Talk:Bell's theorem/Archive 2

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Perhaps we should have less technical introduction instead of talking about local variables couldn't we say "Bells theorem demonstrates that assuming the predictions of quantum mechanics are correct, the world cannot be both deterministic and local." Zvis (talk) 11:26, 5 January 2010 (UTC)

Confusion in description of correlations

The Alice-Bob description appears to be broken. The article states "When Alice and Bob measure the spin of the particles along the same axis (but in opposite directions), they get identical results 100% of the time. But when Bob measures at orthogonal (right) angles to Alice's measurements, they get identical results only 50% of the time." - This not correct for Photons used in Aspect experiments. When the analyzers are aligned the results are 100% anti-correlated and 100% correlated at 90 degrees. It then makes another odd claim in stating ".. and when the angle between them is zero no correlation." The zero correlation occurs at 45 degrees. In fact, the Aspect paper shows that QM predicts perfect anti-correlatoin at zero degrees, correlation at 90 degrees, and zero correlation at 45 degrees (or 135 etc.) Bell's work showed that the QM prediction (QM predicts a correlation curve as -cos(2*theta)) could not be a result of local hidden variables because such local hidden variable models could, at best, give correlations that lie below a straight line between correlation and anti-correlation. This means that, a 22.5 degrees QM gives a .7071 correlation and local theories cannot exceed 0.5 (if Bell has made correct assumptions as to the rotational properties of local hidden variables). Note that the mathematical content below this strange description is correct, and contradicts the descriptions. Also, the tables are misleading because they list alignments (0,45,90) for which local-hidden variable models can match the quantum prediction.

This article would be a lot clearer (IMHO) if it was illustrated with a picture of the QM prediction (Cos 2 theta) and the limit of the classical (Hidden variable) theories as Bell determined. (The latter being a straight-line between correlation and anti-correlation, with the maximum deviation (22.5 deg) pointed out.

The above criticisms are true; in particular, the illustration here near the top is NOT an illustration of the setup for the original inequality. In the original, Alice measures the polarization at an angle and Bob measures the polarization at some other angle. BUT they do not have relative angles. So define the angle that Alice measures it at as z; then Bob measures at some angle relative to z, call it theta. If theta is 0 or 180, they get consistent results (they don't actually know what the other one had since they don't know if their +z/2 (say) is from the other measuring -z/2 with their same orientation or the other having apparatus at angle -z/2 with a result of +z/2.) In between, at theta =90, their results are totally uncorrelated. On the other hand, when the options are doubly uncertain (Alice at 0 or 90 and Bob and 45 or 135 for a total of 4 possibilities) then the second illustration does hold. Here the sqrt(1/2) probability holds; they always get .707...correlation; but the one time they are anti-correlated. —Preceding unsigned comment added by YouRang? (talkcontribs) 21:14, 13 July 2009 (UTC)

If the second illustration is set up for angles A (0 and 90) and B (45, 135) then the relative angle will be an integer multiple of 45 degrees. In this case a local hidden variables model (straight line from correlation to anticorrelation) matches a QM model. I think you will find that both QM and LHV can give the same numbers at relative analyzer angles of 0, 45, 90, 135 etc. Also, in the Aspect experiment, there was no constraint on the initial alignment of the source, so the choice of fixed alignments is illusory, the source alignment is random and therefore only the relative analyzer angle is significant. If you look at Aspect's original paper he plots the correlation for all relative angles, and only convincingly breaks the inequality at 22.5 degrees because of the poor quantum efficiency of the detectors. —Preceding unsigned comment added by Grok42 (talkcontribs) 00:48, 31 August 2009 (UTC)

Re: Valid derivations

Re: CT's claim The derivation of the CHSH inequality that is presented is suspect.

Come on CT, you have to do better than that.

In fact, by the law of large numbers, the proof currently in the article is identical to the proof in Bell's 1971 paper with one small difference: in Bell's paper, he also allows instruments to contain hidden variables. However, this introduces no new ingredients into the proof, since the average values are still going to be in the closed interval -1, 1. In CT's proof this fact is not made clear. Aside from the that, as I said, both proofs are mathematically equivalent.--CSTAR 17:47, 23 July 2005 (UTC)

I strongly dispute your claim that the proofs are equivalent. The failure to allow for components of hidden variable associated with the detectors is no minor matter. A more important failure, though, is the omission of any mention of the hidden variables which were are the heart of the EPR debate. Furthermore, the whole structure of the thing is unrelated to that of the actual experiments. Surely we want a derivation that does apply to the real world? Bell's original version did not quite fit the bill (it required the assumption that when polarisers were parallel you necessarily got 100% agreement. This is a quantum-theoretical idea that, although usually approximately true in practice, is not required to be always true under local realism.) His later derivation was designed for use in real experiments and has been used for this purpose, though, unfortunately, it is generally misinterpreted in a manner that may turn out to be critical (see CHSH inequality page).
I've just been reading part of Nick Herbert's "Quantum Reality" and confirmed my suspicions that there are quite a number of invalid proofs of Bell's theorem around, mainly in popular books but also in serious articles. His derivation is restricted to cases where there is rotational invariance and there are no missing values. It, too, does not explicitly mention hidden variables.
These various derivations, valid only in very special circumstances, will not help readers understand the actual experiments. The derivations will make it hard to understand even the detection loophole, which is intrinsically very straightforward. People reading popular books are left with the impression that something absolutely impossible has been demonstrated, whereas the truth is that no loophole-free test has yet been done and the question of whether or not local realism is true is wide open.
Why does wikipedia have to play this game? Why can't it present an derivation that Bell himself supported instead of a pseudo-derivation invented later?
Caroline Thompson 09:49, 24 July 2005 (UTC)
Owwww, this is indeed painful. There is some merit to sticking to the historical version of the proof, and I admit the formulation as integrals over lambda is how I remember the proofs being given. As to the validity of a derivation... Naively, the derivation given in the current article appears to obey the standard assumptions of probability theory. Admittedly, probability theory has somewhat shaky axiomatic/(non-)constructive foundations; some things, like differential equations involving two or more random variables, remain undefined to this day. The axiomatic foundations of "hidden variables" are probably in much worse shape. However, these concerns don't seem to have a direct impact on Bell's theorem; at the survey level taken here, it all seems to be "sufficiently correct" to me.
Thus, naively, much of the debate here is bordering on "original research": If a derivation is flawed in some subtle way, possibly due to an inconsistency in the axiomatic foundation of probability theory, that deserves a journal article, not a WP article. If alternative derivations, which give the same result, can be distinguished by performing some experiment, then that experiment should be written up and published in a journal. It should not be debated on these pages. This is the wrong forum for such arguments. linas 19:22, 24 July 2005 (UTC)

Re: Valid derivation (bis)

In the previous section I made the claim by the law of large numbers, the proof currently in the article is identical to the proof in Bell's 1971 paper with one small difference. I have made similar arguments in discussions with Drezet. OK here's the proof of this claim. The following text has been produced (using Xemacs global replace mostly) from a portion of the article with additional editing inserting line breaks for readability, replacing use of averaging over sequences with integrals over a sample space and inserting a definition of a hidden variable model. (Note that in the article in its current version, I was using an informal model referring the reader to other articles where formal models were discussed).The proof is mathematically identical to that given by Bell's 1971 paper (e.g. prove that a certain random variable has value ≤ 2 and integrate) Now of course I don't expect CT to accept this, but I argue that any fair minded person with the knowledge of the Law of Large Numbers and access to Bell's 1971 paper will see the equivalence.

Bell's thought experiment

Bell considered a hypothetical setup in which two observers, now commonly referred to as Alice and Bob, perform independent measurements on a system S prepared in some fixed state. Each has a detector with which to make measurements. Moreover, on each trial, Alice and Bob can choose between various detector settings; after repeated trials Alice and Bob collect statistics on their measurements and correlate the results. In one version of this setup, Alice can choose between two detector settings to measure one of A(a) or A(a′), and Bob can choose between detector settings to measure either B(b) or B(b′).

There are two key assumptions in Bell's analysis: (1) each measurement reveals an objective physical property of the system (2) a measurement taken by one observer has no effect on the measurement taken by the other.

In the language of probability theory, one would say that a series of measurements corresponds to a set of samples of a random variable. Depending on how the electrons were prepared, one might expect the measurement of one electron to somehow correlate with a corresponding measurement of the other: the random variables are assumed to not be independent, but linked in some way. None-the-less, because of the independent measurements, there is a limit to the amount of correlation one might expect to see. The Bell inequality expresses that maximum amount of correlation one should expect.

A version of the Bell inequality appropriate for this example is given by Clauser, Horne, Shimony and Holt, and is called the CHSH form:

where C denotes correlation.

Experimental tests of Bell inequalities, such as the above, on quantum mechanical systems, show a correlation in excess of this limit. Thus, one seems to forced to conclude that one or both of the assumptions fails for quantum mechanics. The first assumption is roughly analogous to the assumption of local realism; the second corresponds to the idea that no hidden message is exchanged between the particles or the particle detectors at the time of the measurement, the one particle "telling" the other how it should get measured.

Statement of Bell's theorem

In statistics, the correlation coefficient of random variables X, Y is

where σX is the square root of the variance of X. However, in this article, we will refer to the closely related, but unnormalized quantity

as the correlation.

We assume there is a hidden parameter space Λ and the observed outcomes by both Alice and Bob result by random sampling of the parameter λ ∈ Λ. Moreover, we assume that observed values are functions of the local detector settings and the hidden parameter only.

Value observed by Alice with deterctor setting a =A(a, λ)
Value observed by Bob with deterctor setting b =B(b, λ)

In particular, the hidden parameter space Λ has a probability measure ρ and the expectation of a random variable X on Λ with respect to ρ is written

where for accessibility of notation we assume that the probability measure has a density. Bell's theorem. The CHSH inequality (1) holds under the hidden variables assumptions above.

For simplicity, let us first assume the observed values are +1 or −1. ; we remove this assumption in Remark 1 below.

Let λ ∈ Λ. Then at least one of

is 0. Thus

and therefore

Remark 1. The correlation inequality (1) still holds if the variables A(a, \lambda;), B(b,\lambda;) are allowed to take on any real values between -1, +1. Indeed, the relevant idea is that each summand in the above average is bounded above by 2. This is easily seen to be true in the more general case:

To justify the upper bound 2 asserted in the last inequality, without loss of generality, we can assume that

In that case


Remark 2. With the extension given in Remark 1, CHSH inequality still holds even if the instruments themselves contain hidden variables. In that case, averaging over the instrument hidden variables gives new variables A(a, λ), B(b, λ) on Λ to which we can apply the previous remark

This is a great improvement, but it is still, I'm afraid, misleading. Do re-read Bell on the subject. The variables we measure don't take on any value between -1 and +1. They do frequently, however, take the value 0. The reason that the average for given λ is not always +1 or -1 is not merely that you sometimes get the opposite of what you'd expect but because many of the values are 0.
Incidentally, do we need to have the general definition of a correlation given here? Isn't it enough to say that what Bell looked at was the quantum correlation? We could even (says she, hopefully!) get away from any use of X and Y notation! And (dare I say it?) Latex. Personally, I much prefer the smaller font of ordinary text.
Another point: I've always understood Bell's theorem to have two parts: an inequality that is obeyed under local realism plus the observation that QM infringes it. Caroline Thompson 09:06, 25 July 2005 (UTC)
Misleading? Look at equations (8) in his 1971 paper. Compare them to eq (7). --CSTAR 13:04, 25 July 2005 (UTC)
Caroline, +1, -1 and 0 are elements of the set "any value between -1 and +1". Assuming a smaller, restricted set does not change the math. So I don't understand what you are complaining about. As to quantum correlation, that article is horrid, and needs to be deleted or re-written to actually say something meaningful. Right now, its completely garbled. linas 14:04, 25 July 2005 (UTC)
CSTAR and Linas: I'm interested in physics, not maths. A mathematical truth such as the fact that the set -1, 0 and +1 is a subset of the set of values between -1 and 1 is not relevant to the issue. The theorem concerns what is measured, and the values measured cannot take just any old real value -- they are restricted to the three (well, not quite true, since in actual experiments there is a fourth possibility, in that you can get both + and - at the same time). Anyway, the point is that the mathematical derivation ought to be as clearly related to the experiments it models as possible. It ought not, therefore, to imply that the apparatus used can give non-integer readings.
Re the quantum correlation page, only the first sentence is actually needed. The rest is context. Perhaps this could be made clearer. Caroline Thompson 20:47, 25 July 2005 (UTC)
I take it my opinion that Bell's theorem comprises two parts -- the local realist inequality and the QM prediction -- has been voted out. We really need an independent opinion on this. Apart from this doubt, though, the introduction is now good. In view of the definition you've assumed, does the QM derivation now deserve to be present?
Incidentally, I note that there is a link to the now re-directed "Bell test loopholes" page. Surely we can say a little more on these than is given in Bell test experiments? The topic does deserve a page of its own.
I revised the quantum correlation page late last night and am not sure I've got it right. I found myself inevitably beginning to transform it into a page on the detection loophole. The point is that the local realist formula can cover null outcomes easily, but the QM one (which I probably should have given as well) is only intended for the case where all outcomes are +1 or -1. The only way of making the QM theory apply to experiments with inefficient detectors is to assume that we can take just the set of coincidences and treat these as if they were the set of emitted pairs. In other words, we are forced to assume "fair sampling". This is why the separate definition of "quantum correlation" is important. When there are no null results, because the mean on each side is (under rotational invariance) zero and because all results have absolute value 1, the definition coincides with the ordinary one. When there are some null values, the difference becomes critical. [I'll put this para in the "quantum correlation" talk page as well.] Caroline Thompson 12:52, 26 July 2005 (UTC)


I'm not sure I agree with this:

The desire for a hidden-variable theory was based on the set of psychological preconceptions about "how the real world works" based on one's contact with the macroscopic

It would be as though we could reduce Kant's transcendental aesthetic to psychology. I'm sure it has been tried, but I don't think this psychological reductionism is a good idea.--CSTAR 13:58, 19 July 2005 (UTC)

I was interrupted while writing that sentence. Change it. I can't find the right word to express the idea of "literalist belief in macroscopic-based aesthetics based on ordinary mental conditioning" that I'm grasping for in this paragraph. linas 01:27, 20 July 2005 (UTC)

is memeplex fits the definition:belief ,based on Ordinary mental conditioning, education/indoctrination/propoganda?. a replicated (by mental conditioning ) belief would be such memeplex.

