The paper offers, mainly in its first Sections, a clear, extensive, and detailed presentation of Bell’s theorem, with some emphasis on the locality issue. (The Authors might also outline explicitly that the occurrence of a correlation between observables in a pure state is a typical quantum feature, unfitting the familiar ignorance interpretation about pre-existing values since the state is not thought of as a mixture)
The experimental situation about violations of Bell’s inequalities is carefully accounted for up to recent developments (see Section 7).
Section 10 includes some topics that, strictly speaking, are only marginally called into play by Bell’s theorem. This is the case, for instance, of Subsections 10.7 and 10.8 dealing with Many-worlds and relational interpretations of quantum theory and Consistent histories: it might appear unlike a reader looking for such topics under the title of this paper. The Authors could perhaps evaluate the opportunity of viewing these subjects under another heading of Scholarpedia. In its present form also Subsection 10.4, entitled Classical versus quantum probability (and logic) seems to have weak reference to Bell’s theorem. Here, however, the reader might expect to find also something about the so-called classical (or Kolmogorovian) representabilty of a set of (empirical) probabilities, a topic that has received attention by several authors since the eighties: within that framework Bell-like inequalities appear as necessary conditions for classical representability.
Section 11, entitled Non locality and relativity, mainly consists of a discussion about the notions of empirically relativistic and fundamentally relativistic theories. It ends with the statement that “…it remains unclear what exactly fundamental relativity means or requires” and, similarly, “whether Bell’s theorem … can be reconciled with fundamental relativity thus remains a very much open question”. Also in view of Scholarpedia’s recommendation for short papers, the opportunity of simplifying this Section, or viewing the subject under another heading could be considered.
The paper, as a whole, can be regarded as a significant contribution to Scholarpedia, even better if the above suggestions will find attention.
This article is, in my opinion, a very good review of the topic of Bell's theorem. It not only provides a clear presentation of Bell's theorem itself, comparable to that given in Bell's latest presentation of the topic -- "La Nouvelle Cuisine," Ref. 21 -- but also includes a lengthy section discussing the many misconceptions which crop up in recent discussions of the issues. This is a particularly important service which a Scholarpedia article on this type of topic should provide. I particularly appreciate the explanation regarding "counterfactual definiteness," which contrasts sharply with the wikipedia article on Bell's theorem (the latter still claims that this is an essential assumption of the analysis...). I also appreciate the considerable efforts which the authors undoubtedly invested in creating such a thorough review, and their willingness to volunteer to become curators of the article -- a role which for such a controversial topic will surely require much patience. However, I believe that for this to become *the* definitive updated review of Bell's theorem, an additional effort is still required. In the following, I provide a list of issues which the authors should, in my opinion, consider carefully (my remarks refer to the version of March 24).
1. The statement in the first sentence of the abstract that "our world is non-local," as the (conditional) conclusion of Bell's theorem is, in my opinion, an overstatement. After all, "our world" is not a mathematically well-defined concept (as the authors point out elsewhere). I would expect to see here a statement such as that opening Section 4, "cannot be accounted for by any local theory" (however, I have no objection to the use of "our world is non-local" in the penultimate sentence of the abstract).
2. The word "interactions" in the second sentence is perhaps too strong, and could be replaced by, e.g., "influences." In any case, it is appropriate to mention already in the abstract that quantum phenomena do not allow non-local signalling, in order to clarify the relevant meaning of "locality."
3. The concept of "spacelike separation" is described (but not named) in the second sentence. It could be helpful to mention it explicitly as well.
4. There are two subsidiary assumptions involved in the analysis of Bell's theorem, one is that the instrument settings can be treated as free variables (mentioned here, e.g., in section 5), and the other is that the future cannot affect the past (mentioned, e.g., at the end of subsection 10.3). Again, it is appropriate to mention these already in the abstract (which will remain sufficiently brief even if two or three sentences were added), or at least in the opening sections (the authors have done this in footnote 13, but I do not think this is sufficient).
5. The above-mentioned two assumptions are included in the "no conspiracy" condition discussed in the article. In contrast, the causal arrow of time was included also in the definition of "locality" or "local causality" preferred by Bell himself (see "La Nouvelle Cuisine" or "The Theory of Local Beables," Ref. 15). Should it not be clarified from the outset that Bell's definition of locality is not based exclusively on time-reversal symmetric concepts such as space-like separation? (see, e.g., the figures in section 6 here).
