Talk:Bell's theorem

Article contradicts itself

In the overview..

"Similarly, the results will be anti-correlated (+,-) (-,+) if their analyzers are aligned on orthogonal axes."
But then to the right the table says otherwise.
" Orthogonal axes: pair 1 pair 2 pair 3 pair 4 ...n
Alice, 0°: + − + − ...
Bob, 90°: − − + + ...
Correlation: ( −1 +1 +1 −1 ...)/n = 0.0
"Dave3457 (talk) 02:48, 11 June 2009 (UTC)

This overview was a huge mess, and contained multiple issues. I just spent a fat chunk of time revamping it. Much better now. Isocliff (talk) 08:46, 1 October 2010 (UTC)

Cute theorem but trivially refutable; why so much space?

In outline: Essentially QM argues that rather than having specific values miniscule objects and processes have probability distributions. These probability distributions are collapsed into specific values when various conditions apply. For instance a decaying tauon may decay into this that or the other thing with varying degrees of probability. Once it has decayed into (say) tauon neutrino + X + X antineutrino, then it is a specific value.

For any studied (historical) system there are no relevant probabilities left, there are only realized values. Given an appropriate ordering and mapping these can be expressed as a sequence of values on some more or less continuous interval.

It cannot be proven that the system was not the result of consulting such a sequence.

If the sequence is random then there are no hidden variables (there, at least).

If the sequence is pseudo-random then there are hidden variables.

It cannot be proven that the sequence is not pseudo-random without knowing the pseudo-random period.

So at most Bell has proven that some hidden variable models are insufficiently sophisticated. 76.126.215.43 (talk) 22:43, 11 July 2009 (UTC)

I don't think you really understand Bell's theorem. It has nothing to do with random versus pseudorandom measurement outcomes. You could assume the classical hidden-variable theory is truly random or the quantum-mechanical universe is pseudorandom, it doesn't matter. The important thing is that the quantum statistics of the measurement outcomes can't be simulated by any classical algorithm applied independently to the two subsystems that are measured. It can be simulated by a classical algorithm that gets to choose the outcome for both subsystems together, but you can set things up such that that requires faster-than-light communication. -- BenRG (talk) 21:18, 20 July 2009 (UTC)

Agree with BenRG. You're asserting the very thing that Bell's theorem, and the accumulating evidence and experiments since; a naive philosophical realism. Before you patronize a well-established scientific theory on purely philosophical grounds, it's good to understand the theory you're trying to criticize. If you can demonstrate your thought-experiment in the same manner as the EPR [which is Bell's starting point] to refute it, there's a nobel prize waiting for you. —Preceding unsigned comment added by 209.105.184.93 (talk) 03:00, 20 October 2010 (UTC)

The error of reasoning in naive philosophical realism is captured in "Essentially QM argues that rather than having specific values miniscule objects and processes have probability distributions." This presupposes that quantum wave functions are probability distributions over specific values, but that is not what QM asserts. In QM it is wave functions that are real, not the specific values found from measurement. Measurements lead to mixed states on subsystems that we interpret as probability distributions over specific measured values. However, this is only a mathematical artifact of the projection of a larger wave function onto a measurement subspace of interest. The wave function for the universe as a whole never collapses. Wave function evolution per QM is fully deterministic. It is only our interpretation of specific measured values on subsystems that are probabilistic. Quantum decoherence explains the physical processes involved. Non-local quantum correlations demonstrate that QM is inconsistent with naive philosophical realism. Experiments demonstrate that the universe is consistent with QM. Manawiz (talk) 23:01, 27 December 2010 (UTC)

Experiments: Where's the point?

Assuming that there's a hidden spin (or polarization) variable, then these measurements prove that when your spin (or polarization) detector's axis is not aligned exactly at the axis of an incoming particle, then there is a misalignment-angle-dependend probabilty that it will either output the same result like a correctly aligned device or not.

I can't see why this proves anything more but that a misaligned binary measuring device outputs fuzzy results, and that the statistical distribution of this fuzz is correctly stated by a certain equation. Any device mapping a linear input to a binary output will behave more or less like this. For example when you have a TTL circuit and the input voltage for a 1 signal is below the circuit's specifications, than it will take this input for either 1 or 0 (as is not allowed anything inbetween) with certain probabilities that depend on the input voltage and the circuit's physical properties.

