Bell & CHSH type inequalities and experiments

[minkwe]

Second attempt at answering (short).
Alternative answer: the response you wrote about random sampling is very long and rather complex. Perhaps you can break it down into five of six steps, each with just one yes/no question at the end?

Ok let me break it down:

1. The correlations in the CHSH cannot all be independent random samples of any population. Because once one correlation is set, the others are fixed. Agree or disagree.
2. The issue is not that the correlations which you measure in any sample by random sampling are unfair. Rather, the issue is that the CHSH requires terms which are mutually dependent on each other in a specific non-random manner. Agree or disagree.
3. Therefore, the fair sampling assumption and procedures introduced to make sure experiments are producing fair samples of the population are trying to solve the wrong problem. You can not reduce the number of degrees of freedom to that required the CHSH, by fair sampling. Agree or disagree?
4. The correct upper bound for the number of degrees of freedom present in experimental data, and QM is 4 not 2. Agree or disagree?

[FrediFizzx]

4. The correct upper bound for the number of degrees of freedom present in experimental data, and QM is 4 not 2. Agree or disagree?

I would have to disagree with that if the 1971 Bell derivation of CHSH is correct and given that the Weihs, et al, dataset that violated CHSH was from single measurement run.

[minkwe]

4. The correct upper bound for the number of degrees of freedom present in experimental data, and QM is 4 not 2. Agree or disagree?

I would have to disagree with that if the 1971 Bell derivation of CHSH is correct and given that the Weihs, et al, dataset that violated CHSH was from single measurement run.

Maybe you have a different definition for what a "run" is. To me, a "run" represents all the particle pairs that contribute to a specific Expectation value, whether or not the actual angles where randomly interleaved during the experiment. Pick your favorite CHSH and Aspect type experiment, and look carefully at how each of the correlations E(a,b), E(a,b'), E(a',b') and E(a',b) is calculated. Then remember that each specific particle pair in an experiment is measured only once. You will quickly realize that no particle pair which contributes to E(a,b), contributes to any of the other correlations. So we have 4 disjoint sets of particles for the 4 correlations being calculated in any Aspect-type experiment. This point is never in any doubt and nobody contests it. So we have 8 data streams of 4 pairs. Since each of the values is independent of each other in this kind of experiment and they each have an upper bound of 1 and a lower bound of -1, the linear combination of those 4 correlations has an upper bound of 4.

It is really very easy to see that the expression:

S = E(a,b) - E(a,b') + E(a,b') + E(a',b')

Where E(a,b), E(a,b'), E(a',b') and E(a',b) are all independent of each other, and each has bounds [-1, +1], has an upper bound of 4. The fact that those correlations are mutually independent in Aspect type experiments means the correct upper bound for Aspect type experiments is 4. There is no other way.

As I keep explaining in many different ways, the original CHSH has an upper bound of 2 for that expression precisely because the terms are not mutually independent. It is obvious by looking at the above expression that the only way S can have an upper bound less than 4 is if those correlations are not mutually independent, ie if dependencies exist between them. That is indeed the case in the original CHSH as I have already explained and that is why we have a 2 as the upper bound since according to the derivation, they all are mutually dependent on each other.

It is therefore very clear now why experiments and QM appear to violate the CHSH. Because the terms from experiment and QM are mutually independent while those in the original CHSH are not. So Bell was wrong to conclude that QM violates the CHSH. Which means Bell was wrong to conclude that no theorem of hidden variables can reproduce the QM correlations, as we now know from Joy's work.

What part of the argument as I've outlined here is wrong or unclear?

[gill1109]

Anyways, I guess you agree with Michel on his 3 questions. Now, I don't see a problem with the way say the Weihs, et al, experiment did the CHSH violation by using aprox. N/4 for the data of each E(a, b)'s. And I don't see why it is a problem to do that in a simulation also.

Yes I agree with Michel on his 3 questions, and I have no problem with the Weihs et al. experiment, and I have no problem with anyone simulating an experiment like that in that way. No problem at all. This is the experiment we have been talking about all the time. This is what Bell - CHSH is all about.

Regarding my notation <AB> and your notation E(a, b): both of these notations are common in the literature and they are always both supposed to mean the same thing! A is the spin in direction a of Aiice's particle, B is the spin in direction b of Bob's particle, and the angle brackets stand for expectation or mean value or ensemble average.

