text on probability. I recommend John Rice's "Introduction to Mathematical Statistics and Data Analysis", a standard undergraduate text for mathematicians, physicists, engineers and other "exact scientists".

[FrediFizzx]

... and I don't see where any assumption about mean values zero is needed or used.

I guess when you read something, you only see what you want to see. From Bell's paper that you provided the link for, "One might then count A and/or B as zero in defining P, <A>, and <B>;...". Now if you can get from the first expression to the second expression of his derivation a different way mathematically, please show all the detail. Thanks. We are taking baby steps here.

[gill1109]

He means that if we choose to assign the values 0 to A and B when there is no detection, then the observed measurement outcomes can take the values -1, 0 and 1, and we include the zero's when calcuating their averages <A> and <B>.

It's important to start with baby steps! We probably all need to retrace our steps, to go back to the beginning.

[FrediFizzx]

Well yes, he meant that also but I think he used the notion that <A(a, L)>,etc. would be zero for large enough N to go from his first expression to his second expression. What else is there that could get you mathematically from the first to the second? But that kind blows the whole thing up if everything is zero. That is what is odd about it.

Anyways, I suppose it doesn't matter all that much. I was just very curious about Bell's version of the derivation because it is different. But we can easily see from Aspect's derivation that nothing can violate the bound of 2 for CHSH in that version which is the same as yours. The fact that QM predicts a CHSH value of 2.82 for optimal angles is for the version of CHSH where the bound is 4.

[gill1109]

(1) Well yes, he meant that also but I think he used the notion that <A(a, L)>,etc. would be zero for large enough N to go from his first expression to his second expression. What else is there that could get you mathematically from the first to the second? But that kind blows the whole thing up if everything is zero. That is what is odd about it.

(2) Anyways, I suppose it doesn't matter all that much. I was just very curious about Bell's version of the derivation because it is different. But we can easily see from Aspect's derivation that nothing can violate the bound of 2 for CHSH in that version which is the same as yours. The fact that QM predicts a CHSH value of 2.82 for optimal angles is for the version of CHSH where the bound is 4.

(1): you mean from (4.1) to (4.2)? The principle of repeated averaging. Fubini's theorem. E( E(X | Y) ) = E(X). I can explain in more detail, if you like.

(2): like Michel you seem to be unaware of the huge conceptual difference between population mean (ensemble average, expectation value) and sample average (sample mean, experimental mean). Population versus sample.

Statistics! You need statistics! Experimental science needs statistics!

[FrediFizzx]

(1): you mean from (4.1) to (4.2)?

No, not from (4.1) to (4.2); please read back a few messages. We are talking about from the first line of Bell's CHSH derivation to the second line on page 11.

[gill1109]

(1): you mean from (4.1) to (4.2)?

No, not from (4.1) to (4.2); please read back a few messages. We are talking about from the first line of Bell's CHSH derivation to the second line on page 11.

Do you mean the 11th page of the pdf file or the page numbered 11? Please quote a few words from the start and from the end of what you don't follow.

Where is Michel, and where is the translation of http://rpubs.com/gill1109/Bet into Python and Mathematica and Java? My R implementation of Joy's instructions for calculation of the four correlations, and for the subsequent determination of who wins the bet. Preparations for the experiment and the bet are stagnating. I can't only offer a piece of R code authored by me to the adjudicating committee. Who don't speak R anyway. We need versions in several languages on public display on internet, and authorized both by Joy and me. Everyone can read it in the language they prefer, and test them all (same inputs, same outputs).

This work *has* to be taken in hand by Joy's supporters.
Some of Joy's supporters need to support him by helping to get this done.

[FrediFizzx]

The page 11 where the derivation of CHSH is. Says 11 at the top.

[gill1109]

The page 11 where the derivation of CHSH is. Says 11 at the top.

You agree with (4.4), top of the page named "11"? (12th page of the pdf file)?

You agree with (4.2) on the previous page? cf. footnote 10.

You have to know what overline A and overline B stand for. They stand for the mean value of Alice's outcome given a particular setting and particular value of the source hidden variables lambda.

Or you have a problem with F.S. Crawford's suggestion (see footnote 11) to allow for missing detections by encoding them as an outcome 0?