This all looks like some circle argument about some statistical equations proofing themselves, not like valid a falsification of the local variables hypothesis.

Am I missing something!?

--217.87.172.196 (talk) 19:45, 24 November 2009 (UTC)

Relativity and Determinism

After EPR (Einstein–Podolsky–Rosen), quantum mechanics was left in an unsatisfactory position: either it was incomplete, in the sense that it failed to account for some elements of physical reality, or it violated the principle of finite propagation speed of physical effects. In a modified version of the EPR thought experiment, two observers, now commonly referred to as Alice and Bob, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state. It was equivalent to the conclusion of EPR that once Alice measured spin in one direction (e.g., on the x axis), Bob's measurement in that direction was determined with certainty, with opposite outcome to that of Alice, whereas immediately before Alice's measurement, Bob's outcome was only statistically determined. Thus, either the spin in each direction is an element of physical reality, or the effects travel from Alice to Bob instantly.

Look at this in terms of Relativity. In every experiment like the one described above, for every individual measurement taken by Alice and Bob, there will be a frame of reference in which Bob's measurement happens first, one in which Alice's happens first, and one in which the measurements happen simultaneously. Logically, then it doesn't even make sense to talk about the possibility of one measurement affecting the other. In frames where the measurements happen simultaneously, the two have to be conspiring, simultaneously affecting each other. And since either measurement could come first and at any time after the creation of the entanglement, then the resolution has to be predetermined, even in cases where A or B "randomly" set their detectors or change their detector orientations while the particles are in transit.

Is there a hole in this reasoning? If not, does this mean the universe has to be deterministic? —Preceding unsigned comment added by 70.166.64.3 (talk) 22:19, 30 March 2010 (UTC)

The situation where the two distant measurements are close to each other in time excites a lot of talk about "quantum effects" (people seem to know better than to say "information") propagating faster than light, etc etc. But the times are totally irrelevant. The situation is this: If Alice and Bob measure the two entangled spins in the same direction they will get opposite results. If they measure the spins along orthogonal directions they will get uncorrelated results. It doesn't matter whether Alice goes first, or Bob, by how long. 76.102.80.158 (talk) 01:46, 31 December 2010 (UTC)

What is wrong with the Overview?

In today's edit by 128.143.100.177 (talk) (→Overview: This was just wrong...) a confusion (not a mistake) is revealed, to the effect that in the Overview a measurement is discussed using polarization measurements of two correlated photons (as performed by Aspect and coworkers in 1981). The discussion of these experiments is largely analogous to the discussion in terms of correlated spin-1/2 paricles in a singlet state (which the editor had in mind), be it that angles differ by a factor of 2. The confusion is enhanced by a picture in that section referring to spin rather than photon polarization, but nevertheless meant to illustrate the issue.WMdeMuynck (talk) 22:25, 31 March 2010 (UTC)

Overview 4th par.: 45°?

So far, (when the analyzers are aligned, orthogonal, or 45 degrees) the measurement results can be modeled by proposing physical attributes, -- but there's no discussion of 45°. ABS (talk) 17:19, 12 June 2010 (UTC)

Source lacks authority

In the overview, it is suggested that Bell's theorem is controversial in some way. 'Not everyone agrees with these findings' We'd really need a source of some notability to back this up. The source for this appears completely unreliable. It's not from any kind of peer-reviewed journal, but self-published by someone who appears to be a bit of a crank. I would just delete it but it appears to have already been deleted and promptly re-added.Steady unit (talk) 22:36, 2 September 2010 (UTC)Steady unit (talk) 23:25, 2 September 2010 (UTC)

Agreed. The lead-in paragraph, especially near the end, is using vocabulary that appears to be in doubt of the experiment's results, and then implies further doubt by appealing to a feigned controversy over interpretation. The experiment has been done repeatedly, and each time, the results have been the same. The burden of proof belongs to 'not everyone' and whoever 'interpretation of these experiments is still the subject of some debate' is; they are welcome to step up to the plate and design an experimental refutation, but until then, I will be going through the article and removing un-sourced [and political] assertions. —Preceding unsigned comment added by 209.105.184.93 (talk) 16:11, 18 October 2010 (UTC)