6. Similarly, it would be appropriate to use "relativistic causality" rather than "the theory of relativity" in the last sentence of the abstract, as some would take the latter to imply a time-reversal symmetric theory, whereas in order to "prohibit non-locality" one typically employs considerations which treat past light-cones and future light-cones quite differently. This remark applies also to the end of the penultimate paragraph of section 1.
7. It would be helpful to add section numbers to the section headings.
8. Similarly, add a reference to section 8 (and 10.6) when mentioning non-contextuality (section 1, 3rd para, italicized in parentheses).
9. Ref. 8 could (and I would say: should) be shortened to "Ref. 5, p. 11." This type of remark applies to many of Bell's quotations in the list of references.
10. Section 2 discusses the EPR argument. That paper famously introduced "elements of physical reality," in a manner which might be acceptable only to "realists." It is therefore relevant to emphasize that the part of the logical argument which uses this concept is not the part which is necessary for a discussion of Bell's theorem. In other words, it suffices for Bell's theorem that the "pre-existing values" exist in the mathematical description used by whomever is making the predictions for the experiments, and it is not necessary that they exist in "physical reality."
11. Overall, the first 4 sections are excellent. My only further suggestion is to add a reference to another excellent general exposition of the subject "Is the Moon There When Nobody Looks? Reality and the Quantum Theory," Physics Today 38(4), 38-47, April 1985, and possibly additional references.
12. Section 5 formulates the "locality condition" mathematically, using Bell's lambda variable. However, this variable is introduced as the "set of data ... made available to both systems, say, by a common source." Here one might argue that some type of realism is being taken for granted by the authors. In contrast, Bell introduced lambda from the outset as the "parameters" providing a "more complete specification of the state" (Ref. 6). The point is that Bell discussed mathematical descriptions of physical systems, rather than the physical systems directly, and used the variable lambda as a general notation for the relevant set of variables which is available within any particular mathematical description (as a concrete example, he used lambda for the position variables of Bohmian mechanics, in his "Introduction to the hidden-variable question" of 1971). When defined in this manner, it is clear that no assumption of realism is involved (beyond the assumption that mathematical descriptions may be of relevant). Furthermore, the discussion of mathematical descriptions can (and in my opinion, should) be used to clarify with mathematical accuracy (rigorously) the issues of locality, causality and free variables, possibly along the following lines:
A detailed mathematical description of a physical system consists of (a) a list of variables, each associated with a set of values it can take (describing various properties of the system), (b) an identification of which of these variables are free variables (input), and which are output variables, and (c) a set of mathematical rules, which may be probabilistic or deterministic, according to which each of the dependent variables is evaluated. The rules must be ordered in a manner which allows their execution, e.g., they must be free of causal loops. For such a description to be locally causal, or "Bell local," some of its variables must be associated with particular points (or regions) in space-time -- these are the "local beables." The list of local beables must include physically local quantities such as device settings or experimental results [the description may also involve non-local variables such as P(A_1,A_2), the joint distribution function of Eq. 3, but the rules determining the values of the local beables may not depend on the non-local variables]. The description is "local" if each one of the rules involved in specifying the values of the local beables depends only on the values of other local beables which are time-like (or light-like) separated from it (I refer here to two variables as time-like separated if the space-time points they are associated with are time-like separated). Similarly, the description is "causal" (in some reference frame) if the rule for each local beable includes a dependence only on local beables from its past. As the conjunction of these two, a description is locally causal if each of its rules allows only for dependencies of beables on other beables which belong to their past light cone.