[thray]

people do not realise that genuine CHSH type experiments, although imaginable in the abstract theory, are unperformable. And performable experiments are governed by different inequalities which have never been violated, not even by experimental error.

Once they realise it, they will stop believing that LHV theories are impossible and they will stop discouraging others from looking for such theories.

Michel is dead on accurate. Violations of inequalities tell us that the analysis is wrong, not that an analytical theory is impossible.

[Joy Christian]

people do not realise that genuine CHSH type experiments, although imaginable in the abstract theory, are unperformable. And performable experiments are governed by different inequalities which have never been violated, not even by experimental error.

Once they realise it, they will stop believing that LHV theories are impossible and they will stop discouraging others from looking for such theories.

Michel is dead on accurate. Violations of inequalities tell us that the analysis is wrong, not that an analytical theory is impossible.

I very much second that. The high-horse on which Richard Gill has been chasing down the ghosts of Bell is itself a hollow mirage, created by his own wishful delusions.

[gill1109]

people do not realise that genuine CHSH type experiments, although imaginable in the abstract theory, are unperformable. And performable experiments are governed by different inequalities which have never been violated, not even by experimental error.

Once they realise it, they will stop believing that LHV theories are impossible and they will stop discouraging others from looking for such theories.

Michel is dead on accurate. Violations of inequalities tell us that the analysis is wrong, not that an analytical theory is impossible.

Michel's point is that the only certain bound one can place on an observed value of - E(0, 45) + E(0, 135) - E(90, 45) - E(90, 135), where each "E" is the average of products of measurememt outcomes encoded +/-1 in a different subset of runs, is 4. He is absolutely right. One needs to distinguish between population mean values, and sample averages.

Secondly, performable experiments tend to have ternary outcomes (-1, 0, +1) where one outcome means "no detection". For those experiments one needs generalized Bell inequalities for a 2x2x3 experiment (two parties, two settings per party, three outcomes per party and setting combination). Last year two groups for the first time ever violated generalized Bell inequalities using polarization of photons. The photon polarization is the first quantum system for which each experimental loophole has been closed.

At last the experimenters are using the proper statistical methods advicated for years by Caroline Thompson and championed by people like myself and Jan-Ake Larsson. This is a very gratifying development.

But in the context of Joy's experiment there will be no non-detections and there will in fact only be one set if runs: we'll calculate the four correlations seoarately, but each on the same set of N measured spin directions, exactly according to Joy's instructions in his experimental paper.

The key ingredients are: 1) http://arxiv.org/abs/0806.3078
Joy's experimental paper (and just one page out of that), 2) the agreed terms of the bet, and 3) http://rpubs.com/gill1109/Bet
my R script implementing them.

The paper describes an experiment which results in two computer files each containing N (= 10 000, say) directions. Unit vectors in R^3 encoded e.g. with spherical coordinates theta, phi.

The experimental paper now defines the measurement outcomes as sign(a dot u), sign(b dot v). Actually, we'll have v = - u. Correlations are defined as usual. Joy and my bet is determined by just four of them.

Namely: Alice's angles 0 and 90 degrees; Bob's 45 and 135.

rho(alpha, beta) is +/- 0.5 under LHV, +/- 0.7 under QM

John Reed is working on a Mathematica translation of the script. Michel is working on a Python version. This way, we have complete transparency.

Joy said at the FQXi forum that he already had a simulation experiment, which makes the real experiment superfluous. So he just needs to ask his programmer to adapt the simukation code so as to generate two computer files on N directions in R^3, encoded in spherical coordinates theta, phi. I asked Joy if he could supply the two files within 6 weeks, in time for the Vaxjo conference. I'm willing to raise the stakes to 10 000 Euro but then the terms of the bet are tightened: no files before Vaxjo means I win. Apart from that, everything else remains the same.

[FrediFizzx]

The page 11 where the derivation of CHSH is. Says 11 at the top.

You agree with (4.4), top of the page named "11"? (12th page of the pdf file)?

You agree with (4.2) on the previous page? cf. footnote 10.

You have to know what overline A and overline B stand for. They stand for the mean value of Alice's outcome given a particular setting and particular value of the source hidden variables lambda.