I regret the changes recently made to the overview, referred to above, because they substitute a majority view endorsing a nonlocality explanation of the Bell inequalities for a view leaving room for alternative explanations. An alternative explanation has been discussed in chapter 10 of my book, Willem M. de Muynck, Foundations of quantum mechanics, an empiricist approach, Fundamental theories of physics, vol.~127, Kluwer Academic Publishers, Dordrecht, Boston, London, 2002, based on peer reviewed publications (see my website [1] for a list). I am not suggesting to explicitly deal with this alternative on the present Wikipedia page (which is not really necessary since at several places in Wikipedia there are references to my website), but I agreed with the previous version precisely because of its cautiousness, which is now lacking.WMdeMuynck (talk) 20:10, 18 October 2010 (UTC)

I appreciate the interest of the above two commenters in this article, but both of you seem to have gotten the wrong idea about the major rewrite I did of the overview on October 1st. To be abundantly clear, I ABSOLUTELY agree that any suggestion of controversy in the physics community with respect to Bell's theorem is ridiculous. The one sentence at the very end of my overview rewrite that refers to dissenting views was included solely in the hopes of avoiding future back-and-forth editing, due to the the presence of (as far as I can tell) a single user dedicated to casting doubt on Bell's theorem. If you check, you will see that the previous version was far more supportive of such dissent, to the point of being seriously misleading. The overly conciliatory sentence at the end was not ideal, but it is at least correct. So again, since we seem to be in agreement on this point, it should be fine to begin revising any phraseology that casts undue doubt on the theorem. (Edit: On further examination, it appears you are actually referring to the first section, before the overview, in which case we definitely agree. This part is even worse than I realized so I will edit it immediately, to at least be accurate.)

Now, WMdeMuynck, your comment does not seem to make much sense to me. Please be specific. I don't see how you can argue that my revision does not "endorse a nonlocality explanation" when I plainly said that the results demonstrate "the existence of superluminal effects".

Sorry, if my mastery of the English language is insufficient. When I said "a is substituted for b", I meant to say that "b is replaced by a". Hence, I certainly implied that your edit is endorsing a nonlocality explanation (as you plainly said). My criticism was about the suggestion of exclusivity of that view (which view, indeed, was widely held within the physics community, but currently meets with increasing criticism), and about the neglect of the possibility of alternative explanations.WMdeMuynck (talk) 09:12, 19 October 2010 (UTC)

My rewrite of the overview on Oct 1st was primarily to address three problems with the prior version: 1) the presence of errors and contradictions 2) the greatly exaggerated "disagreement" with the standard interpretation of BT, and 3) the fact that the description of BT was overly specific to the point of being incorrect. If you'd like to explain what you meant, Id be very interested to hear what you have to say, but please keep in mind that I cannot read your book. Isocliff (talk) 03:22, 19 October 2010 (UTC)

I agree that the former presentation of the problem was far from ideal, and should be improved. However, I found your edit not an improvement. The alternative you chose is the one developed by John Bell, introducing a kind of nonlocality that has triggered phantasies of superluminal communication, the latter, however, being recognized by the physics community to be unobservable. Hence, the nonlocality view has its own problems, which undermines the possibility of plain statements as presented by you. Unfortunately, a complete analysis of the problem needs an analysis of both the mathematical formalism of quantum mechanics as well as its interpretation, probably going beyond the level of sophistication aspired by a Wikipedia article.WMdeMuynck (talk) 09:12, 19 October 2010 (UTC)

I see. We'll your english seems pretty good to me, except for the one sentence that gave the reverse impression than was intended. Im certainly interested to hear your argument, but I dont see how you can reasonably describe the nonlocality of entangled systems as an "explanation", considering we experimentally observe the instantaneous statistical changes in measurement outcomes. Avoiding this conclusion, it seems to me, would require a departure from traditional assumptions so radical that it would be incumbent on you to state so explicitly (i.e. Superdeterminism, Solipsism), or else, in the words of someone much smarter than myself, a weird "conspiracy of nature". I don't think its necessary to think of the nonlocal effects as "communication" except in the most general sense, since I've never heard anyone suggest that entanglement results from particles sending signals to each other. I think the more common view is the one that parallels the quantum formalism: that both parts of the entangled system are simply described by the same information (the same quantum state). Your use of the word "phantasies" to describe this nonlocality leads me to believe that you flatly reject the mainstream consensus (perhaps that was your intention). But this wikipedia article's primary responsibility should be to convey the meaning and significance of Bell's theorem as it is understood and accepted by the physics community as a whole, otherwise very few articles would be able to say much of anything. Now I haven't conducted a formal survey of the literature, but the quantum mechanics textbook used at MIT, Harvard, my own school WPI, and a lot of other universities (Griffiths) refers to Bell's theorem and these experiments as demonstrating nonlocality, without any qualification, exactly as I did in the revision. So again, feel free to state your case tersely if you can, since Im interested. But I doubt I'll be comfortable accepting a major revision of physics understanding into the article without the demonstrated support of some major people in the field (who Im sure would also defend the mainstream view much better than I ever could). I dont mean disrespect, since your credentials dont seem insignificant at all, but you're still just citing yourself. Isocliff (talk) 04:42, 25 October 2010 (UTC)