It may be necessary to add that the "number" of variables in the "list" may well be infinite. For example, in the present context the description of the motion of a point particle as r(t) involves a different local beable for each instant t, rather than a single time-dependent variable (here the input would typically be the initial position and velocity, and the rules would include Newton's equation). As examples of non-local classical descriptions, one could include not only the scalar potential in the Coulomb gauge but also the phase-space density of the Liouville equation (I think that these two correspond to the two possibilities mentioned in a much-quoted paragraph of Einstein, which have sometimes been misinterpreted as representing "locality" and "realism"). Thus, if a description is non-local, but equivalent results can be obtained from a local description (based on Maxwell's or Newton's equations in the above examples), one would not say that the physical theory is non-local. In contrast, when a theory (or an experiment) gives results which can not be reproduced by *any* locally causal mathematical description that it would be called "Bell non-local" (one must use such a name to preserve the distinction from, e.g., "signal non-local"). I think that with such definitions one can state as a mathematically rigorous theorem that "the predictions of quantum mechanics are Bell non-local" (because the "no conspiracy" condition, in the sense that the use of free variables for the device settings is appropriate, is built into quantum mechanics), and also that "a hypothetical loophole-free experiment would falsify local causality in a theory-independent manner, and in particular, independent of quantum mechanics" (because the choice of free variables is built into the conception of the experiment and into the report of its results, and hence accepting something like superdeterminism is tantamount to rejecting the experimental data, or perhaps even the very concept of experimentation).
Several of the remarks below point out where, in my opinion, the level of rigor of the presentation may be improved by adopting such lines (specifically, in relation to section 6).
13. The first sentence following Eq. 4 appears to describe causality rather than "free parameters." Indeed, it is required directly by causality that lambda (or its distribution) can not depend on alpha_1 and alpha_2. It is the other way around -- alpha_1 and alpha_2 do not depend on lambda -- that is prohibited by the requirement that alpha_1 and alpha_2 are free variables (again, if one thinks about lambda and the alphas as elements of reality, then issues related to their interdependence may be somewhat mysterious, but if one thinks of them as variables in a mathematical description then everything is very simple and clear). Of course, probability theory allows one to interchange between dependent and independent variables, and so the "superdeterministic" claim that P(lambda) could depend on the alphas if the latter were not free variables does make sense. But if the way you deal with this is to simply state that we assume that the alphas are indeed free variables, then I do not see the point in using probability theory in this manner (even if one rejects the concept of "a locally causal mathematical description" discussed above).
14. Ref. 18 derived the CHSH inequality for deterministic hidden variables. Bell pointed out that the result would hold for stochastic variables as well in footnote 10 of the above-mentioned 1971 work, "Introduction to the hidden-variable question." Clauser and Horne gave a full discussion (however, I think they did not mention Bell's footnote) in "Experimental consequences of objective local theories," Phys. Rev. D, vol. 10, p. 526 (1974). It would be appropriate to mention these additional references.
15. Section 6 is entitled "Bell's definition of locality," and merely notes parenthetically that "Bell sometimes also used the term "local causality." In contrast, it seems to me that Bell quite consistently used the term "local causality," beginning in 1975 in the titles of sections 2 and 3 of his "The theory of local beables" (Ref. 15). Furthermore, in his very first article (RMP 1966, Ref. 5) the first paragraph of the section on "locality and separability" brings "The ideas of space, time, and causality..." into the discussion. Bell used the term "locality" alone only when causality had already been implicitly introduced into the discussion, e.g., by appealing to the EPR argument or by describing Bohmian mechanics. Therefore, I believe that Bell's usage of "locality" was, in essence, always an abbreviation of "local causality," and that this should be clarified in the text.
16. Section 6 discusses the definition of local theories in terms of conditional probabilities, including theories for which Bell's definition is not directly applicable. It seems to me that this is not very effective, and that it could be preferable to simply define a theory to be "non-local" (or perhaps "necessarily non-local") if the predictions it makes can not be made by any local theory for which Bell's definition does hold. Furthermore, I would think it helpful to simply label those theories for which Bell's definition does not hold as "non-local mathematical descriptions."
17. I disagree with the authors when they say that "it does not seem possible to write down a clean mathematical definition of 'non-conspiratorial' theory..." I would think that the mathematically clean concept of "free variables," is applicable to the alpha_1 and alpha_2, and that this captures precisely what Bell meant. Taken together with causality, and with the fact that lambda represents local beables associated with a time earlier than the time associated with the alphas, this implies that lambda is statistically independent of the alphas.