Or you have a problem with F.S. Crawford's suggestion (see footnote 11) to allow for missing detections by encoding them as an outcome 0?

Go back and read the whole thread then you will hopefully understand what I am asking about. I'm not going to keep repeating myself like you do.

[gill1109]

Go back and read the whole thread then you will hopefully understand what I am asking about. I'm not going to keep repeating myself like you do.

I did read the whole thread, and I don't understand what you are asking about. Too bad then.

Off topic but related: John Reed is going to make a Mathematica version of the bet resolution script which I wrote. I hope you'll be interested to test it. It needs to be agreed on by Joy's supporters.

I hope it is becoming clear that the whole CHSH inequality business is a bit of a red herring.

We have a theory which says rho(0, 45) = rho(90, 45) = rho(90, 135) = - 0.7, and rho(0, 135) = + 0.7.
We have a theory which says rho(0, 45) = rho(90, 45) = rho(90, 135) = - 0.5, and rho(0, 135) = + 0.5.

"rho" refers to ensemble averages or population means, and are derived on the basis of different mathematical frameworks.

We have experiments in which we can calculate E(0, 45), E(90, 45), E(90, 135) , and E(0, 135) as averages of products of finitely many observed values of binary measurement outcomes, when the experimenters have imposed some different measurement settings on the measurement devices. Usually the four correlations are of course calculated on four different subsets of runs of the experiment, since the measurement devices do not allow to measure according to two settings at the same time. Those four sample averages can in principle take any value between - 1 and + 1, independently on one another, so the only thing one can say for certain is that - E(0, 45) + E(0, 135) - E(90, 45) - E(90, 135) lies between - 4 and + 4.

Joy's proposed experiment is rather remarkable in that it does allow observation of the spin in different directions at the same time. After all, he instructs the experimenter to *measure* u and then to *calculate* A(a) = sign( a . u) and B(b) = sign( b . -u ).

A computer simulation of a LHV model is similar to Joy's experiment in that in the code, a value of lambda is created and a value of A (and of B) is calculated, for particular values of settings a and b. Whether or not one calculates and prints or saves the values of A and B for other values of a and of b, they still do "exist" in a mathematical sense. You may recall how Michel got very angry when I proposed to add some lines to his code doing some different things with the "hidden variables" than he had already written in his code. Because they were "hidden" we were not supposed to look at them. Well, the word "hidden" simply means that in the real world we do not get to see them directly, but just because some physical variable is hidden doesn't mean that we are forbidden to think about it. Einstein, Podolsky and Rosen started this whole thing by thinking about counterfactual outcomes of not performed measurements, and using their thought experiment to infer the existence of hidden variables, hence to logically prove the incompleteness of quantum mechanics - those hidden variables really did exist hence it was the task of the physicist to build models which described them. Which is actually what Joy Christian has done, of course.

[FrediFizzx]

I did read the whole thread, and I don't understand what you are asking about. Too bad then.

As I said earlier, it doesn't matter much as CHSH has been successfully exposed as a fraud on this forum. And... since Bell's original inequality is a just a special case of CHSH, then it too is a fraud.

And also... the quantum experiments support Joy's model microscopically (and his model also supports the quantum experiments -- we expect loophole free tests to be possible) so all that is left is for a proper macroscopic test to see if the torsion of space does in fact manifest macroscopically. If not, then we just have to figure out why it manifests only microscopically and not macroscopically. None of this has to do with the fact that Bell's theorem is a completely dead issue.

[gill1109]

I agree, CHSH inequality and Bell's theorem are dead issues. Frauds, if you like. The only bound is +4.

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).

A really huge experiment (or really huge simulation experiment) can explore the whole surface, observing the sample correlations E(alpha, beta) = sum_{measurements k with settings alpha, beta} A_k B_k /N(alpha, beta) , where N(alpha, beta) is the number of runs at those two settings. In real experiments, each run k is performed under just one pair of settings. You cannot measure the polarization of one photon in two directions at the same time.