Depending on the context, we might be talking about experimentally observed correlations, or we might be talking about theoretically defined correlations. People tend to use the same notation for theoretical and experimental quantities. It's natural. But it can be confusing, and this confusion has led Michel to his position that the whole Bell CHSH literature is mistaken.

One should distinguish apples from oranges! They are both fruit, for sure, both good for you, for sure, one can sometimes safely substitute an apple for an orange ... and sometimes one can't. One can and does compare them!

[gill1109]

1. The correlations in the CHSH cannot all be independent random samples of any population. Because once one correlation is set, the others are fixed. Agree or disagree.

Are you talking about correlations defined mathematically within some mathematical model, or correlations defined empirically and measured and computed numerically in an experiment? Please clarify.

[gill1109]

2. The issue is not that the correlations which you measure in any sample by random sampling are unfair. Rather, the issue is that the CHSH requires terms which are mutually dependent on each other in a specific non-random manner. Agree or disagree.

This is your issue, not mine. I disagree that this is an issue. We have to distinguish theory from experiment. And in experiment, we have to take account of statistical error, as well.

3. Therefore, the fair sampling assumption and procedures introduced to make sure experiments are producing fair samples of the population are trying to solve the wrong problem. You can not reduce the number of degrees of freedom to that required the CHSH, by fair sampling. Agree or disagree?

The question starts with the word "therefore". But I did not agree with your premise.

4. The correct upper bound for the number of degrees of freedom present in experimental data, and QM is 4 not 2. Agree or disagree?

I disagree with your use of the word "degrees of freedom" here. It's a technical term having precise (different) meanings in different fields. It is not applicable here.

I agree that if you want to write down a deterministic upper bound for what can come out of a finite amount of experimental data, with each correlation computed on a disjoint subset of data, the smallest upper bound one can give is 4. Proof. One run for each of the four usual measurement setting pairs. In three of the runs, the outcomes are equal. In the fourth, the outcomes are different. Three measured correlations equal +1. One equals -1. CHSH = 4.

[gill1109]

First attempt at answering (long).

Heinera already spent a long time trying to explain to you why this random sampling stuff is relevant.

I'm still waiting for Heinera's response about relevance, if he has shared that with you, please share it, I haven't seen it yet. All he did was ask me to confirm the obvious fact that a random sample of a population should have almost the same mean as the population. I continue to say it is irrelevant to my point and neither you nor he has demonstrated how random sampling enables you to restore the missing relationship between terms mandated by the CHSH. So I'm sorry your first attempt misses the mark by two miles.

First mile: you say "Nobody here is saying that exactly the same relationship holds between the correlations computed from four disjoint (but each separately, random)". But you surely are, that relationship is embodied in the CHSH inequality. Everytime you claim that an experiment violated the CHSH you are saying you expected that relationship to hold. Everytime you say QM violates the CHSH, you are saying you expect that relationship to hold. But I've been trying hard for months to get you to see that the correct relationship for experiments and QM is the one with an upper bound of 4.

Second mile: you say "Heinera and I are both trying hard to explain to you that they will all three be close to one another, if N is large. the three things here being (1) the correlation based on a random subsample of size about N/4 which we actually get to see, (2) the correlation which we would see if we would compute it on all N, and (3) the correlation which we would see in the second case if N were infinite." So you are saying again that the sample mean of a fair sample is close to the population mean which is uncontroversial and still irrelevant to the point I'm making and has never been contested by anyone.

I'm saying that the point which you are making (that the only certain bound we can claim for experimental data is 4) is irrelevant.

I demonstrated how, to use your words, "random sampling enables one to restore a missing relationship between terms mandated by the CHSH" in my paper. The relationship is no longer certain. The relationship is not restored perfectly, it is only restored approximately, with large probability. It is statistical. If N=10^8, the probability that CHSH will be above 2.001 is less than 0.001. Look carefully at what my Theorem 1 actually claims, and work through the proof in the appendix.

[FrediFizzx]

4. The correct upper bound for the number of degrees of freedom present in experimental data, and QM is 4 not 2. Agree or disagree?

I would have to disagree with that if the 1971 Bell derivation of CHSH is correct and given that the Weihs, et al, dataset that violated CHSH was from single measurement run.