Local realism

CSTAR, I'm afraid your present definition is still somewhat misleading. You say:

In order to formulate Bell's theorem, we formalize local realism as follows:
There is a probability space Λ and the observed outcomes by both Alice and Bob result by random sampling of the parameter λ ∈ Λ.
The values observed by Alice or Bob are functions of the local detector settings and the hidden parameter only. Thus
Value observed by Alice with detector setting a is A(a, λ)
Value observed by Bob with detector setting b is B(b, λ)

I think we need to explain that the "natural" interpretation of the above, in which the hidden variable space is entirely associated with the source, might apply to Bell's original inequality but not to those used in practice. The latter employ allow for the fact that there will be factors other than the actual detector settings that are associated with the polarisers and detectors and determine the actual outcomes, hidden variables associated with the source determining only the probability of detection. Though the local factors can be regarded as further components of our hidden variable, they play a logically different role and Bell, in 1971, derived his version of the CHSH inequality on the assumption that, because they were independent on the two sides, they could be averaged out and the derivation of the inequality could proceed by replacing the individual outcomes A and B by their averages over these strictly local components. In 1974 Clauser and Horne produced a derivation that better reflected what is being assumed. They worked directly in terms of probabilities of detection, not individual outcomes.

Anyway, the long and the short of it is that the definition you give is, in practice, too restrictive. If you interpret hidden variable space as including the local components, I don't think your derivation will make sense. You need to replace outcomes by either average outcomes or (as in CH74) probabilities. Caroline Thompson 09:48, 30 July 2005 (UTC)

The structure of the proof as it now exists in the article, follows exactly Bell 1971. As to your comment about hidden variables including local components, this is just a remark in Bell´s paper (and is a remark in the article). I will be happy to put in the mathematical details, which are fairly straightforward.CSTAR.
This matter of including local components is more than just a remark! It is the reason why he proceeded to do the derivation in terms of averages for given detector setting and &lambda:, and the reason why Clauser and Horne worked in terms of probabilities for the same parameters. The definition of local realism as it stands is definitely misleading, as is the fact that you work through the whole derivation as if what is measured in a given trial can take any real value and only later remark that the derivation applies to averages. I haven't looked carefully at the mathematical details. They are probably correct. It's the wording I'm objecting to and the notation.
Incidentally, I used underscores in the CHSH inequality page only because I could find no way of doing overscores. They really ought to be overscores -- the standard notation for averages. Caroline Thompson 09:14, 1 August 2005 (UTC)
The page is steadily improving! What it needs now is clarification of what the local realism assumption implies. How about includingi somewhere (before the second derivation) the following:
Though the important components of the hidden variable λ from the point of view of Bell's logic are the ones associated with the source and shared by Alice and Bob, there are others that are relevant to the separate detectors, the these others being independent. This argument was used by Bell in 1971, and again by Clauser and Horne in 1974, to justify a generalisation of the theorem forced on them by the real experiments, in which detector were never 100% efficient. The derivations were re-worked in terms of the averages of the outcomes over the local detector variables. The formalisation of local realism was thus effectively changed, replacing A and B by averages and retaining the symbol λ but with a slightly different meaning. It was henceforth restricted (in most theoretical work) to mean only those components that were associated with the source.
Caroline Thompson 09:32, 12 August 2005 (UTC)

Normalised or non-normalised correlation?

CSTAR, I'm afraid there is still a problem here. You are careful to state, correctly, that the correlation C used in Bell test derivations is an "non-normalised" one, but do not mention the fact that the estimated used in practice is normalised. This is a real problem and lies at the heart of the detection loophole. Caroline Thompson

You are using "normalized" in two different senses.--CSTAR 14:19, 14 August 2005 (UTC)
True, which all goes to show that we are not talking about "correlation" but about a rather different statistic, quantum correlation. The way in which this is estimated in practice is not the way you'd do it if you were estimating ordinary (normalised) statistical correlation. The literature often used the term "normalised correlation" but this refers to special kind of normalisation -- division by the total number of coincidences. It's perhaps better to avoid the term and say that the calculation is all done on the post-selected set of data for which coincidences are observed, i.e. selecting those trials for which outcomes on the two sides happen to be simultaneously non-zero. Caroline Thompson 08:36, 15 August 2005 (UTC)

Did Bell break Quantum Mechanics?

I'm a new user here, so please forgive any breaches of protocol on my part. I started looking into this topic a few days ago. What I have discovered so far is that Bell's Theorem implies at least one of three things:

1) Logic does not apply to the quantum world.

2) There is no intrinsic physical reality in anything. (The existence of objects depends upon our observation of them.)

3) Quantum effects are non-local.

And now you have all sorts of folks parading all over the internet arguing like demented politicians for their candidate for the truth. The truth is with a dilemma, or rather a trilemma in this case, for a conclusion no one knows what the truth is.

I suggest that Bell's theorem is not so profound as it is generally believed to be. EPR was published to prove that quantum mechanics is incomplete. Bell took that suggestion and forced completeness upon it by inclusion of the parameter . (In all of this few seem to recall Goedel's proof that no logical system; e.g. quantum mechanics, that obeys Peano's arithmetic can be both complete and self-consistent.) As quantum mechanics is complete with the inclusion of Bell's hidden variable it is no longer self-consistent; i.e. it gives nonsense results because because we have exceeded the bounds of the set it describes -- that of observable objects.

--Mr EE 11:43, 2 September 2005 (UTC)

Well, careful there. You have untrained novices with little or no formal education in mathematics parading all over the internet, arguing about quantum. I think the actual academic establishment, i.e. professors, those with PhD's who have worked with QM for decades, these people have a broadly accepted orthodoxy that has not really been "broken" or overturned or challenged in 80 years. There are definitely many extremely interesting things in QM, but the subtle parts are considerably more subtle than what can be reached/understood informally, by the layperson. linas
P.S. the standard orthodox answers are 1) false. 2) its not physics, its philosophy (although see einselection) 3) true. But thinking about locality hinders true understanding. linas 14:46, 2 September 2005 (UTC)
Completeness in the sense of Godel's theorem and in the sense of physics are two entirely different things. --CSTAR 14:34, 2 September 2005 (UTC)
The appeal to authority doesn't fly as far as I'm concerned. I have studied QM myself for years. When originally exposed to Bell's Theorem my reaction to it was one of disbelief and dismisal. My current interest in it stems from a remark I encountered in the course of another discussion to the effect that "Bell's Inequality" is a cracked pot notion. Bell's Inequality as far as I'm concerned is purely a math problem with no necessary recourse to physics. However, its implications for physics if true, i.e. if Bell's Theorem is true, are admittedly bizarre. I agree that 1) is false. If we say that logic is invalid in any space, we kick the legs out from under all reason. While true that 2) is a question of philosophy, you draw a false distinction between philosophy and physics when you say "it's not physics, it's philosophy." Physicists in this sense are a subset of philosophers; if not, then what are you doing here? 3) I would not say that thinking about locality hinders understanding, though I do say that thinking about locality is irrelevant. The wave function is not itself an observable, and thus I will never be able to measure the effect of non-locality upon it. Now as to CSTAR's comment, Bell included specifically to provide the kind of completeness that EPR had said quantum mechanics lacked by extending the domain of quantum mechanics to include "unobserved observables;" i.e., hidden variables of the kind that EPR, and particularly Einstein, insisted must exist.
Goedel's idea of completeness was that the logic in question, L, be able to definitively prove or definitively disprove any question to which it properly pertains. In this case, , and that QM answer any question about physics was specifically what Einstein had in mind when he stipulated that QM was incomplete. If you don't think so, read the original EPR paper.--Mr EE 18:15, 2 September 2005 (UTC)
The technical meaning of completeness in Goedel's completeness theorem is that a theory is complete iff a proposition is provable precisely when it is true in every model. Goedel's incompleteness theorem has various versions:model theoretic and proof theoretic, the proof theoretic being the more familiar one. But in any case, completeness in QM DOES NOT mean that it is able to answer every quesion about physics. Completeness of QM has a much more limited meaning, which is, whether properties that in the EPR terminology, are elements of physical reality correspond to elements (such observables) in QM.
Re: Your comment The appeal to authority doesn't fly as far as I'm concerned.
Who's making this appeal? I'm just telling you what I know, you can believe it or not.--CSTAR 18:42, 2 September 2005 (UTC)

Well, as to appeal to authority: most mainstream physicists have been living with and have accepted Bell's results for four decades. QM won, local theories lost. I don't think anyone serious and knowledgable runs around and says "Bell is broken"; QM isn't broken either.

RE: point 2) -- I was stating the current orthodoxy, not my beleifs. Its currently very hard to refine the notions of observer vs. observed in QM (i.e schroedingers cat/ wigners friend). The work on einselection is the *only* serious attempt I know of to work on this question. Hmm, actually, not quite true. There are some fascinating things going on with quantum chaos, but its not currently clear what bearing this will have on the measurement question.

For example: one problem that was solved recently is that of the wave functions of an ideal gas in a box. The wave functions are insanely fractal, and permeate the entire box (even if you've confined all the atoms to on side; the wave functions interfere destructively on the empty side). However, since the functions are fractals, there are all sorts of difficulties regarding support and lesbegue measure and all that. I suspect that these fractal wavefunctions permeating all space help account for einselection, but no one has been able to write these down explicitly.

RE: point 3) -- There are interesting things one can do to attempt to experimentally test locality, and Alain Aspect's experiments in France were one such. I am not aware of any groundbreaking experiments in this area since Aspect's work in the 1980's.