If resources are limited it is wise to restrict attention to a judicious set of settings. The optimal choice turns out to be the four combinations obtained from restricting alpha to the two angles 0 and 90 degrees, and beta to the two angles 45 and 135 degrees. The only reason for this is statistical efficiency. For a fixed number of runs N, if you want maximum discriminatory power between Bell-LHV on the one side, and QM or Christian-LHV on the other side, you should put all your efforts into recovering the correlation surface at cleverly chosen points where the difference between the two theories is largest. Obviously, you want at least two different angles for alpha, and at least two different angles for beta. It turns out that just two of each is best, provided they are these two pairs. This maximizes the power of the experiment. It's a kind of statistical version of Tsirelson's inequality, which as you know can also be proven within Christian's S^3 framework (Van Dam, Gill, Grunwald in IEEE-IT (2005), on arXiv in 2003http://arxiv.org/abs/quant-ph/0307125)

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; 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).

The 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.

However Christian appears to have backed off this unorthodox experimental design, thanks to warnings given by M. Fodje. I wonder if he is going to revise http://arxiv.org/abs/0806.3078. Since he's disowned his own experiment, I conclude that he has also withdrawn from our bet.

[Joy Christian]

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?

[gill1109]

Apparently there are two J. Christians. One at the University of Oxford who, in http://arxiv.org/pdf/0806.3078v2.pdf , wrote "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"

Must have been a different J. Christian.

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 j (or −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 (j · a) and sign (−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 (j · a) remains independent not only of the remote parameter b, but also of the remote outcome sign (−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 (j · a)} {sign(−j · b)}, (16)
where N is the number of trials.

[gill1109]

If not, what exactly is the difference? I'm on vacation right now so can't really do the detail of Bell's derivation of CHSH but it would be nice if someone could show the extra detail steps for the derivation. And keep in mind that the A and B functions are averages so show them like this, <A(a, lambda)>, etc.

From Michel's thread I gather that Bell's LHV CHSH derivation and Gill's version of CHSH are both "rigged" and nothing can violate them as far as violating a bound of 2. It is also easy to see from that thread that for both QM and quantum experiments the bound is 4 and not 2.

Did we ever resolve the questions at the start of this topic?

Seems to me that Fred Diether doesn't realize that every mathematical theorem is a tautology. It has to be true, by definition. He also doesn't define what exactly he means by "Gill's CHSH" and "Bell's CHSH". Gill derives several auxiliary results and a theorem about a spreadsheet, he then leaves mathematics and discusses experiments and meta-physics. At the end of his paper he turns to computer simulation models and argues that the spreadsheet picture can be a useful tool when exploring the limitations of LHV simulation models, under constraints designed to prevent the simulation from exploiting known loopholes. (Which are of course irrelevant when discussing simulation models of past experiments, because so far every experiment which had been done was badly flawed in one way or another ... something well known for decennia).

He also says "the A and B functions are averages". No: the A and B functions are functions. One can go on to talk about averages which one might denote like <A(a, lambda)>, meaning that A(a, lambda) is averaged over values of lambda in some domain. One had better explain what domain one is talking about and if there is a probability measure involved, what probability measure it is, and what it is supposed to stand for. eg are we averaging over the hypothetical population of all possible values of lambda which might arise in one run of an experiment, weighted according to their relative frequence in this population; or are we talking about an experimental average involving n particular values of lambda realised in n runs of the experiment as n single, independent, draws from that population.

Not making this distinction turns out to be the root cause of endless confusion about Bell, CHSH and all that. Most people participating in increasing the confusion (increasing the noise to signal ratio) did not read Bell's later works where he explicitly paid attention to this distinction, with the aim of getting the signal to noise ratio back on its feet.

The trouble is, theoretical physicsts know nothing about this distinction; experimental physicists know it my instinct but really only know about experiment, not about theory. Hence the bridge is missing in the physics community itself because of inadequate training both of theoreticians and of experimentalists. Neither has a good grounding in probability theory and neither has a good grounding in modern applied statistics. They think they know everything because Gauss already explained that errors always have a normal distribution and that straight lines should be fitted by least squares. It is a very sorry situation, and I have been fighting against it all my career. The younger physicists are beginning to notice that they are missing something. The old ones are fortunately dying out. But before they do so, they are still very influential.