Maybe you have a different definition for what a "run" is. To me, a "run" represents all the particle pairs that contribute to a specific Expectation value, whether or not the actual angles where randomly interleaved during the experiment. Pick your favorite CHSH and Aspect type experiment, and look carefully at how each of the correlations E(a,b), E(a,b'), E(a',b') and E(a',b) is calculated. Then remember that each specific particle pair in an experiment is measured only once. You will quickly realize that no particle pair which contributes to E(a,b), contributes to any of the other correlations. So we have 4 disjoint sets of particles for the 4 correlations being calculated in any Aspect-type experiment. This point is never in any doubt and nobody contests it. So we have 8 data streams of 4 pairs. Since each of the values is independent of each other in this kind of experiment and they each have an upper bound of 1 and a lower bound of -1, the linear combination of those 4 correlations has an upper bound of 4.

For me, a measurement run is you run the experiment (or simulation) for a certain total amount of particle pairs. Or this can be for a certain amount of time. Yes, each pair is measured once only. Correct, that we have 4 disjoint sets of pairs taken from the total of pairs in the measurement run.

It is really very easy to see that the expression:

S = E(a,b) - E(a,b') + E(a,b') + E(a',b')

Where E(a,b), E(a,b'), E(a',b') and E(a',b) are all independent of each other, and each has bounds [-1, +1], has an upper bound of 4. The fact that those correlations are mutually independent in Aspect type experiments means the correct upper bound for Aspect type experiments is 4. There is no other way.

Then Bell's 1971 derivation of CHSH is wrong. Where exactly is it wrong?

[gill1109]

It is really very easy to see that the expression:

S = E(a,b) - E(a,b') + E(a,b') + E(a',b')

Where E(a,b), E(a,b'), E(a',b') and E(a',b) are all independent of each other, and each has bounds [-1, +1], has an upper bound of 4. The fact that those correlations are mutually independent in Aspect type experiments means the correct upper bound for Aspect type experiments is 4. There is no other way.

Then Bell's 1971 derivation of CHSH is wrong. Where exactly is it wrong?

It is not wrong. It is a derivation of a relationship between ensemble averages.
When you take a finite sample, you observe some experimental averages.
They can be anything.
If the sample is very large and the experiment is very good, the experimental averages should start to get close to the ensemble means. Comprendes, mi amigo?

[FrediFizzx]

Ok, when you do a CHSH calculation for an experiment you count all the +1+1 coincidences and the -1-1 coincidences and add them together. Then subtract all the +1-1 coincidences and -1+1 coincidences and then divide by the total number, N, of all those coincidences for that angle pair. That is for E(a, b). Then you do it for the other angle pairs, E(a', b), etc. So basically with NE for the outcomes of A and B equal and NU for A and B un-equal,

E1 = (NE1- NU1)/N1
E2 = (NE2 - NU2)/N2
E3 = (NE3 - NU3/N3
E4 = (NE4 - NU4)/N4

Then you can do like in the QRC for the CHSH best calculation,

CHV=N[Max[Abs[E1+E2+E3-E4],Abs[E1+E2-E3+E4],Abs[E1-E2+E3+E4],Abs[E2+E3+E4-E1]]]

The Max function just takes the "Abs" set that has the highest value. The N function spits out the value. So... for sure there is no dependence on any of the E's with each other in the calculation for an experiment. But in the Bell CHSH derivation there is. I think I am now finally seeing Michel's point. That can only mean that CHSH as Bell derived it is in fact baloney.

[gill1109]

So... for sure there is no dependence on any of the E's with each other in the calculation for an experiment. But in the Bell CHSH derivation there is. I think I am now finally seeing Michel's point. That can only mean that CHSH as Bell derived it is in fact baloney.

No, this can only mean that you don't understand the difference between experimental physics and theoretical physics. And you don't see the role of statistics. Ever heard of error bars? Are you saying that all theoretical physics is baloney?

[gill1109]

Here you confuse the results of an experiment, and the output of a model, which are two different things. In an experiment, the E(a,b) from one set of particles is independent from the E(a,b’) from a completely different set of particles, as you say. The point is that it is impossible to construct a LHV model so that the same is true for the model.