"Well, as to appeal to authority: most mainstream physicists have been living with and have accepted Bell's results for four decades. QM won, local theories lost. I don't think anyone serious and knowledgable runs around and says "Bell is broken"; QM isn't broken either."
I accept Bell's results. What they imply to me however is that QM is a theory about what we observe, and only about what we observe. Bell's theorem proves that QM cannot provide us any information about things we cannot observe. Are there hidden variables? Bell did not prove that there are none. He did prove however that it does not matter if there are any; that was my point. As far as physics goes if you can't measure it, it might as well be figmentary.--Mr EE 00:25, 3 September 2005 (UTC)
You folks seem to have forgotten the other option: that QM is wrong re entanglement. Once you take account of the loopholes in the various Bell tests (and Aspect's were by no means free of these) you find that alternative classical-style theories can explain all that is actually seen. Can I prove these theories are correct and QM wrong? Well, I believe this would become obvious were the experimenters to explore wider ranges of parameters. I think it could be shown that QM does not describe correctly all that we observe.
For those tests that are open to the detection/fair sampling loophole, all that is needed is to look and see what happens to the results as you vary the "discriminator" level -- the voltage level above which your instruments decide that the electrical pulse represents a "photon". I am currently studying some experiments that have already in fact shown what happens: the visibility of the coincidence curve increases as the discriminator threshold increases. In QM language, increasing the discriminator threshold means decreasing the "quantum efficiency" of the detectors. The observed increase is precisely what Bell is famously quoted as considering to be unlikely, that QM would be less successful at explaining experiments with high efficiency than with low.
The experiments in question are reported in:
S. A. Babichev, J. Appel, and A. I. Lvovsky, “Homodyne Tomography Characterization and Nonlocality of a Dual-Mode Optical Qubit”, PRL 92, 193601 (2004)
Have a look at the graph, fig. 4b. Unfortunately, the experimenters concerned currently think the observations do agree with the QM prediction, but they don't seem to be using the standard assumptions about singlet states. What we need now is a similar investigation but based on one of the already-published Bell test experiments in which the QM prediction is already specified. The usual prediction definitely does not agree with what was seen. [For more on the loopholes see my user page.] Caroline Thompson 09:15, 3 September 2005 (UTC)

This is becoming rediculous with the monotone sequence of indentation to indicate who is speaking. Because of CSTAR's objection (I believe it was CSTAR. You all will have to correct me if I'm wrong.) I read Gödel's paper: On Formally Undecidable Propositions. Having read it, I am now firmly conviced that my assessment that Bell stuck Quantum Mechanics on the horns of Gödel's dilemma is correct. The rub comes because if that is true then this entire debate about locality vs. non-locality is interminable because the question of locality itself is an exemplar of an undecidable proposition under quantum mechanics.--Mr EE 14:39, 3 September 2005 (UTC)

Reply: Yes I was the source of the comment that you responded to. Thank you for your response. However I disagree with your conclusion that the Gödel incompleteness theorem is the reason for endless and unresolved debates here between local realists and mainstream physicists. Just to make sure that we are talking about the same things, and to clarify my notation, let me go over briefly what Gödel accomplished in his paper. Please forgive me if I'm being redundant and are going over things you know.

First Gödel addresses a question about formal systems, that is systems given by a formal language together with axioms and rules of inference for formulas in that language. As you point out Gödel was addressing the question of whether "these axioms and rules of inference are also sufficient to decide all mathematical questions which can in any way at all be expressed formally in the systems concerned".

Now quantum mechanics is not ordinarily thought of as a formal theory, but in fact it is possible to provide a reasonable system for quantum mechanics. Call this theory QM. To get an idea how this formalization can be done, see for instance

  • Gerard Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley Interscience, 1972
  • Varadarajan, Geometry of Quantum Theory, Springer-Verlag, 1985.

These formal theories are sufficiently rich so that Gödel's theorem applies, so indeed there are "formally undecidable propositions" in this theory QM (as you correctly point out). The question that is being discussed, however, is whether the property of local realism can be settled within QM. To to do this, we need to formulate the property of local realism as a formula that can be written down within the language of QM. Again, this can be done; call the formula LR. However, it can be shown that within the formal system QM LR is false.

Now this does not disprove local realism, because what we have actually shown is an implication

QM implies not LR

But whether QM is actually true or not is an empirically determined fact. And local realists will argue that QM is not empirically verified (which of course I happen to think is absurd..there is abundant empirical support for QM).

It can be argued that both notions of "incompleteness" established a leitmotif for much of the intellectual development in the the latter half of the 20th century. In that sense, there is a cerain eerie similarity between the results. But my claim is that they are fundamentally different. --00:11, 5 September 2005 (UTC)

I agree. They are fundamentally different.
Incidentally, surely one should be able to ditch the part of QM that deals with entanglement without throwing away the whole thing? Personally, I think QM embodies several fundamental mistakes, invalidating a lot of the formalism. The whole idea of a theory that deals only with "observables" is bound to lead to paradoxes. One that bugs almost all of the "quantum optics" experiments I've studied is the interpretation of an interference pattern. When a quantum theorist "observes" no interference pattern, he deduces his beams were incoherent. But his theory does not, it seems, allow him to think about the other possibility: that the beams were in fact 50-50 mixtures, half with one phase difference and half differing by 180 deg. Just this one tiny "hidden" variable -- the actual relative phase of each beam -- makes all the difference, doing away with any need for magic.
Such mistakes are now deeply embedded in the theory, yet often it contains clues about what really happens. It has, as people do not tire of telling us, produced many valid predictions. We just need to look again at its successes, reject its false claims (e.g. that of entanglement) and build a new theory that is not constrained by the strictly observable. Caroline Thompson 08:58, 5 September 2005 (UTC)


First, I am a total pragmatist when it comes to this stuff. QM is experimentally valid and on that basis it is foolish to discount the theory as hokum. However, as you pointed out QM can be (and indeed is) formalized in the sense of Gödel. Since this is the case, QM is either incomplete or inconsistent. Completeness formally defined according to Gödel is the property that any proposition belonging to the domain of a given logic is provably true or provably false under the axioms of said logic. This was exactly Einstein's lament with regard to QM: That the HUP left propositions belonging to QM in an indeterminant condition. Bell came along and supplied , providing an element that completed quantum mechanics -- he stipulated what was otherwise a formally undicidable proposition. By Gödel's result this meant that QM as a logical system is inconsistent. If QM is inconsistent (with the inclusion of Bell's theorem) it is impossible to know that LR is false; likewise, it is impossible to know that it is true. Rather than cutting my own throat with Occam's razor by saying that "within the formal system of QM, LR is false," I think it more prudent to say the existence of hidden variables shows every evidence of being formally undecidable.--Mr EE 17:52, 5 September 2005 (UTC)

Mr. EE,

All meaningful logical questions should be formally decidable. As Wittgenstein said: 'Where there is a question, there is an answer'. If you can't formally get an answer, then you are simply asking logically meaningless questions. As a matter of fact, as I have shown on my page , Bell test experiments can well be explained in terms of classical physics, so QM does not enter into the interpretation of these at all.


The true explanation of Aspect's results

Hi Thomas, I agree with your conclusion -- that QM does not enter into the interpretation of the actual Bell test experiments -- but not, I'm afraid, with your ideas as to the true explanation of the observations. If you study the actual experimental reports and consider the likely behaviour of real photodetectors etc., you find that the exact time of detection is rarely, if ever, critical. The evidence is that the various sources used all produce short pulses of light. There is no need to assume that each necessarily comes from one individual atom. Bell-type hidden variable ideas work just as well if we substitute "light pulse from complete source region" for "photon".
The main component of the hidden variable in Aspect's experiments and other early ones would have been the direction of polarisation of the individual pulse -- something that, under QM, is not supposed to exist until the moment of detection, when it changes from a superposition of states to one particular one. See local hidden variable theory for a general picture of how this works. The basic model does not automatically reproduce the QM correlations, but there are two ways in which allowance for actual conditions improves matters:
  1. The response of the detector to a signal of given intensity would not have been strictly a matter of proportionality. Aspect's apparatus involved "discriminators" that cut out all output voltages below a chosen threshold. Quantum theory says that the variations in voltage would have been random, but classical theory strongly suggests otherwise -- that though there would be random elements, the general trend would be towards higher voltages for stronger input signals. A recent experiment seems, to my view, to have unwittingly confirmed this to be the case. See:
S. A. Babichev, J. Appel, and A. I. Lvovsky, “Homodyne Tomography Characterization and Nonlocality of a Dual-Mode Optical Qubit”, PRL 92, 193601 (2004).
  1. Aspect adjusted his data by subtracting "accidentals" before analysis. It can readily be shown (see my 1999 paper) that this adjustement makes a huge difference, changing results that agree (too well?) with the basic hidden variable model into ones that rule the model out. The matter of whether or not the subtraction can be justified all depends on your model of how the source works and whether or not the many atoms present there at any given time act independently or, being simultaneously illuminated by the same coherent laser beams, tend to all act together. This question has almost been ignored, probably because so few have been aware of its importance.
Until both of these points (and a few others) have been thoroughly investigated and debated, the question of the true explanation remains open. Caroline Thompson 09:04, 20 September 2005 (UTC)


Obviously there isn't a perfect correlation between the photoelectron emissions in both detectors as otherwise the coincidence rate should be identical to the individual count rates of the detectors, whereas in fact the former is about 2-3 orders of magnitudes less than the latter, i.e. the correlation is actually less than 1%. This is apparently because a) the only thing that is correlated between the light pulses in both channels is the arrival time of the front of the light pulses (whereas their frequencies and lengths are not) and b) there is likely to be a further de-correlation in the detectors due to the circumstance that the photoionization process is merely stochastic for the low light intensities involved (as outlined on my page regarding the Photoelectric Effect). However, by the very circumstance that a coincidence stage is being employed, all the uncorrelated events are obviously being filtered out, so one is just left with the small amount of correlated events. Because one is only interested in the relative variation of the latter, one might therefore as well assume that the light pulses in both channels are perfectly identical in the first place and examine the dependence of the coincidence rate on the polarizer orientation under this assumption (as done on my page regarding the Classical Interpretation of Bell Test Experiments).


Important faults in the page

Hi Thomas

I don't think here is the right place to debate the issue of the particular LR model that applies to Aspect's experiments. I agree with you that there is one, but let's leave it at that for now.

There are other important faults in the page, rendering it still substandard, and these are what we should be concentrating on. For instance, the diagrams in the present version are both misleading, the second revealing ignorance of what a two-channel polariser typically does: it outputs two signals at right angles, and the device is a cube, not (as shown) merely a cuboid. Is Bell-test-photon-analyer.png reasonable? Why was this diagram used rather than the one currently in Bell test experiments, namely Two channel bell test.svg? Caroline Thompson 21:06, 27 September 2005 (UTC)

Dr Chinese' revert games

I'll leave you to it! You're being petty. Why fuss about this particular reference to the loopholes? Why act as if you had never heard of them and did not take them seriously? I seem to remember you telling a different story a year or so ago and, indeed, you have been known to hedge your bets, admitting that local realism could, after all, be right. Caroline Thompson 21:06, 27 September 2005 (UTC)

For the uninitiated: Caroline Thompson denies the existence of photons, a fundamental element of modern physics. I am doing my best to remove her non-standard POV from Wikipedia pages related to Bell, EPR, and quantum mechanics in general. She also attempts to use Wikipedia to reference her own site in an attempt to gain search index rankings and generally to use Wikipedia as a forum for her ideas.-DrChinese 01:40, 29 September 2005 (UTC)

Re: hedge your bets, admitting that local realism could, after all, be right.
Well yes it could be "right", but it isn't.--CSTAR 00:48, 30 September 2005 (UTC)

How can the photoelectric effect be explained without photons? GangofOne 03:19, 30 September 2005 (UTC)

Dr. Chinese, you are being disruptive and rude. Please stop. While I personally beleive that in the end, QM will be upheld, it is none-the-less true that there are loopholes in the current set of experiments. Caroline is making a concerted effort to try to understand these in her own way. In general, it is vital that those who are interested in this topic understand the loopholes, as this should lead to beter experiments that can close them. Do remember that the michelson-morley experiment and the photoelectric effect were but tiny warts on the edifice of physics; if Caroline thinks she's found another wart, let her investigate. It is perhaps unfortunate that she does not have a stronger understanding in QM, but judging from your resume, you have no formal credentials in this either, and it may well be that she understands this topic better than you.

Caroline, please do continue your studies. You should probably be more conservative with your edits. I think we've got plenty enough pointers saying there are loopholes. A more comprehensive coverage of what those loopholes are would be a good thing to have, but we don't need 101 links interpenetrating all theese articles.

Among other things, it is vital that you cultivate CSTAR for his knowledge and to approach him as a freind and not an enemy. (and CSTAR, please, the same). He really does understand the math behind QM, and so I think its critical that you work collaboratively so that he gets comfortable with what you are trying to do. Be prepared to discuss, not argue, and be clear when you think there's something you don't understand, and ready to admit the same. If he says that something is this certain way, he's probably right.

In particular, I found the comment about the diagram above quite interesting. The beam splitters do indeed work at right angles, and so the black-and-white diagram is "more right" than the color one. Now, I can't imagine why this would make a difference (nor could many/most physicists), but, as they say, "the devil is in the details", and physics is founded on many subtle and supple confusions. linas 22:03, 30 September 2005 (UTC)

Sorry, linas, that ain't gonna happen. Loopholes relate to experiments, not the Theorem. CT can continue her studies in the proper forum, and that isn't here. This is for established science, and that is the beginning and end of it. And if there are loopholes in Bell tests, there are equal loopholes in every scientific experiment that exists. But that's another topic. I will be continuing the enhancement of the Bell page working with CSTAR as appropriate. He has put a lot of time into this. I plan to help polish a few rough edges which may make it more readable to the average reader. After all, that is the audience.-DrChinese 00:35, 1 October 2005 (UTC)

linas is trying to be a good citizen and he is to be commended for trying to mend fences. Although DrChinese could have been more "polite", his intervention here had no ad-hominem aspects to it and I think he is legitimately acting to restore the page to a status approved by a consensus reached long ago. I also note that CT has had standing requests on her webpage urging readers to this page to support her position. I'm not sure what WP policy is on this (and I don't intend to act as a WP policeperson) but this smacks of rabble-rousing to me.