You say: I dont see how you can reasonably describe the nonlocality of entangled systems as an "explanation". My answer: Nonlocality explains von Neumann projection in an EPR experiment (this is an explanation, although a bad one). Nonlocality is not part of "description" (entanglement is). In the wake of logical positivism physicists have learnt to make no difference between explanation and description: in talks you will often hear that something is explained when an adequate description is found (i.e. when it is subsumed under a physical theory, as is the case in the Deductive-nomological model).

You are right that I have strong doubts with respect to nonlocality precisely because it is unobservable (you can't use it to send messages). Violation of the Bell inequalities in EPR-Bell experiments only occurs if incompatible observables are involved. But incompatibility is a local affair (mutual exclusiveness of measurement arrangements). So it seems that violation of the Bell inequalities has a local origin.

I have discussed this extensively on my website [2] and in my book Willem M. de Muynck, Foundations of quantum mechanics, an empiricist approach, Fundamental theories of physics, vol.~127, Kluwer Academic Publishers, Dordrecht, Boston, London, 2002. Unfortunately, the problem of nonlocality is entangled with many other aspects of the interpretation of the quantum mechanical formalism. I tried to find the most weak interpretation possible, which I refer to as the `empiricist interpretation', which does not try to bestow on the formalism more physical meaning than is strictly necessary. It is not my intention to burden Wikipedia with all these intricacies, but every now and then I try to venture a moderating word when a presentation in Wikipedia in my view becomes too one-sided.WMdeMuynck (talk) 17:01, 31 October 2010 (UTC)

First let me say, In regards to your last sentence, I'm not objecting to what you're doing on here, just expressing my own disagreement and trying to understand. I do think you've expressed your dissent in a respectful and responsible way, so I definitely appreciate that. At first you certainly piqued my interest with what you described as alternative formulations of Bell's inequalities, however after looking more closely it looks like there is a fatal flaw in your reasoning. Before I get into that, what I find most bewildering about your position, and what I have yet to find a satisfying explanation for in your work, is why you can ignore the physical evidence for nonlocality that has been demonstrated over and over. We directly observe the statistics of a particle changing dramatically after measuring its entangled partner in a spatially-dispersed system. The possible mechanisms for getting around this, which already seemed somewhat far-fetched (i.e. the communication loophole), have been all but experimentally ruled out. It just seems to me to me that an argument against nonlocality should be under some obligation to provide a specific explanation for the otherwise unambiguous evidence. You say that "incompatibility is a local affair". This is incorrect, and it precisely because it is a nonlocal affair that we can discern the existence of nonlocal correlations. Compatibility is determined by the detector settings of the respective distant observers, which can be chosen randomly after the photons (or whatever) are in flight, in mutually-exclusive lightcones. If you want to argue that the elaborate efforts undertaken to ensure this randomness have failed, okay. But then as more and more ways are found to randomize the settings, the more contorted and unlikely it seems to me any such theory would have to become. The above statement, combined with your describing nonlocality as "unobservable" makes me increasingly wonder if there isn't a fundamental misunderstanding somewhere. We can "observe" that the nonlocality occurred when, after an experiment, classical communication is used to exchange information about measurement results that occurred in the past.