No I'm not. I asked 3 questions, 2 about Theoretical Bell and CHSH constructs and one about experiments. You objected only to my answer about experiments. That's what I'm clarifying to you that the same is true for a LHV model of the experiment.

The experiment uses a completely separate set of particles, why would you insist on using a single set of particles when modelling it? Even the QM predictions are for separate sets of particles.

Why would you insist on using a single set of particles when modelling it? Answer: ask Einstein, Podolsky, Rosen. Or read Bell. Or both, and more.

A local hidden variables model is a *model* for one pair of particles and it allows you to define *within the model* all the different measurement outcomes of all the different possible measurements in the model, at the same time. After all, in the model, there is a function A(a, lambda)! (I'm not saying you would want to do this. Of course not. I'm just saying that you could do this.)

Since the birth of quantum mechanics, people have asked themselves whether such a model could underlie quantum mechanics. In other words, though QM appears weird, maybe there is a "normal" hidden layer, behind the scenes. All that quantum weirdness would not then really be weird, it would only be apparently weird.

This is not a stupid question since till quantum mechanics such models were always possible and indeed everyone believed that deep down the universe is just a bunch of billiard balls bumping into one another in deterministic fashion. Everyone believed that all instances of randomness in physics are just instances of unknown initial conditions, uncontrollable initial conditions, chaos theory (sensitive dependence on initial conditions causing exponential divergence between trajectories even when they start very close together). According to this point of view, every physical random generator is a pseudo random generator.

Bell's genius was to take the EPR argument (that QM was incomplete, because QM predictions itself showed that such a hidden layer exists), turn it on its head, and turn it into an experiment which might actually decide whether or not a hidden layer can explain the predictions of QM.

Note the EPR argument talks about an entangled state of two particles and considers the two pairs of maximally incompatible observables position, momentum on each particle. One can rewrite EPR for spin half system and again one consider the maximally incompatible observables sigma_x, sigma_y, sigma_z on both particles. Or take just two spin observables for each particle. Bell showed that something really interesting turned up when you played a bit more with this idea. Clauser-Horne-Shimony-Holt played a bit more and ended up with the CHSH set-up where the most exciting observables to work with are sigma_x, sigma_y on one of the particles, and (sigma_x+sigma_y)/sqrt 2 and (sigma_x-sigma_y)/sqrt 2 on the other particle. Later confirmed by Tsirelson to give the largest possible deviation from what could happen classically.

Later, "A tight Tsirelson inequality for infinitely many outcomes" by S. Zohren, P. Reska, R. D. Gill, W. Westra http://arxiv.org/abs/1003.0616, Europhysics Letters 90 (2010) 10002, took the canonical generalization of CHSH to higher spins, took the spin number to its limit, and got an EPR-like experiment back again. The observables are of course position and momentum on one particle, and (position + momentum)/sqrt 2 and (position - momentum)/sqrt 2 on the other particle.

[Joy Christian]

Bell's genius was to take the EPR argument (that QM was incomplete, because QM predictions itself showed that such a hidden layer exists), turn it on its head, and turn it into an experiment which might actually decide whether or not a hidden layer can explain the predictions of QM.

This is wrong and it should be corrected every time it is mentioned. Bell intended to take the EPR argument and turn it on its head. But he miserably failed in doing so. He made fatal errors in attempting to do so. One of his mathematical errors is in his very first equation, as I have explained on my blog. The other major mistake of his is the same mistake that von Neumann had made which Bell himself contributed in popularizing. von Neumann's mistake was not discovered by Bell alone---it was already known to Einstein and was formally published as a "disproof" by Grete Hermann 30 years before Bell. There are also other problems with Bell's argument, such as the super-determinism "loophole", which was pointed out to him by Clauser, Horne, and Shimony in Epistemological letters. Thus Bell's argument was stillborn.

[gill1109]

There are also other problems with Bell's argument, such as the super-determinism "loophole", which was pointed out to him by Clauser, Horne, and Shimony in Epistemological letters.

The super-determinism loophole makes physics meaningless. I don't think we can blame Bell for not taking it too seriously.

We also can't blame him for not noticing that one or two other people had earlier seen von Neumann's mistake ... just about nobody else was aware of it, either. In fact at the time when Bell wrote about it, the communis opinio was that von Neumann was right (which all goes to show that an entire scientific community can be stunningly wrong).