As to the diagram, it's just a diagram, which means it's not to scale nor is it a "conformal representation" of the setup. Isn't that perfectly legitimate for a diagram?

Linas could you please help me in rewriting quantum correlation? It's really awful.--CSTAR

linas, CSTAR: We all need to work to make this group of pages better. The only person I am asking to cease and desist is CT, whose biased agenda is well documented. There is no point to trying to compromise with her, and believe me I have tried - including the private communications which she is now using to mis-portray my stance. I too want everyone to play nice and work towards the common Wikipedia goals. linas, I hope you can assist CSTAR if you have the time. My focus is trying to make sure the novice reader who comes to these pages walks away with a reasonable understanding of the subject; and that the more knowledgeable reader gets a fairly up-to-date version of where things are at. After all, there have been a lot of important new experiments just in the past 10 years. (Aspect is approaching its 25th birthday!) I do not plan to debate the scientific merits of EPR/Bell/Aspect/et al in these talk pages; this should really be done elsewhere as the science is well accepted. As to which diagram to use, they both look pretty decent to my eyes.-DrChinese 14:19, 1 October 2005 (UTC)
"My focus is trying to make sure the novice reader who comes to these pages walks away with a reasonable understanding of the subject", you (Dr Chinese) say! You will not, I think, be too surprised to find that my aim is the same. The difference is that I do not wish to see students unnecessarily misled into thinking the experiments have been more conclusive than they actually have been.
The issue is of vital importance! Suppose, after all, QM is wrong here ... it scarcely bears thinking about ...
See my comments in the Bell test experiments talk page, justifying my reversion. I fear you are suffering from major illusion regarding a matter of logic. It is in the nature of "loopholes" that every single one must be blocked at the same time in order to count.
You have been behaving as if your victory in getting my original loopholes page deleted gives you carte blanche to curtail all references to the subject. This has never been the case. It is possibly fair enough, in the current climate, to say that wikipedia does not need a separate page on the subject, but that is quite a different matter from spreading the false impression that they don't matter.
Linas, your more balanced outlook is much appreciated, as is CSTAR's, though quite why he thinks local realism "isn't right" defeats me!
Caroline Thompson 22:30, 6 October 2005 (UTC)
Caroline, note that QM will never be "wrong", although some day there may be corrections to it. By analogy, Newton's theory of gravity didn't become "wrong" just because Einstein showed up. And, in a certain sense, we already know that QM is wrong, and that quantum field theory is the more correct explanation for reality. The reasons CSTAR (and myself) don't believe in local realism is because there are these HUGE branches of inter-related, inter-twined and very beautiful mathematics, that happen to (by accident? on God's purpose?) share a lot in common with the math used in QM. I am thinking of topics ranging from the ergodic hypothesis and chaos theory where things like the transfer operator is a crude example of something from a C* algebra, which is something that occurs in functional analysis. Never mind the rotation group and its representation by SU(2) and SO(3) and the Spin algebras. The stuff fits together so marvelously, so perfectly, that it makes much more sense than local realism offers.
However, as a careful physicist, I think its important to identify and close all the "loopholes" in the EPR-type experiments. For one, its important to be able to say "now we are finally done", and for two, one may get lucky and find something new. Personally, I wouldn't bet on it, but "shit happens", such as e.g. the photoelectric effect, which was this absurd little tiny problem in the classical theory that ushered in the era of QM. linas 23:31, 7 October 2005 (UTC)
Hi Linas: I've just seen what Dr Chinese has done to the Bell test experiments page. He has reverted again, with the result that he has eliminated the standard reference (namely to Pearle, P, “Hidden-Variable Example Based upon Data Rejection”, Physical Review D, 2, 1418-25 (1970)) on the one loophole he now condescends to mention. He has totally ignored my discussion of the fact that the Rowe et al experiment (which did indeed block the detection loophole) had other serious loopholes and hence is irrelevant to the discussion. He has ignored the basic fact that all loopholes must be blocked at once if you are to claim clear success for quantum entanglement.
Is such arrogant behaviour consistent with wikipedia policy? Doesn't logic matter? Doesn't giving the best possible refs matter? Caroline Thompson 09:04, 8 October 2005 (UTC)

Mathematical models and "reality"

But Linas, physics and maths are two separate things! Your above appeal to all this "inter-related, intertwined and very beautiful mathematics" is, I fear, totally irrelevant. When we talk of local realism we necessarily mean the real physics of the real world, not a mere mathematical model that happens to fit the real world so far as certain phenomona are concerned.

I'm afraid what you wrote is such a wonderful example of something I was reading yesterday I just had to smile!

  • "In terms of the power it wields, Quantum Theory is today the Pentagon, if Washington DC were the world of physics. It is a formidable place that you dare not criticize. If you do, they will throw so much mathematical mumbo jumbo at you that you will forget your mother’s maiden name by the time they are done with you." [From]

It's not as if this was news to me. I just happens that I realised 10 years ago the full implications of what I'd been taught in my maths degree back in 1965: that mathematics is no better than the assumptions that lie behind it. Only if every single assumption involved in a mathematical model agrees with something that is physically plausible am I going to even consider the possibility that it will make correct predictions of anything physical. Failing that, it is just part of a fantasy world. Caroline Thompson 08:52, 8 October 2005 (UTC)

Yeah, well, what can I say? It is almost certain that "local realism" is a part of a fantasy world, and that QM is not. As to the analogy, QM is not so much the Pentagon as it is Mount Everest. Those few who ascend come back with tales of tremendous vision. The search for local realism is like a search for a vast cave inside of Mount Everest: it may be there, and it may not. You must excuse those who might think Everest is not hollow, and even if there does prove to be a cave here, its bulk cannot be demolished and is grandeur is not diminished. It is what it is. linas 22:48, 14 October 2005 (UTC)
The pentagon? Geesh, if that analogy's true, local realists should be rubbing their hands in anticipation.--CSTAR 00:57, 15 October 2005 (UTC)
Ah well, each to his own opinion, but as to it's being "almost certain" that local realism is a fantasy, I beg to differ. I don't think you can reconcile the quantum world with the kind of model Newton might have devised, made of solid particles obeying laws similar to his gravitational one. You need something very much more subtle, in which there are probably no "solid particles" there at all, only waves, acting in ways that no waves on the everyday scale can quite match. [See my own pet "Phi-Wave Aether" idea, in which there are semi-permanent "wave centres" that play the part of particles. There are a couple of essays on it on my web site.]
Anyway, the point is that these waves or whatever are real and local, regardless of any mathematical model that says they have to be nonlocal. There is (in view of the various loopholes) no conclusive evidence that any nonlocal effects ever happen. Yes, quantum theory may be Everest, but it is a structure built effectively by a committee.
I've been looking back at the history, and everyone who was anyone would have been to one of Bohr's conferences or (more recently) had a spell working with Anton Zeilinger. A completely different group of people who had never even heard of QT might be able to build not just Everest but the whole universe! Caroline Thompson 09:22, 15 October 2005 (UTC)
Caroline, I am starting to suspect that you have never studied the mathematics of group theory, and in particular, the group representations of the rotation group SO(3) and SU(2). These are not terribly complex objects: SO(3) is a certain collection of 3 by 3 matrices, and can be readily comprehended with a bit of study and practice. I think its also critical that you learn how to add together spins, using either Clebsch-Gordon coefficients, or possibly some more modern mechanism. If you get a good book on this topic, you will find that it is not at all hard or ethereal or abstract, its pretty darned concrete as math topics go. linas 00:31, 16 October 2005 (UTC)
Linas: I do have a degree in mathematics and at one point considered a career in the subject! It's not that I couldn't do the maths if I thought it relevant but, at least in the case of light, I'm quite certain it is such a crude approximation as to be useless. If you want the "real" matrices that are appropriate for dealing with light, go back to the pre-quantum text books. Read:
  • Ernst Mach (John S Anderson and A F A Young trans.), “The Principles of Physical Optics”, E P Dutton and Co., Publishers, New York, 1926 (recently reprinted)
for hints of the true complexity of how light gets refracted by solids. Read:
  • Shurcliff, W A and Ballard, S S, "Polarized Light", Van Nostrand 1964
for the matrices and more recent interesting facts.
You could even look at papers such as:
  • Krasa, J and Jiricka, J and Lokajicek, M, “Transmittance of a laser beam through a pair of crossed polarizers”, Physics Letters A 186, 279-281 (1994)
Tell my how your mathematical models apply to that! Caroline Thompson 09:20, 16 October 2005 (UTC)

Web links.

I have removed the links in the following references as they were 404 erroring.

  • A. Aspect et al., Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities, Phys. Rev. Lett. 49, 91 (1982), available at
  • A. Aspect et al., Experimental Test of Bell's Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. 49, 1804 (1982), available at
  • J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local hidden-variable theories, Physical Review Letters 23, 880-884 (1969), available at

Of curse they can be re-inserted if they come back. Rich Farmbrough 10:18, 10 November 2005 (UTC)

They seem to have gone for good -- a pity, as they were free. You can buy individual articles from but that's not quite the same thing. Caroline Thompson 09:53, 11 November 2005 (UTC)

Simply connected spacetime

The following was recently removed from the article:

No physical theory of local hidden variables (on a simply connected spacetime) can ever reproduce all of the predictions of quantum mechanics.

While I agree with the removal ... it does bring up a very interesting point that I suppose should be discussed somewhere. Suppose one modelled Planck-scale spacetime as a filigree of wormholes and what-not. What form would Bell's theorem take on such a manifold? The point being that geodesics have highly ergodic trajectories on surfaces of negative curvature, so one would expect to get a giant mess of random behaviour. I think this could make for an interesting article (even if this article is not the place for that). linas 02:24, 15 November 2005 (UTC)

If finite propagation speed fails, Bell's theorem fails.--CSTAR 03:05, 15 November 2005 (UTC)
I don't understand what simple connectivity has to do with Bell's theorem at all, other than that you need a suitable spacetime (i.e. need a causal structure, which maybe requires a noncompact manifold with a global timelike vector, or some such topological restriction. Whatever the restriction, I'm sure there are non simply connected spacetimes with the appropriate causal structures.) Is there some relationship? Is the point that on multiply connected spacetimes, local hidden variable theories can reproduce quantum mechanics? I have trouble believing that. -Lethe | [[User talk:Lethe|Talk]] 03:13, 15 November 2005 (UTC)
As far as I know, nothing. But that's beyond the limits of what I know.--CSTAR 03:21, 15 November 2005 (UTC)
Yes, well, I guess its too late to retract my remarks. I convinced myself during a walk around the block that this will lead to nowhere. Maybe the point was that with appropriate hand-waving, there are still places such as this were "Bell-test loopholes" can be imagined. linas 23:44, 15 November 2005 (UTC)


Commenting on this quote:

It was a conclusion of EPR, that once Alice measured spin in the x direction, Bob's measurement in the x direction was determined with certainty, whereas immediately before Alice's measurement Bob's outcome was only statistically determined. Thus either spin in the x direction is not an element of physical reality or effects travel from Alice to Bob instantly.

But, at least in case of photons, not only the two measurements, but any two (space-time) points on the "trajectories" of the two particles are separated by a space-like interval. Hence, they can't be said to be "before" or "after" one another. Each measurement can be said to be after photons were emitted, but nothing more in terms of temporal sequence is defined. No point on the trajectory of Alice's photon is definitely before Bob's measurement, and no point on the trajectory of Bob's photon is definitely before Alice's measurement.

Sorry for some repetitiveness, but isn't it clear enough that the dilemma in the above quote is a false one?

If I'm not mistaken in this, quantum-mechanical nonlocality and relativistic blurring of synchronicity are so intricately intertwined here that one is tempted to think that they are but two sides of the same coin.

- M (Dmanin 06:43, 19 November 2005 (UTC))

Of course you're right. The article should say something like "so that the measurements are causally disconnected"; --CSTAR 07:18, 19 November 2005 (UTC)

Implications of violation of Bell's inequality

I'd feel a lot better about the section "Implications of violation of Bell's inequality" if violation of the reality assumption (called here "counterfactual definiteness") were put on an equal footing with violation of locality. It's really pretty silly to give up locality, since it's a basic assumption of all mainstream physics. By comparison, I've always felt it's easier to give up statements about (some) counterfactual observations. I don't think denying the reality assumption necessarily entails the many-worlds interpretation, but I'm happy to be corrected on this point. Dave Kielpinski 05:37, 14 December 2005 (UTC)

PS. If no one objects in the next few days, I may edit the relevant paragraph (in a minimally intrusive way) to accomodate my view. Dave Kielpinski 05:38, 14 December 2005 (UTC)

I'm not quite sure of what you're refering to. Can you post the suggested edits here first? "Local" is a tricky word in QM, since one must first state "local with respect to what topology in which space". Clearly, the correlation of spins in a singlet state is non-local in Euclidean space ... linas 19:42, 14 December 2005 (UTC)

In fact, it's not at all clear that QM is nonlocal. Certainly quantum field theory is local in the usual sense of the word, and attempts to extend nonlocal Bohmian mechanics to the relativistic domain have so far been unsuccessful, creating a strong presumption that QM is local. Bell inequalities for local realistic theories can be falsified by 1) giving up locality or 2) giving up realism, i.e. "counterfactual definiteness." In the article, these two options are not treated on an equal footing, and I propose to remedy that.