Now here is where either I have misunderstood your argument, or your analysis has made a serious mistake. Bell's inequality is an expression of a condition placed on the expectation values of the products (correlations) of measurement results. It is derived by assuming definite values of physical quantities, whose expected value, as we see it, is given by the integral of the measurement results times the probability density of the hidden variable over all possible values of the hidden variable(s). On the other hand, you're inequality apparently involves probabilities computed as a function of which subsets of results $(S^{i})$ the various observables are confined to exhibit. So it seems that your result expresses something else entirely. To put it another way, BI is a limit on the expectation value of the product of two spin measurements made on two separated particles, as a function of the detector orientations, with the only other assumption being that when the axes of measurement are aligned (or anti-aligned), the results must be perfectly anticorrelated (correlated) which is the property we observe with pairs prepared in the state ${1 \over {\sqrt {2}}}(\left|\uparrow \downarrow \right\rangle +\left|\downarrow \uparrow \right\rangle )$. Your derivation doesn't seem to require this condition, or any hidden variables, but you apparently derive something that looks like Bell's Inequality expressing a limit on the probabilities of results as a function of the results, so again it seems both the domain and the codomain of the functions $P(S^{1},...,S^{n})$ which are the subject of your paper, are fundamentally different from those of Bell's theorem. As a final remark, because I cant seem to find any references to hidden variables in your derivation, your conclusion could be said to apply to any theory, realistic or not, which would plainly put it at odds with what we observe, since the BI is violated experimentally. Even if this is not a correct assumption, and you do reference hidden variables somewhere, and I assume you are correct that this result demonstrates the impossibility of hidden variables to account for quantum mechanical behavior, then the mere existence of Bohmian mechanics would seem to contradict your conclusion, since it is a hidden variable theory that reproduces the results of experiments and of QM. For this reason I am especially interested to see what remarks of Bohms you are referencing in this article, apparently in support of your thesis, but I cannot seem to find/access that article. Care to help provide the details on this, or otherwise explain where I go wrong in my analysis of your work? Isocliff (talk) 09:10, 1 November 2010 (UTC)

Let me first say that I appreciate very much your extensive reaction on my previous answer to you. Since this discussion does not seem to be about the editing process I will explain my point of view in more detail on your talk page.WMdeMuynck (talk) 03:35, 2 November 2010 (UTC)

Excellent edit Isocliff, and to Mr. WMdeMuynck; I'm not quite sure this is the proper place, either, as this page applies more concerning Bell's theorem and it's effects, as the lead-in states, than an explanation on whether the interpretation of quantum mechanics and it's overall philosophy is factual on particular points. However, I am by no means an important editor here, so I appeal to Isocloff's comments above. As a side-note: I am currently reading your work on your personal homepage (for personal enlightenment), and it is interesting thus far. —Preceding unsigned comment added by 209.105.184.93 (talk) 03:19, 20 October 2010 (UTC)

Bell's assumptions

Throughout this article, references are made variously to tricky philosophical concepts like hidden variables, realism and counterfactual definiteness. Indeed, these are almost all mentioned as soon as possible within the introduction. I have found a similar issue on the page "Nonlocality" and it doubtless occurs on many other pages discussing quantum foundations.

There is some debate in the literature over the exact assumptions of Bell's Theorem and many, many physicists use the term "local realism" in their papers (without really thinking about it, in my opinion). However, after some thought and research it seems apparent that the only real assumption Bell makes in his theorem, is Locality. That is, the statement of Bell's Theorem should really be:

no physical theory satisfying locality (or local causality) can reproduce all of the predictions of quantum mechanics.

To quote Tim Maudlin (cited below):

"Bell did prove that no local deterministic theory, no local hidden variables theory, and no local realistic theory can make The Predictions [i.e. the same predictions as quantum theory] because he proved that no local theory can make The Predictions. But the addition of the unnecessary adjective yields a highly misleading result."

Counterfactual definiteness is just as misleading. The confusion appears to have arisen out of Bell's inference that results of spin measurements in a local theory predicting EPR correlations, must be predetermined. To quote Maudlin again (he expresses the argument very succintly, so I don't feel bad about this!):

"Bell cites exactly the EPR correlations (for any chosen direction a to measure spin) and a locality condition (“if two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained by the other”) and concludes (“it follows that”) that the theory must postulate an initial state for the particles that predetermines the results of all possible spin measurements and therefore must assign a more complete state than the singlet state. No invocation, either explicit or implicit, of any assumption of counterfactual definiteness appears in this argument. Rather, a form of counterfactual definiteness (that is, the claim that the initial state of the particles must determine what the result of any spin measurement would have been) follows from the argument."

Also, for good measure, a quote from Bell in "Bertlmann's socks..." which has the same meaning if you replace "determinism" with "counterfactual definiteness":

“It is important to note that to the limited degree to which determinism plays a role in the EPR argument, it is not assumed but inferred. What is held sacred is the principle of ‘local causality’ or ‘no action at a distance…’ It is remarkably difficult to get this point across, that determinism is not a presupposition of the analysis."