Indeed, Joy Christian and a few other people think Bell made some kind of fatal error. Richard Gill and a whole lot of other people think that Joy Christian is badly mistaken. See for instance http://www.scottaaronson.com/blog/?p=1028. Could an entire scientific community be stunningly wrong? Yes, of course. People should judge for themselves, not follow the crowd. IMHO both cases (von Neumann's no go theorem, Joy Christian's no Bell theorem) are pretty transparent.

But this is going seriously off-topic.

[Heinera]

Good. So let's just assume that Joy's experiment has been performed. I am now presented with a two column list, where the first column is a collection of vectors lambda, with the other column equal to minus lambda. I now pick two angles a and b. For each row in the list, I compute the two outcomes according to Joy's prescription (they will take values among +1 and -1). So now I have a two column array of length N, with values in the set +1 or -1. Now I compute the E(a,b) correlation of these two variables on all the N rows. Then I randomly (according to the description I previously described) select a subset of approximately size N/4. I then compute the correlations again on this subset. Let's call that correlation E'(a,b). I guess you agree that E(a,b) is approximately equal to E'(a,b)? (and sorry, I'm afraid it's yet one more yes/no question).

Yes, if you are doing proper random sampling, E'(a,b) from the population will be almost the same as E'(a,b) from the random sample.

Good. So I still have that list where the first column is a collection of vectors lambda, with the other column equal to minus lambda. I now compute four correlations E(a,b), E(a,b'), E(a,b'), E(a',b'), all on the entire set. Then I randomly select a subset of approximately size N/4, and recompute E(a,b) on this subset. I assume I still get the same value as the E(a,b) computed previously (i.e., the first of the four correlations I computed on the whole set)?

I guess you see where I am going with this now. Which of the four correlations do you assume will change when going from the entire set to 4 random, disjoint subsets?

[Joy Christian]

Indeed, Joy Christian and a few other people think Bell made some kind of fatal error... Could an entire scientific community be stunningly wrong? Yes, of course. People should judge for themselves, not follow the crowd.

I am disappointed that you would link Scott Aaronson's racist blog in this forum. There is no place for racism in science. On Scott Aaronson's blog there are racist comments about my ethnic background and ad hominem attacks on me by people who should know better. Scott Aaronson, for instance, did not read a single line of my argument before criticising it, and to this very date Richard Gill has not corrected his silly mathematical errors in the misguided straw-man attack he launched against my work to protect his deep-seated prejudices: http://arxiv.org/abs/1203.2529.

In any case, the fact remains that Bell's argument was stillborn. Bell's argument is no sillier than von Neumann's argument. You can read all about Bell's mistake here.

This is wrong and it should be corrected every time it is mentioned. Bell intended to take the EPR argument and turn it on its head. But he miserably failed in doing so. He made fatal errors in attempting to do so. One of his mathematical errors is in his very first equation, as I have explained on my blog. The other major mistake of his is the same mistake that von Neumann had made which Bell himself contributed in popularizing. von Neumann's mistake was not discovered by Bell alone---it was already known to Einstein and was formally published as a "disproof" by Grete Hermann 30 years before Bell. There are also other problems with Bell's argument, such as the super-determinism "loophole", which was pointed out to him by Clauser, Horne, and Shimony in Epistemological letters. Thus Bell's argument was stillborn.

[minkwe]

Good. So I still have that list where the first column is a collection of vectors lambda, with the other column equal to minus lambda. I now compute four correlations E(a,b), E(a,b'), E(a,b'), E(a',b'), all on the entire set. Then I randomly select a subset of approximately size N/4, and recompute E(a,b) on this subset. I assume I still get the same value as the E(a,b) computed previously (i.e., the first of the four correlations I computed on the whole set)?

I guess you see where I am going with this now. Which of the four correlations do you assume will change when going from the entire set to 4 random, disjoint subsets?

Please you have not been reading what I've been saying. Do yourself a favour and read my last detailed post to you where I clearly explain what I'm talking about. You keep repeating the uncontroversial accepted fact that a random sample of a population should have the same mean as the population, completely missing the point.

The terms in the CHSH are not independent. But those from a random sampling experiment are. Even Richard accepts this.