I realize that all sorts of crackpots are attracted to this subject - a glance at my publication record will show that I am not in that category. Dave Kielpinski 06:54, 15 December 2005 (UTC)

Added publication record to my user page. Dave Kielpinski 07:20, 15 December 2005 (UTC)

Very good, then, glad to have you here, and quite correct about the observation about crackpots. We need the help :)
Funny you mention QFT: in a typical Largrangian, the interactions are "local" in the sense that interactions are at a point or involve no more than second-order derivatives: e.g. is local because everything happens at point x. However, one then promptly integrates over all space-time to get the action, and then over all field values to get the functional, and so in that sense QFT is highly non-local. So, for example, any given Feymann diagram (part of a scattering matrix element) involves one or more integrals over all space-time. More simply put, Green's functions are "nonlocal", and Green's functions are "physical" in the sense that they get measured in the lab. So I still maintain that "locality" is a tricky word to use correctly in QM.
Actually most textbooks say that it is the S-matrix elements that are observable, not the Green functions (which are subject to gauge ambiguities, amongst other things). --Michael C. Price talk 08:31, 13 September 2007 (UTC)
So would another valid way to look at it be that the universe is deterministic? —Preceding unsigned comment added by (talk) 20:26, 11 September 2007 (UTC)
I suppose I should encourage you to edit the article, and if I don't like what I'll see, I'll scream. I'm not sure which sections you are even referring to; they may be left-over cruft from a earlier edit battle. I'm not convinced that the various articles we have in Category:Quantum measurement or even Nonlocality, Principle of locality, Action at a distance (physics), Quantum entanglement etc. are consistent, or are free of murky, wacky statements, etc. And finally, I'll admit that I'm not sure I know how to distinguish locality from realism; I've happily thrown both to the wind. I'm not even sure there is a clear definition of these terms... linas 23:24, 16 December 2005 (UTC)
I understand the distinction you are making between different uses of the word "locality" - it is a little ambiguous. I believe the standard meaning of "locality" in QFT is the first sense that you mention, i.e. interactions must be written in the form c \overline{\psi}(x) \psi{x}, while interactions like \overline{\psi}(x - x_0) \psi{x} are forbidden. Without this type of locality, the whole notion of causality is in question and QFT falls to the ground. This is also the sort of "locality" referenced in Bell's theorem. I have never encountered your second definition of "locality" in discussion or in the literature. Incidentally, your first definition seems perfectly clear to me, as all closed classical and quantum systems are in principle describable in the Hamiltonian formalism. I guess I could have edited the article by now... Dave Kielpinski 02:23, 17 December 2005 (UTC)
Definitions of locality are problematic when talking about Bell. I have taken to talking about Bell locality and Bell reality in many discussions because of the confusion. I then map those to the specific assumptions Bell made about each in his original paper. That allows you to use "local" to refer to a theory in which causes cannot propagate faster than c.--DrChinese 19:03, 3 March 2006 (UTC)

Consistent QM??

Griffiths and Omnes apparently propose some kind of "middle ground" concerning Bell's theorem, but I don't understand it and the article doesn't even mention it (except from an uncommented reference to Griffith). They propose a reformulation of the "Copenhagen" interpretation. As Griffith formulates it, "[Consistent Histories] provides a realistic picture of the atomic realm [...]. the CH appraoach removes any need to look for alternatives to standard quantum theory"

I don't understand the essence of it, but it sounds relevant for this article as well as all those that depend on this article -- can someone summarise their ideas and motivate why they are not included, or why instead they should be included?

Thanks, Harald88 22:04, 17 December 2005 (UTC)

PS I now see that there is a consistent histories page and I'll read it -- but it is currently not linked... Harald88 22:22, 17 December 2005 (UTC)

Misunderstanding of Heisenberg?

"since the act of taking the measurement changes the state." Isn't this a common misinterpretation? —The preceding unsigned comment was added by (talkcontribs) 2006-04-03 00:35:16 (UTC)

In quantum mechanics, measurement operators do change the system state. See quantum operation.--CSTAR 01:38, 3 April 2006 (UTC)
(Unless the system state was already an eigenstate of that measurement operator. So a measurement doesn't ALWAYS change the state...) --GangofOne 03:04, 3 April 2006 (UTC)
"The measurement changes the measured" is the Observer Effect. "The more precisely you pin down X, the less precicely you can know Y" is the Heisenberg Uncertainty Principle. The Observer Effect is analogous to measuring the position of a billiard ball by colliding it with another billiard ball. The Uncertainty Principle is because of wave/particle duality and would hold even if you could measure the billiard ball without disturbing it. Which is why billiard balls are poor analogies for weird quantum wave/particle thingies. —Preceding unsigned comment added by (talk) 20:21, 11 September 2007 (UTC)

Accessibility problems

As this is an encyclopaedia entry, I think it is only fair on the casual reader to define all variables used in the article. Articles like this should aim to educate those who have no experience of the topic, rather than those who understand it before they even started reading.

I have the same feeling as expressed above. In particular, the meaning of a' and b' is not explained in the article. I also believe that one should be careful in using the word "system". Does it refer to the whole system {A+B} or only to one of the two particles (either A or B) ? In fact, the article would become clearer if the section on "Bell's thought experiment" was rewritten to describe exactly the way it has been tested by Aspect (two photons, each travelling in its own direction and two polarizers measuring their spin in two different basis,...). Not precising the physical quantities measured tries to show the generality of Bell's theorm, but it makes the presentation confusing.-- 18:28, 15 April 2006 (UTC)


Bell Theorem is Religious Indoctrination(it may be called "atheistic",religion doesnt imply anything,just faith),it denies all theories that may arises in the future and denies any future science frameworks chance to break free because earlier theory and framework restricts it.Its a statement of orthodoxy to restrict science,making it more mainstream and conforming. You can argue that it true today,prfectly logical or explained by evidence but its out current understanding of events and it MAY CHANGE IN THE FUTURE. If you accept such dogma,you accept a non-theistic religion based on faith in Quantum Mechanics.Its a logical consequence of this theory.—Preceding unsigned comment added by (talkcontribs)

All your base are belong to us.--CSTAR 22:21, 4 May 2006 (UTC)
You have no chance to survive make your time. Dave Kielpinski 14:19, 6 May 2006 (UTC)
My dear, that's what makes Bell's Theorem so brilliant. Byrgenwulf 12:31, 28 July 2006 (UTC)
For great justice. —Preceding unsigned comment added by (talk) 04:08, 11 December 2007 (UTC)

MWI counter claim

The statement:

Hence many worlds can adhere to both the properties of philosophical realism and the principle of locality and not violate Bell's conditions -- the only interpretation that can do this.

had the final clause deleted with claim that it is false. If true I would like to see a source cited, specifying which other interpretations of QM satisify these requirements. --Michael C. Price talk 12:19, 28 July 2006 (UTC)

"MWI is the only interpretation which can hold to philosophical realism and the principle of locality".
A few things here:
  1. That's POV, and possibly even OR.
  2. Where's the citation for that statement?
  3. Relational Quantum Mechanics similarly holds to realism, inasmuch as the moon is there when we aren't looking, and locality, inasmuch as the influence of objects is restricted to their lightcone.
  4. It accounts for EPR/Bell's Theorem very similarly to the relative state formulation...because we have to specify relative to which observer a given quantum mechanical description holds, and because observers need to interact (presumably they move at speed < c) in order to compare their results, there is no non-locality but all participants, equipment, photons/electrons etc. are real (they will have a "state" relative to some observer).
  5. There are two references in the article in my sandbox: Rovelli (1996) and Rovelli (2006). I would cite 'em as footnotes or inline, but that would seem odd, because I am not citing a statement but a lack of one! So should I just add them to the bibliography?
  6. Once the RQM article is done, I shall include a brief paragraph here about how it relates to Bell's Theorem.
Byrgenwulf 12:31, 28 July 2006 (UTC)
The claims for RQM I leave in your capable hands, but they will be subject to the same requirements as assertions about other interpretations. The references for the MWI claims are at Counterfactual definiteness -- they should probably appear on this article's page as well. Do you accept that MWI violates CFD? --Michael C. Price talk 14:24, 28 July 2006 (UTC)
This article has a very long history, some of it very bitter. One of the original editors for this page was User:Caroline Thompson a local realist who vigorously claimed QM was flawed and in fact, Bell test experiments had not disproved local realism. Dealing with her required a great deal of compromise. Please see the talk page history (and its earlier archives). She died earlier this year.--CSTAR 15:06, 28 July 2006 (UTC)
Hmmm, the issue is a subtle one; I suppose the most simple answer is that MWI violates CFD precisely because there are no counterfactuals...every possibility is actualised, so it is meaningless to contemplate what would have happened if the electron had spin up instead of down, say, because it does, just maybe not "here". I don't, however, see counterfactual definiteness as having any special value or meaning though.
The EPR/Bell section is next up in the RQM article, and obviously any claims made here will be both referenced and sound (after all, I am quite pedantic about that). Specifically this paper deals with the phenomenon, but the results follow quite simply from the original RQM's a matter of straightforward application.
So, can we take that last clause out now? :P
CSTAR, I am not quite sure what relevance the history of this article has to with my quibble over this 7 word clause...could you perhaps explain?!? Neither of us are disputing Bell's Theorem or QM here. Is it to do with my comment above? I'm very confused... Byrgenwulf 15:32, 28 July 2006 (UTC)
This soup was concocted by chefs who had to negotiate and argue for what ingredients to put in. Not only that but ingredients kept getting taken in and out. After a while, what exactly was in the soup was never very clear. The 7 word clause was hers.
Re: Is this Thompson the anon IP address to whom I left that flippant message? I doubt it. --CSTAR 15:42, 28 July 2006 (UTC)
I see, thank you; but I really do think it should go. I just checked the unfortunate lady's userpage, and she apparently passed away in February, which means that since the message was left in May, if it was left by her we have rather more spooky action at a distance than any interpretation of QM can handle. Byrgenwulf 15:51, 28 July 2006 (UTC)

Notable quotes

The quotation from Heinz Pagels basically consists of a string of inflammatory pejoratives ("rubbish," the ad-hominem "wish-fulfilling fantasy," "really weird," and even the really weird term "beast-like") with no mention of argument or evidence. It therefore conflicts with the Wikipedian requirement of NPOV, so I propose to delete it. Since it's the only quotation listed, I propose to delete the whole "Notable quotes" section. Of course, if anyone finds a more neutral quotation, the section can be added again. —Were-Bunny 19:38, 9 October 2006 (UTC)

Since it is a sourced quote then the NPOV rule does not apply; it is not presented as a statement of fact but as a statement of Pagels' opinion. Don't delete it, but do find other quotes that express contrary opinions if you feel the section is unbalanced. --Michael C. Price talk 00:08, 10 October 2006 (UTC)

I'm not a physicist, so I'm loathe to edit the original article. Based on my slow reading of Gribbin (Schrodinger's Kittens, 1994), however, this comes across as a particularly messy article.

For example, the article refers to von Neumann's proof against local variables, but Gribbin claims that proof was demolished by Bell. If that is true, it should not be mentioned here as of its only historic significance.

I'll paraphrase below how Gribbin describes Bell's Theorem and its modern consequences. If this makes sense I suggest it be incorporated with a citation to Gribbin (Schrodinger's Kitten). I think I can see pieces of Gribbin's lucid summary in the article, but the message fragmented and expressed in unnecessarily formal language.

Bell's Theorem showed that if non-locality were found to occur, irregardless of any interpretation of quantum mechanics, then physics had to abandon one of two cherished beliefs:

1. That the world exists independently of our observations of it. 2. That there is no communication faster than the speed of light.

Subsequently non-locality has been shown, several times, to occur. That means we have to give up on either the "persistent world" or faster than light communications. Not surprisingly, physicists have decided the lesser evil is to accept a faster than light communication -- as long as that communication carries no "meaning". In other words, "meaning" cannot travel faster than light.