I propose that the article replaces any mention of hidden variables, realism, counterfactual definiteness etc... with the word "locality", unless it is otherwise relevent (for instance, when discussing exactly this kind of confusion). It might be argued that since a lot of physicists explicitly use the words "local realism" this is too one-sided for Wikipedia - this annoys me since then Wikipedia is contributing greatly to the confusion. At the very least, I propose to add a section clarifying these assumptions in the context of the theorem, and outlining arguments from both sides (however much it pains me).

Sources:

- Bell's "Speakable and Unspeakable...": The papers "La Nouvelle Cuisine" and "Bertlmann's socks..." have many pertinent comments along these lines.

- Travis Norsen "Against Realism" http://arxiv.org/abs/quant-ph/0607057: cites a barrage of arguments against using words like "realism" and "counterfactual definiteness". Convincingly argued.

- Tim Maudlin "What Bell proved..." http://ajp.aapt.org/resource/1/ajpias/v78/i1/p121_s1?view=fulltext

Sabri Al-Safi (talk) 14:37, 21 March 2011 (UTC)

J-Wiki's edits

There have been a number of useful edits by J-Wiki. But it seems a pity to lose the mention of Aharonov. Bell refers to the paper by him and Bohm in 1957, not just Bohm. I think I linked to the wrong thing and so got him axed. Myrvin (talk) 10:51, 26 April 2011 (UTC)

Hello, Myrvin. Although Bell points to the Aharanov-Bohm paper of 1957 as advocating the example he uses, Bohm originally published his version of the EPR argument, using the example of quantum spin, in the conclusion of his 1951 book, Quantum Theory. I'm not sure of the best way to handle this, but for now I've now added a reference for the 1951 book to the article. J-Wiki (talk) 00:25, 28 April 2011 (UTC)

I have to agree with J-Wiki on this one... it seems that Bohm came up with his spin-based variant of EPR well before his contact with Aharanov. However, their 1957 paper does seem to be the first to put its finger on the importance of correlations and suggest that resolutions of the EPR paradox might be experimentally testable.--Sabri Al-Safi (talk) 09:00, 28 April 2011 (UTC)

Joy Christian

I put this man's name in to replace a tag. It has now been removed and the man has been called a "fringe lunatic". I don't think such language should be used about an academic. Faith versus heresy is back. Myrvin (talk) 10:27, 4 May 2011 (UTC)

The "fringe lunatic" part was meant to be humourous. Is it enough to add references for the statement concerned, or do we really have to put down names in the article itself? The reason I ask is that there aren't any reputable practising academics who doubt Bell violations (as far as I can see). For example, looking at the arxiv (since I don't think he has any peer-reviewed published work on the matter), Christian's papers since about 2007 have repeatedly propounded the same argument with varying degrees of comprehensibility. There have been several papers by more reputable authors with simple refutations of his argument, after which the vast majority of academics have moved on. Essentially, it seems bizarre to explicitly state names, since it appears to promote their thoughts on the matter to a level of significance equal to those of Einsten, Bell, CHSH etc... --Sabri Al-Safi (talk) 11:13, 4 May 2011 (UTC)

I read on Wikipedia's style guidelines that "The templates , , , or are available for editors to request that an individual statement be more clearly attributed." I think the attribution is fairly clear as it stands? --Sabri Al-Safi (talk) 11:24, 4 May 2011 (UTC)

Vandalism or Error?

Remark 1, third indent, under the CHSH Inequality section has the word "JEFF" on the left side of an equation. This was introduced in the Revision as of 07:20, 27 August 2011 which, otherwise, seems sensible (although IANAP so take that evaluation for what it's worth). — Preceding unsigned comment added by 24.239.181.139 (talk)

The diagram of the source, Bob and Alice has two typos: the y axis for both Bob and Alice should be a "z" axis in order for the diagram to be consistent with the rest of the text. In addition, in the text, please change "The operators B(b'), B(b)" to "The operators B(b), B(b')" so that it corresponds to the x-z plane being rotated 135 CC degrees into the x'-z' plane, with b corresponding to Bob's measurements along x' and b' to Bob's measurements along z'. Perhaps the figure should also prime the axes on Bob's side. Clejan (talk) 18:17, 14 September 2011 (UTC)