He knows a thing or two about statistical degrees of freedom and he knows that the CHSH bound does not apply in this case. The only certain bound one can give is 4. I must say, that he's absolutely right there.

[gill1109]

Good. So I still have that list where the first column is a collection of vectors lambda, with the other column equal to minus lambda. I now compute four correlations E(a,b), E(a,b'), E(a,b'), E(a',b'), all on the entire set. Then I randomly select a subset of approximately size N/4, and recompute E(a,b) on this subset. I assume I still get the same value as the E(a,b) computed previously (i.e., the first of the four correlations I computed on the whole set)?

I guess you see where I am going with this now. Which of the four correlations do you assume will change when going from the entire set to 4 random, disjoint subsets?

Please you have not been reading what I've been saying. Do yourself a favour and read my last detailed post to you where I clearly explain what I'm talking about. You keep repeating the uncontroversial accepted fact that a random sample of a population should have the same mean as the population, completely missing the point.

The terms in the CHSH are not independent. But those from a random sampling experiment are. Even Richard accepts this.

He knows a thing or two about statistical degrees of freedom and he knows that the CHSH bound does not apply in this case. The only certain bound one can give is 4. I must say, that he's absolutely right there.

Michel, everyone knows this! You don't need to keep repeating what we all already know.

What you don't realize, is that what you see as the death blow for application of CHSH to real experiments (or simulated real experiments) is not a death blow at all.

If you want to apply the CHSH bound to an experiment, you have to take *two* steps.

The first step is, you assume that local realism is true. It's a straw-man you could say: a hypothesis which you want to test. You take it as a working hypothesis and you see if the experimental data fits or not to the hypothesis.

The second step, is that you take account of statistical error. A bound on a population expectation value obviously doesn't apply to an experimental average. Infinite theoretical population versus finite actual sample.

It has to be seen as a statistical bound. It's a bound on the expectation value. Which we don't know. But sample averages are close to population mean values, if the sample is large and random. The N/4 runs on which a particular correlation is based, are a random sample of the N runs of the whole experiment, which is a random sample of the infinite ensemble of runs which we obtain when we imagine an experiment with infinitely many runs. (One run = one pair of particles).

What do experimenters do? They don't look at whether or not "observed CHSH" is bigger than 2. That would be stupid. Even if local realism is true, "observed CHSH" can exceed 2 with probability half. After all, we can think of situations with LHV where the population value of CHSH is exactly equal to 2.

No, they observe a value of CHSH, say they observe the value 2.70, and they compute a standard error (error bar), say 0.07, and they say "wow, we observed a deviation of 10 standard errors above the LHV highest possible mean value of 2". They calculate (2.70 - 2)/0.07 = 10, right.

Just like the Higgs Boson. The observed signal was so many standard deviations above the *mean value* of the signal which you would have if there were no Higgs, that they could calculate that were there no Higgs boson, the chance of getting such a large signal is 1 in a billion (or whatever it was). I believe particle physicists throw a party when the signal is 5 times as large as the standard error. Obviously, they know for sure that observed signals have a Gauss distribution.

[minkwe]

S = E(a,b) - E(a,b') + E(a,b') + E(a',b')

It is truly astonishing that a bunch of mathematicians are having difficulty seeing clearly that if each of the terms in the above expression has bounds [-1,+1], then the expression has an upper bound of 4 when the terms are all independent. But if and only if the terms are mutually dependent, can the upper bound be reduced below 4.

Astonishing indeed.

[Heinera]

Good. So I still have that list where the first column is a collection of vectors lambda, with the other column equal to minus lambda. I now compute four correlations E(a,b), E(a,b'), E(a,b'), E(a',b'), all on the entire set. Then I randomly select a subset of approximately size N/4, and recompute E(a,b) on this subset. I assume I still get the same value as the E(a,b) computed previously (i.e., the first of the four correlations I computed on the whole set)?

I guess you see where I am going with this now. Which of the four correlations do you assume will change when going from the entire set to 4 random, disjoint subsets?

Please you have not been reading what I've been saying. Do yourself a favour and read my last detailed post to you where I clearly explain what I'm talking about. You keep repeating the uncontroversial accepted fact that a random sample of a population should have the same mean as the population, completely missing the point.

Sorry for being dense, but I guess your answer to my question is that it is uncontroversial that none of the four correlations would change?