I'm going to go ahead and delete the notable quotes section. It's been a while since the deletion was proposed above yet no one has followed up with other quotations. I'd like to point out that just because it's cited doesn't mean it is NPOV, which it isn't. --Android Mouse 07:23, 30 July 2007 (UTC)

Restored section for reason I originally gave. If you feel other quotes should be given, fair enough. "Add - don't delete" is policy. --Michael C. Price talk 11:34, 30 July 2007 (UTC)
It's sourced, yes, but where is the source that asserts the quotations notability? Anyone can write a book, pushing their POV, and claim that they are an authority of the subject. Who asserts this quote is notable, and why isn't that sourced? --Android Mouse 16:54, 30 July 2007 (UTC)
  1. Heinz Pagels was a notable reliable source on quantum mechanics.
  2. No, not "anyone can write a book", and certainty not on QM with the authority and clarity as HP, see #1.
  3. And if I have to provide source2 to assert the notability of source1 do I therefore have to provide source3 to etc etc. Look, the book itself is notable. --Michael C. Price talk 21:03, 30 July 2007 (UTC)
The book itself may be notable but what about the quote itself? Merely being featured in a notable book doesn't mean every and any quotation from it is notable. --Android Mouse 21:40, 30 July 2007 (UTC)
True, but lots of articles have quotes from notable books, papers etc, without demonstration that the quote itself in isolation is notable. There we just have to use our common sense. And I say it is relevant to the article. --Michael C. Price talk 21:48, 30 July 2007 (UTC)
Many articles do use notable quotes without referencing their notability. The difference is those quotes help improve the reader's understanding of the topic and bring clarity to the subject matter. In this article the quote doesn't bring clarity to what Bell's theorem is, it is merely devisive. There is no explanation as to who considers it notable or why or if the quote has had any real world impact on the subject. I've searched for others who might have thought the same as you and used the quote [1] but I haven't found any page that quotes it verbatim except for this page which ironically links back to this exact article. Please give me a single reference for why this quote is notable above any other quote from the book and I'll be happy. --Android Mouse 22:01, 30 July 2007 (UTC)
Since I disagree with your assessment I see no need to provide a cite. There is no requirement for the quote to explain Bell's theorem (hopefully the rest of the article does that). What the quote does show is that Bell's theorem is central to much disputation and philosophical musings -- which this physicist (HP) regards as unsubstantiated.--Michael C. Price talk 22:32, 30 July 2007 (UTC)
Because you agree with the quote or because you feel it is notable doesn't exempt it from requiring a citation like everything else on Wikipedia. --Android Mouse 18:43, 2 August 2007 (UTC)
It is sourced - that counts as a citation. Many other articles have quote taken from books. Oh, and BTW, whether or not I agree with the quote is irrelevant. --Michael C. Price talk 09:48, 6 August 2007 (UTC)
The quote itself is sourced. The quote's claim to notability isn't. You haven't even provided a source for why the book is notable. Becuase other articles don't have adequite citations isn't a reason for inclusion --Android Mouse 20:31, 7 August 2007 (UTC)
Don't be ridiculous. Of course the book is notable. It was a best seller. --Michael C. Price talk 06:49, 11 August 2007 (UTC)
Out of the entire book, what makes this quote notable over any other random paragraph in it? I've searched around, as I pointed out above, and haven't found it used before but in this article. So I'm wondering, if the quote is notable, why am I not finding it used anywhere but here? I might understand if it helped explain the subject but it offers no explanation and doesn't help the reader understand Bell's theorem at all. It's merely divisive. --Android Mouse 07:36, 11 August 2007 (UTC)
I've already addressed your divisive claim. And there is no requirement for quoted text to appear elsewhere in the internet. The pages of a bestseller will do.
You've obviously got some beef with the contents of the quote. If you feel it needs balance then find a quote that sings the praises of telepathy and Bell's theorem. It would be a much more productive use of your energies and interest in the topic. --Michael C. Price talk 10:31, 11 August 2007 (UTC)
The quote is clearly relevant as it illustrates a mainstream interpretation of the theorem. If anyone wants to balance that with an alternative point of view, just find a notable quote (e.g. one from another best seller) with such a point of view. Rod (A. Smith) 16:22, 11 August 2007 (UTC)

deindent-- I think you are confusing a notable opinion with a notable quotation. If this quotation is notable, why can't anyone point me to a reference to where it has been quoted and used before? Again, what makes this quote more notable than any random quote I could pull from the book? Surely you aren't asserting everything that is in this book could be pulled and called a notable quotation? --Android Mouse 19:03, 11 August 2007 (UTC)

No, that's where common sense comes in. Oh and BTW notability is not a criterion for inclusion, contrary to popular belief.--Michael C. Price talk 19:32, 11 August 2007 (UTC)
Notablity is a criterion for inclusion if you claim it to be, as the section does. --Android Mouse 19:34, 11 August 2007 (UTC)
A lot of other criteria for inclusion have also been claimed. See above. --Michael C. Price talk 20:15, 11 August 2007 (UTC)
The criteria you've given would make any quote taken from a bestseller as being notable. This is what you are claiming? --Android Mouse 20:24, 11 August 2007 (UTC)
No. --Michael C. Price talk 20:42, 11 August 2007 (UTC)
So why not provide more to show the quote is more notable than just any other quote from the book? A simple reference to its usage will be fine by me. --Android Mouse 20:48, 11 August 2007 (UTC)
Still banging on about notability. Do not interpret my future silence as agreement, but until you address the many points raised, or raise new points, instead of just recycling past issues, I shall not respond further.
2:1 the consensus is against you. --Michael C. Price talk 21:08, 11 August 2007 (UTC)
You haven't yet addressed my previous point. I've pointed out that the criteria you've given would indeed make any randomly selected quote from a bestseller on the topic worthy of being a "notable quote". I then asked if you agreed with what your criteria implied and you said no. If it is notable, then why hasn't it been quoted before? If it has, why won't you provide me a reference for this?
You seem to be confusing consenus with a majority opinion. --Android Mouse 22:01, 11 August 2007 (UTC)

Is this correct?

"then hidden variable theories cannot be correct; not unless information is being transmitted between the particles faster than light, or the experimental design is flawed."

I thought that information couldn't be transmitted faster than light? The two particles can be instantly correlated, but thats not the same as information being instantaneously transmitted, true?

I think just using the word "information" there is wrong. Am I right? —The preceding unsigned comment was added by (talk) 19:40, 30 April 2007 (UTC).

Yes, that is true. If information is meant in the sense of Shannons information theory (or its generalisation to quantum information theory), QM doesnt imply information transfert faster then light. Using QM you can even prove that the entanglement of the two qubits (photons) of the EPR experiment doesnt allow information transfert! For any more questions refer to Deniz195 13:52, 27 June 2007 (UTC)

Bell Theorem loop hole

I remember vaguely that there may be a loop hole that basicly invalidates the whole theorem. Something about pre-determinism.

  • Yes, a hidden variable could exist at the time of the universe's creation. Let's call this variable U, for universal simulator. If everything in the universe were proximate at this time of creation, then all particles may share this variable U, and any new particles may copy U whenever it comes across a U-carrying particle. Thus, all particles with high statistical confidence are U-carriers. Now, since U contains all information from the time of the universe's creation, then, if everything in our universe is pre-determined, we can know seemingly non-local information by merely querying the local hidden variable U, which, as a universal simulator, knows everything about our universe if events are pre-determined from initial conditions. -- Thoreaulylazy 23:26, 24 August 2007 (UTC)

Just a note on Thoreaulylazy's response. While this is a response to Bell's Results, it is not an answer to the problem. The problem says that any theory that admits that photons are correlated in such a way that they do not violate the Bell inequalities, then that theory must be nonlocal. A theory in which the results of polarization measurements on photons are all predetermined and the correlations between them are not consequences of the fundamental mechanics of the theory does not admit these correlations. A pre-deterministic theory of this kind basically says that the correlations obeyed by photons in Bell experiments are mere accidents which arose out of particular initial conditions. This is not a way of winning the game, this is a way of not playing the game. —Preceding unsigned comment added by (talk) 01:52, 12 June 2008 (UTC)

What is measurement?

See my user page. --Pateblen 13:11, 1 August 2007 (UTC)

Main article is terrible, I'm sorry

I spent three days trying to make sense of the main article and either my comprehension is poor or it was meandering jibberish. To its credit, the main article does convey the conclusion (local hidden variables cannot explain the experimental phenomena) very well and in a straightforward manner, but I'm afraid the motley of contributors putting in tables and random formulas have made it impossible to follow. Imagine watching a bad action film with no character development or story arc. That's what it's like to pour over LaTeX equations without adequate explanations about what all the variables and functions represent and what the objective is.

So fed up, and doubting my own reading comprehension, I went to the page for Bell's theorem (link) and it made perfect sense the first time I read it and I went "aha!" The main Wiki article, on the other hand, is frightfully lengthy yet it doesn't elaborate enough on the BCHSH inequality in understandable terms like the Stanford page has. I'm still learning quantum mechanics so I do not feel comfortable enough to rewrite an article as important as this one on Bell's theorem, yet I do implore someone more knowledgeable to take my criticism under advisement when volunteering to clean the article up. For passerby who wound up as confused as I once was, the following paragraphs are, in my own words, what I gleaned from the Stanford page, and I'm quite content that I now understand the matter clearly:

If local realism is true, then there is no impact of a distant actor's actions when measuring a local particle's spin. The phenomenon where two particles under quantum entanglement can have perfect correlation when measured at identical angles (let X be this angle) is a phenomenon that can be explained under the model of local realism by employing local hidden variables, as follows: A deterministic mapping could be programmed into these entangled particles at the time they were entangled, which is feasible because at the time they were entangled they were proximate to each other. The deterministic mapping, call it m, would map an angle, from 0 to 360 degrees, to spin, -1 for counter-clockwise and +1 for clockwise. Therefore, even if these entangled particles are now apart by great distances, one actor measuring one of the particles at angle X would see a spin m(X) and a distant actor measuring the other particle at angle X would also see spin m(X). Since m is a local property carried by each particle, nothing non-local affects the measurement

While local realism seems to work at explaining entanglement, Bell's theorem devises a situation where it fails. Let Q and Q' be two angles the distant actor can measure at, and R and R' be two angles the local actor can measure at. Therefore, we need concern ourselves with only a reduced m, one which doesn't handle 0 to 360 degrees but instead only four angles, Q, Q', R, and R'. Since m is deterministic, there are only 16 possible mappings it could be -- e.g. m(Q,Q',R,R') could be (1,1,1,1) or (1,1,1,-1) or (1,-1,-1,1) et cetera. Let M be the set of these 16 possible mappings. Let s be the spin the local actor observes and let t be the spin the remote actor observes. We can decompose E[s*t|Q,R], the expected value for the product s*t for when angles Q and R are chosen but m is free to be anything in M, into "sum p(m)*m(Q)*m(R) for all m in M", where p(m) is the probability of m occurring, which may be 1/16 if all 16 mappings in M are equally likely to occur, but we place no such restriction on the distribution.

Next, let W = E[s*t|Q,R] + E[s*t|Q,R'] + E[s*t|Q',R] - E[s*t|Q',R'], which decomposes into "sum p(m)*(m(Q)*m(R) + m(Q)*m(R') + m(Q')*m(R) - m(Q')*m(R')) for all m in M". We know that any expression S of the form qr + qr' + q'r − q'r', where q, q', r, and r' are real numbers in the closed interval [-1, 1], must be a real number in the closed interval [-2,2] due to algebra. (Explanation: qr + qr' + q'r - q'r' is linear in all four variables, in other words, the partial derivative of S with respect to any single variable is an expression lacking that same variable; so, S must take on its maximum and minimum values at the corners of its domain. Thus, some integer inputs q, q', r and r' each in { -1 , +1 } must yield the expression's min/max bounds. Either employing further algebra or applying brute-force on all the 16 possible integer inputs, we find -2 and 2 are the bounds.) Therefore, W is bounded by "sum p(m)*S for all m in M" where S is an unknown in the interval [-2,2]. This bound simplifies to "S * sum p(m) for all m in M" which simplifies to S and therefore W is in the interval [-2,2]. This inequality, -2 <= E[s*t|Q,R] + E[s*t|Q,R'] + E[s*t|Q',R] - E[s*t|Q',R'] <= 2, is the BCHSH inequality, which gives us a formal mathematical constraint when believing spin is determined by local hidden variables and not a distant actor.

Bell's theorem continues, this time finding a constraint for W using quantum theories. The wave function in quantum mechanics simplifies E[s*t|A,B] to cos(2B-2A) for any real angles A and B. By letting Q = 2pi/8, Q′ = 0, R = pi/8, and R′ = 3pi/8, then W simplifies to cos(-pi/4) + cos(pi/4) + cos(pi/4) - cos(3pi/4), which is approx. 0.707 + 0.707 + 0.707 - (-0.707) = 4 * 0.707 = 2.828, thereby violating BCHSH. This is perfect, it places contradictory mathematical constraints, between [-2,2] with local hidden variables and 2.828 with the wave function, on W. Due to the law of large numbers, the running average, obtained by repeatedly rerunning experiments, rapidly converges to the expected value, meaning we can empirically measure E[s*t|A,B] for any angles A and B at arbitrarily high confidence levels. As we now know, empirical evidence shows with high statistical confidence that the expected value is above 2.7, easily violating BCHSH and approaching 2.828 predicted by the wave function.

-- Thoreaulylazy 22:28, 23 August 2007 (UTC)

If it's any help, I agree it is not very clear. I suggest someone stick the technical banner on the article itself.--Michael C. Price talk 19:11, 24 August 2007 (UTC) 16:33, 20 August 2007 (UTC)

Overview Question

First of I am not an expert on anything quantum, I'm just a newcomer to this section.

However, when reading the overview page on Bell's experiment I cannot get through the section on hidden variables. The explanation in combination with the hidden variable table is confusing me.

My 1st question is:

In the article it says "Using that scoring system, any possible combination of hidden variables would produce an expected average score of at most +0.5." So I'm looking at the table and out of 4 hidden variables I see 8 combinations, while expecting 16 total combinations of hidden variables. Could someone expand the article explaining why combinations like (+ - - +) ( + - + -) (- + + -) (- + - +) (+ - + +) (+ + - +) (- + - -) (- - + -) are not shown in the table since their expected average scores are not +0.5

My 2nd question is:

Why is the score negated on a-b' but not on a'-b?

I'm not saying that the article is wrong, just that It causes confusion for me and I'm assuming for other people too.

Existance is a struggle between life and death. I just like to watch. --Zeruski 19:53, 5 November 2007 (UTC)

Yes, the presentation is a bit confusing, but I don't know how best to clarify it. The table lists all of the hidden variable combinations that produce the maximum average score (+0.5) under randomized observation axes. The missing hidden variable combinations would give expected average scores of less than +0.5. The scoring system ensures that no matter what hidden variables there are, there is no way to achieve an average correlation (after many trials with randomized axes) of more than +0.5.
The experiment reveals observed values with more than a +0.5 correlation. It's shocking because it exceeds the maximum correlation one would expect with hidden variables and randomized observation axes. Does that help? If so, how can that be integrated into the article? Rod (A. Smith) 21:02, 5 November 2007 (UTC)
Well maybe another column should be added saying: "Any other combination" at the top and "<0.5" at the bottom. But this still does not answer my second question. Existance is a struggle between life and death. I just like to watch. --Zeruski 12:07, 6 November 2007 (UTC)
Also the table right after that does not contain all the combinations, I think it would be beneficial to provide all the possibilities.
Yes, it would probably be better to show all the possibilities. When this discussion concludes, I'll see whether I can fit the other ones in somehow. But, what is the question that remains unanswered? If it's the question of why the scoring system was set up to negate the score on a-b' but not on a'-b, note that the scoring system is purposely set up to ensure that any local hidden variables model will score at most +0.5. If we don't negate on a-b', local hidden variables of (+ + + +) and (- - - -) each would show greater correlation (+1.0). Similarly, if we negate on a'-b, local hidden variables of (+ + - -) and (- - + +) each would show greater correlation (+1.0). So, the scoring system is set up to limit the expected correlation by local hidden variables. Or, did you have a different question? Rod (A. Smith) 18:12, 6 November 2007 (UTC)
Actually, that lower table is supposed to represent the actual experiment being performed, show it's supposed to look somewhat random and should definitely not contain all possibilities. Reading the text again, though, I see that it's not much benefit to readers, so it should probably just be removed. Right? Rod (A. Smith) 18:32, 6 November 2007 (UTC)
Ok, now the scoring questions is actually answered. Thank you. Go ahead and modify the tables. Existance is a struggle between life and death. I just like to watch. --Zeruski 14:30, 12 November 2007 (UTC)
Good. The tables are now modified. Better? Rod (A. Smith) 23:35, 12 November 2007 (UTC)

Original Research?

I don't know much about the intricacies of wikipedia policy, but the prominant treatment of Joy Christian's paper seems to run afoul of (my understanding of) the No Original Resarch policy. As far as I can tell, Christian's paper has not appeared in any peer-reviewed forum and has received quite a bit of criticism. Now, perhaps the paper will get published somewhere, and perhaps it will turn out to be correct and will inspire a new way of thinking about Bell's theorem and QM, but until such time as it gets (at least) some minimal acknowledgement from a reputable, independent source, it should not be presented so prominently, above the cut and with the implication that it is necessarily legitimate (though controversial). It seems to me that removing the reference until such time as the paper gets accepted for publication by some peer-reviewed journal would not be any substantial injustice if the result turns out to be valid. Even then, I would think it should be mentioned further down in the article, rather than the third sentence, unless/until time and the community of experts determines that Christian's result is in fact of central importance to the modern understanding of Bell's theorem. 19:20, 12 November 2007 (UTC)

Yes, you're right. I moved that challenge down into a "Challenge" section that now also contains the "counterfactual definiteness" challenge. Rod (A. Smith) 23:35, 12 November 2007 (UTC)

I have removed all reference to Christian's articles, since they have been rejected by the scientific community and will confuse innocent readers (e.g. quantum mechanics students). I do not feel that wikipedia is an appropriate medium for the debate. —Preceding unsigned comment added by (talkcontribs)

Incomplete reference

In the section "Importance of the theorem", the statement that "This theorem has even been called 'the most profound in science'." is referenced only as "Stapp, 1975". Does anyone know more details such as journal name, volume, and pages? If not the reference (and maybe the statement) should really be deleted as a proper bibliographic reference should provide enough detail to enable a reader to find the paper (or book). Dirac66 (talk) 22:51, 12 January 2008 (UTC)

Encoding of all possible results upon leaving source? (Any takers?)

From the Overview section: "If local hidden variables determine the outcome of such measurements, they must encode at the time of leaving the source a result for every possible eventual direction of measurement, not just for the results in one particular axis."

Do results like those presented in the table (which violate the Bell inequalities) conclusively rule out this possibility? If so, how? Some additional explanation for the layperson on just how such massive encoding of results by local hidden variables is ruled out by these results would help. Why is it impossible for a very complex set of local hidden variables to encode for all the observed results at the time the particles leave the source? Why, e.g., can't local hidden factors encode in advance for results showing a correlation of .71 when the two measurement apparatuses are at 45° to each other? If you can explain this in a perfectly clear and unambiguous way for the layperson, you'll have a brilliant article. —Preceding unsigned comment added by (talk) 10:42, 30 January 2008 (UTC)

There's a completely intuitive explanation in the section called "Bells Original Inequality". It's sort of buried in the article though. The argument is this: imagine that you have two spin measurements in two directions A and B, which are determined by hidden variables and are 99% correlated. Imagine now that the two directions B and C are 99% correlated. Then convince yourself that A and C are at least 98% correlated. The idea is that the number of mismatches between the hidden variables determining A and B plus the number of mismatches between the number of hidden variables determining B and C is the greatest number of mismatches between A and C.
But in quantum mechanics, if A and B are 99% correlated, even distantly, and B and C are 99% correlated then A and C are only 96% correlated, because the probability goes as the wavefunction squared.Likebox (talk) 15:05, 30 January 2008 (UTC)
Thinking about this helps. Thanks. (Although more explanation is still called for.) It could be explained more clearly in the article, though. I concur with those who say (above) that the article in its current state is muddled and too technical for an educated general audience. This illustrates some of the hazards endemic to Wikipedia itself, although it's a noble project.

Clearing up a popular misunderstanding with the No-communication Theorem: a thought experiment (Any takers?)

Suppose Alice's and Bob's measurement apparatuses are in separate rooms and the particle source is housed in a wall between them. Suppose further that Bob and Alice have no means of communication available for transmitting their respective results to each other. Now, suppose that Bob keeps the angle of his measurement device constant at all times and hopes to infer from his results alone whether Alice has changed the angle of her apparatus. A widespread misunderstanding of experiments of this type (and of Bell's theorem itself) has it that Bob would be able to do this. But he couldn't. If he could, then it would be possible to harness entanglement for practical purposes of transmitting signals faster than light. According to the No-communication theorem, it can't be done.

Suppose Bob and Alice agree to a lengthy but very simple experiment of the kind just described, each with his or her measuring device in a separate room and with the particle source between the two rooms. Before beginning the experiment, they tell each other the initial angles of their measuring devices and agree that Bob will keep his angle constant throughout the experiment. The experiment is divided into two phases, each lasting twenty years. Alice and Bob agree that for the first twenty years, neither of them will change the angle of their measuring device. Furthermore, they agree that after exactly twenty years have passed, Alice will make an arbitrary decision to either change the angle of her apparatus by an agreed upon amount and leave it that way thereafter or let the angle stay the same, while Bob's angle will remain constant as before. They will then continue measuring for another twenty years. After a total of forty years have gone by, Bob will attempt to determine whether Alice changed the angle of her apparatus at the halfway point by analyzing his results before and after that time. Will he be able to tell this soley from his own results? He won't. He'll never be able to know whether Alice changed her measuring angle until he looks at Alice's results. And this would be true no matter how long the experiment lasted and no matter how many trials Bob had to work with.

One natural if naive response to this would be the following: "If Bob can never infer anything about the state of Alice's measurement apparatus from his own results, no matter how large a data set he has, then how can it be that these experiments prove non-locality?" Explain in layman's terms why this naive response is mistaken. Explain in a non-technical way how Bell test experiments are thought to prove non-locality in spite of the impossibility of using entanglement for communication. A simple, straightforward explanation along these lines would improve the article greatly. —Preceding unsigned comment added by (talk) 03:59, 31 January 2008 (UTC)

There's a way of thinking about the no-communication theorem which makes it obvious. If you take a many-worlds interpretation, then each observer is split in two at each measurement, and each copy sees a different value of the spin. The correlations between the distant observers are only caused by the meeting process, which copy meets which other copy. There is an angle between the copies, determined by the experimental setup, that tells you how likely each one is to meet up with each other, and this angle reproduces the quantum correlation statistics. The statistics of each one in the absence of any meeting up is completely local.
The many-worlds view is local in the sense that the description of the physical state of a system is only determined by the past light cone. It is not local in the sense that it includes these "angles" that describe the correlations between the distant observers, but these angles only are visible when the observers go to talk. The Bell inequalities can obviously be violated in a many-worlds setup, because you don't have a definite other person to go meet. The "hidden variables" for the other observer are undetermined from your point of view until you talk to the other observer.
So it is only when you demand that the other outcome is fixed and definite before the meetup (that's the hidden variable assumption) that the Bell correlations start to look nonlocal. So in some sense, the naive statement seems to me correct. If it is impossible to communicate signals, there should be some way of thinking in which the theory is obviously local. But in Quantum Mechanics, that interpretation is many-worlds. So if you want a single-world interpretation which isn't solipsistic, the Bell inequality violations require a nonlocal process, wavefunction collapse or whatnot. Note that this nonlocal process can go backward in time along the light cone, so it doesn't have to violate relativity. But whatever the process, it is then mysterious why this nonlocality is conspiring to not allow communications. I think to philosophically get on, it is most convenient to just take a many worlds interpretation and be done, leaving the mind open to future modifications of physical law.Likebox (talk) 23:58, 31 January 2008 (UTC)
I think bringing in the many worlds interpretation just confuses the issue. Anyway, it seems clear that most professional physicists working in this area would deny that the impossibility of harnessing entanglement for purposes of communication entails the truth of some local interpretation. What would such a professional have to say about the thought experiment sketched above? How would he or she answer the naive localist response in a non-technical way? Any takers? —Preceding unsigned comment added by (talk) 04:15, 1 February 2008 (UTC)
I think you have the prejudice that bringing in the many-worlds interpretation confuses the issue, but I think you will agree that it makes things clearer if you contemplate it for a bit.
Not only a many-worlds interpretation, but also a solipsistic Copenhagen interpretation gives the same local resolution. In the solipsistic Copenhagen interpretation only one observer, that's you, is responsible for the collapse of the wavefunction, and there is no other observer which is capable of collapsing anything. Again, your friend's wavefunction doesn't collapse until you go to meet him or her, and then there is no restriction on the correlations because your friend's state isn't definite until you talk. The nonlocality only comes in when you assume that you and your friend have a definite state at the same instantaneous time.Likebox (talk) 17:43, 1 February 2008 (UTC)
"If it is impossible to communicate signals, there should be some way of thinking in which the theory is obviously local." This is the conditional that interests me. You accept it and try to argue on its basis for either a many worlds interpretation or a radically solipsistic Copenhagen interpretation. Fair enough. You're entitled to your view. This conditional is denied by many physicists working and publishing in the field (which doesn't necessarily mean it's false, to be sure) who deny that any local interpretation can account for the data, including physicists who are committed to neither a many worlds view nor a purely solipsistic Copenhagen view. I'm especially interested in how physicists of the latter sort would argue against this conditional, particularly if they can present an argument that is readily intelligible to an educated lay audience. The argument might be very simple and straightforward. If so, it would improve the article greatly. Any takers? —Preceding unsigned comment added by (talk) 03:19, 3 February 2008 (UTC)
Just two little things: First, Likebox, I think you might be mistaken in your appeal to a solipsistic Copenhagen interpretation. Specifically, while no other observers are capable of collapsing the wavefunction, when you instigate a collapse by observing the state of the particle on your side, the particle on your partner's side instantaneously collapses as well. This is what it means for the two particles to be entangled, if they are two electrons in the singlet state, when you measure one's spin in a certain direction, the particle's partner on the other side must have a spin in the opposite direction. So non-local influences are rather inescapable in any orthodox collapse theory.
As a response to the general talk topic, the short answer would be that what is required to achieve the quantum correlation is not the ability to send superluminal signals, but merely superluminal causation and superluminal information transfer (between particles). So while one could not send information this way, whether or not Bob's particle passes or not will rely, at least in part, on the angle of Alice's measuring device, which is at space-like separation to it. (talk) 21:15, 17 June 2008 (UTC)

(deindent) The "solipsistic Copenhagen" interpretation is not a real interpretation, it's a straw-man version of the Copenhagen interpretation in Everett's thesis. It's a way of reformulating many-worlds to show that it is obviously compatible with a Copenhagen interpretation. You pick one observer, and then always talk about the state relative to that observer. Since there is no faster than light anything in many-worlds (there's no collapse), there should be no faster than light in solipsistic copenhagen either, because its the same thing.

Then again, the solipsistic collapse caused by entangling yourself with a local electron, part of a singlet pair, could cause an "instantaneous collapse", even in the solipsistic version. But you could also call it a collapse which moves out at the speed of light with no contradiction, because you don't care about the nondefiniteness in the description of other observers. You never have to make your story consistent among the memories of two different observers. A way of saying that is that you formulate your QM by defining the "now" state on a light-cone going forward in time centered on the observer, and the hamiltonian pushes the cone tip upward along the path of the observer. The collapse collapses the state on the cone. That's a pretty normal formulation of QM.

The example I was thinking about, where I thought this was useful, was for the case of Bell's inequality. That's when the other observer's state is not fully determined even after the collapse. That's when you get Bell inequality violations. The other observer could have measured spin states at a small angle relative to you, so that after your measurement, the other outcome is still (probabilistically) uncertain even after you know your own state. Then the other person's state is indefinite even after the collapse reaches the other person. You have to go there and talk to find out what the result was.

But to be honest, I never thought about the issue of collapse in the solipsistic Copenhagen interpretation that deeply before. I only ever used it to translate back-and-forth between many-worlds language and Copenhagen language.Likebox (talk) 21:59, 7 August 2008 (UTC)

Leggett's inequality link, call for edits to Leggett's Inequality link from Bell's theorem.

Hello everyone.

I followed the link from here to "Leggett's Inequality" and found that the page had not been created yet. So, I had the temerity to make one with what little notes I had.

I am a physicist, but do not work in this field, so I should imagine that what I have put up needs the attention of somebody more competent with all this than I am.

So, please start to edit!

Tethys sea (talk) 12:04, 26 November 2008 (UTC)

Isn't it ultimately all about locality?

Forgive my ignorance, but doesn't the Bell Inequality show that ANY theory which can account for the relevant correlations between results of experiments has to violate locality? I thought that all those "impossibility" proofs about Hidden Variable theories were all just based on assumptions of locality which were, ultimately, unfounded. I mean, it is true that Bell's Theorem proves that "Local Realism" won't be able to reproduce the quantum mechanical predictions, but isn't this a little misleading? That would be like if there was a country where people were arrested for wearing green, and I told you "In that country, blue-eyed people are arrested for wearing green!" I mean, certainly what I said was true, but it's crazy misleading. Same thing here, Bell's theorem shows that no theory which reproduces these correlations can be local. Certainly if no theory can do it, a hidden-variable theory can't do it, but that doesn't make this an anti-hidden variable theorem. I mean, clearly Bell's Theorem implies that no local hidden-variable theory can be true, but it also implies that no local theory thought up by a Scientologist can be true, but that doesn't mean Tom Cruise should sue Bell's corpse.

(note: Many-worlds theories are outside of its scope since, on these theories, there are no definite results of experiments, just a set of worlds corresponding to results. Also, those "pre-deterministic" or whatever "loopholes" are outside of the scope too, since they don't reproduce a predictable correlation; they just reproduce our present data.)

Thanks in advance for your replies. (talk) 05:14, 25 December 2008 (UTC)

Bias and lack of specificity of introductory sentence.

The article begins: "Bell's theorem is a theorem that shows that the predictions of quantum mechanics (QM) are counter intuitive, touching upon several fundamental philosophical issues related to modern physics."

Bell's Theorem itself says nothing about counter intuition as it is not a psychological or epistemological claim. Rather, Bell's Theorem states that hidden variable theories are irreconcilable with quantum mechanics. Philosophical implications are important, but should be addressed later. The first sentence of the Bell's Theorem page should discuss Bell's Theorem and Bell's Theorem alone.

The sentence violates NPOV. The use of "counter intuitive" makes several assumptions about intuition that are not necessarily shared by all readers. Newton's first law is highly counter intuitive to many people before they take high school physics.

Lastly, the part of the sentence, "touching upon several fundamental philosophical issues related to modern physics," says essentially nothing at all. —Preceding unsigned comment added by (talk) 05:10, 20 April 2009 (UTC)

I partly agree with this remark. The first sentence should preferably be changed into "Bell's Theorem states that certain hidden variable theories are irreconcilable with quantum mechanics" (note that the theorem does not exclude nonlocal theories). I have no objection against some reference to "philosophical implications" already at this introductory stage, since it is an important aspect of the theorem.WMdeMuynck (talk) 22:20, 20 April 2009 (UTC)
I revised the introduction taking your comment into account. What do you think? (I am the user who posted the original comment, sorry for not signing in.)Jwpitts (talk) 04:35, 30 April 2009 (UTC)
I am afraid I liked the original version better precisely because it is a bit vague. In particular it did not refer to determinism, which was judged important in the original version of Bell's proof. But later this proof was generalized to statistical hidden variable theories, thus changing the attention from determinism to locality (note, however, that also locality is not a necessary condition for a proof of Bell's inequality). Since this subject is still under discussion, I think Wikipedia should not make definite statements on this. Therefore I prefer a certain vagueness in the introduction (in which counterintuitivity might be referred to because this is a historical fact). In the bulk of the article different views on the Bell inequality might be mentioned.WMdeMuynck (talk) 10:09, 30 April 2009 (UTC)
OK, let's ditch the determinism talk. I still think that the use of "counterintuitive" is a bit biased. It assumes specific psychological attitudes toward intuition. I would welcome the use of the word "surprising" because this simply means that the results were not anticipated. Intuition differs widely from person to person and can change drastically over a persons lifetime. In fact, Feynman says that it is the job of physicists to invent ways of thinking about new physics that make the results intuitive (however surprising the results may have been at the time).
The crucial point is, I think, that Bell's theorem demonstrated the untenability of the Copenhagen idea that quantum mechanics is a complete theory, since it allowed nonlocal hidden variables. Probably this result has been experienced as both counterintuitive and surprising by those believing in the Copenhagen interpretation (which was the large majority of physicists at that time; I was one of them). I don't know of any category of physicists who would have judged this conclusion to be not counterintuitive because we did not have any experimental evidence of such a nonlocality. It is not difficult to think of a category who would not be surprised: it would contain anyone recognizing that Bell's theorem is just an application of Kolmogorov's classical probability theory, there being no reason to assume that this latter theory is applicable to microscopic physics. Although among physicists this certainly is a small minority, I don't think that substitution of `counterintuitive' by `surprising' would be an improvement.WMdeMuynck (talk) 23:51, 3 May 2009 (UTC)
Sure. I have no problem with that use of the word, but you were taking a great deal more care than was used in the original introduction. My qualms were simply that Bell's theorem does not (categorically) prove that "the predictions of quantum mechanics (QM) are counter intuitive," although it may prove that under the accepted (Copenhagen) interpretation of the time, the results are counterintuitive. While the theorem predates experiment, nonlocality has been experimental fact for my entire life. My intuition has been and continues to be built from such a world.Jwpitts (talk) 04:56, 4 May 2009 (UTC)
I think in contrast to the original, mention of the philosophical implications can include what topics are relevant without taking a stance either way.Jwpitts (talk) 21:38, 3 May 2009 (UTC)

Bell's Theorem distribution false?

I've been reading a bit on this subject, and a certain Georges Lochak, from the "de Broglie Foundation", argues that the distribution used:

is incorrect because it implies a classical distribution of chances, that is to say, one is able to find the joint probability, which is, of course, the main problem of QM, because it is not possible in the classical way. It is based on G. Lochak's section "De Broglie initial conception of de Broglie waves", in the book "The Wave-Particle Dualism", subtitle: "A tribute to Louis de Broglie on his 90th birthday". The point is, if his statement and math is correct, it should be included. But because I'm not capable and familiar enough with this subject, someone else should write it, and, obviously, check its validity.

This is the math behind this statement. First let me introduce my notation.

is the probability of A, if B is already true or measured. The rest of the notation should be obvious. The original reasoning contained a different notation I was not completely familiar with (I've converted it to this notation because I've copied the math for a different purpose than wikipedia. I did understand the original notation).

This and the following two are probabilities that don't hold true in quantum mechanics, for it defines a joint probability, and thus supposes one is able to simultaneous measure Quantum Entangled objects without affecting the measurement.

This suggests hidden classical probabilities. g. Lochak argues that is different from predicted quantum probabilities, which must prepare the system before measuring, and thus modifying the state of that which is measured. I believe it basically comes down to this question: can't local hidden parameters produce QM probabilities if the measurement modifies it, but the measurement doesn't affect the other 'entangled' state, which is still to be measured? Measuring the unmeasured particle could modify it in the same way the already measured particle is modified, without introducing information that is shared between the particles at infinite speed, affecting the state of the unmeasured.

I hope that all this is clear. Cheers, Boreras (talk) 15:36, 3 February 2008 (UTC)

After carefully studying of the quoted paper (, I find that he uses different arguments and math to come to the same conclusion, namely that Bell's Theorem doesn't hold true at all, and that some sort of local variable theory is possible.
I am, however, not sure if its true that Christian's non-commuting numbers are the same as Lochak's predicted probabilities (that is to say, the numbers aren't already realized before measuring).
the question is also whether Lochak's reasoning should be quoted, explained or referred to in this article if it holds true. Boreras (talk) 22:08, 4 February 2008 (UTC)
Christian's idea is that the statisitics of variables are to be determined by a new type of averaging procedure. He says to find the expectation value of two variables "A" and "B" you must integrate over all quaternion hidden variables.
and then notes that if A is a unit quanternion and A(q)=Aq and B is a unit quaternion and B(q)=Bq, the noncommutativity of the quaternion algebra reproduces the quantum mechanical predictions when this averaging procedure is used.
While the mathematics clearly gives the same answer as quantum mechanics, the quaternionic algebra is just another representation of spin 1/2, the physical interpretation of this gimmick as "local hidden variables" leaves something to be desired. The hidden variable q produces correlations, but it cannot be interpreted as giving the actual values measured by the experiments, or their probabilities. It is just a mathematical device to compute expectation values, like the wavefunction. That's not a flaw in itself, but all the difficulty of intepretation in the quantum mechanics translate directly to any scheme that reproduce the statistics.
It's not the formalism of QM that's at fault, it's the predictions themselves. Following Mermin, one can state the issue with hardly any mathematics:
"If coin-flips A and B secretly conspire to be 99% correlated, and coin-flips B and C secretly conspire to be 99% correlated, then coin-flips A and C will never be 96% correlated."
That's it. It's an obvious true fact about secret conspiracies. If you are put in rooms to take a yes/no test, and you give your list of answers "yes and no" to your friend B, and your friend B makes a list that is 99% identical, and gives his list of answers to his friend C and tells C to make a list that's 99% identical, you and C will have the same answer at least 98% of the time. There's no way around it.
If two entangled electrons are interrogated by measuring their spins in nearby directions A and B, the spin values you get are random but 99% correlated, likewise B and C are random and 99% correlated, and C and A are 96% correlated. That can never be reproduced by a secret conspiracy where the electrons got together and made a list of all the answers they were going to give.Likebox (talk) 17:51, 6 February 2008 (UTC)