I think that here too one could profit from a discussion of the mathematical description of the theory, rather than the goings on in physical reality. Consider for example a situation where pseudo-random number generators are used to make the random choices of alpha_1 and alpha_2. According to Bell, the variables alpha_1 and alpha_2 in the mathematical description will still play the role of free variables (unless one is discussing superdeterministic theories of the type he did not expect). Therefore, the seeds of the random number generators, despite their being present in the physical reality of region 3, are not to be included in the lambda which is used in the mathematical theory. As an example of this, consider standard QM, and the fact that the seeds of the generators are certainly not included in the description of the quantum state of the system.
18. Section 7 on experiments is again brief and excellent, although I would have expected a discussion of the detection loophole, rather than just a mention, and perhaps a reference to a more detailed discussion of the experimental status.
19. The last sentence of section 7 includes parenthetically "some hypothetical preferred frame," which I find confusing, as one is apparently referring to the lab reference frame.
20. Section 8 discusses non-contextual hidden variables. It seems to me appropriate to mention that one may describe Bell's achievement in his first paper (the 1966 RMP, Ref. 5) as identifying that locality can be used as a strong argument for non-contextuality in some cases, namely in those cases where the commuting variables refer to experiments performed at different locations. His second paper, Ref. 6, is then an identification of a specific prediction of QM which is incompatible with non-contextual hidden variables which belong to this "protected" category (this remark has some overlap with the first paragraph of section 10.6, but it is appropriate not to make it only there, i.e., not to mention it only in the section discussing misunderstandings).
21. The quote of footnote 65 in section 10.1 is already provided in the last sentence of section 4 (footnote 10). It would thus be appropriate to add "as mentioned above" here.
22. The titles of sections 10.2 and 10.3 are somewhat vague. There is no problem in using "local realism" as a description of Einstein's point of view (which was proved untenable by Bell). The problem is with the implication that one must give up on either locality *or* realism. I can suggest 'Discussing "locality" vs. "realism"' as a title for section 10.2. Section 10.3 actually discusses two different topics, with the first two paragraphs discussing EPR's realism. This could be included in 10.2 instead. The title of 10.3 could then refer to "counterfactual determinism."
23. Footnote 71 introduces two very important and relevant concepts - signal locality and whether a theory describes a physical system or the information one has regarding the system. As already mentioned, it would be appropriate to discuss these concepts (also) in the main text, in the sections presenting the theorem itself rather than the controversies.
24. Near the end of section 10.3, again, the "no conspiracy" condition is used in a manner for which it would be more apropriate to call it the "causal arrow of time" condition.
25. In the context of the first paragraph of section 10.5, I would mention that local commutativity is a time-reversal symmetric notion (see, e.g., the first equation in section 6.5 of "La Nouvelle Cuisine"), whereas the concept of a retarded Green's function (see the third equation there) involves a theta-function which breaks the symmetry.
26. The third paragraph of section 10.5 discusses "controllability" and "observability" as fuzzy anthropocentric concepts, but, at least for mathematical descriptions, these can be given a sharp mathematical meaning as "input" and "output" variables (this occurs again in the last paragraph of the section). In fact, the mathematical analyses of the information-carrying capacities of various "communication channels," with or without quantum entanglement, is a topic of much activity in current research. It would seem that the discussion here could benefit from inclusion of a definition of the no-signaling condition (presumably, a lambda-averaged version of parameter independence).
27. A misprint: the word 'the' appears before "the CHSH-Bell inequality" in footnote 88.
28. As pointed out also by reviewer A, the last few sections do not seem to achieve much. Section 10.7 makes some rather vague statements. Instead, what one should say about the many worlds interpretation is that it violates Bell-locality from the outset, because it follows the evolution of the wavefunction. The wavefunction is a non-local variable, and it is simply not allowed as a central element of a "local" description.
29. Section 10.7 begins with a presentation of a further assumption, which may be called "factual definiteness" (in contrast with "counterfactual definiteness"). This assumption is again formulated in terms of the physical reality being described. However, in the context of Bell's theorem, strictly speaking, one should be discussing mathematical descriptions rather than physical reality. Thus, it is sufficient to assume that the description includes rules for generating values of the outcomes, rather than making vague statements about elements of physical reality.
30. Regarding section 10.8, it is not clear to me that consistent histories can be considered a causal description at all. It seems to me that it takes the "block universe" view so seriously, that you must tell the theory which experiments you are going to perform "in advance," because that affects the families of histories which can be regarded as decoherent. In any case, it clearly does not provide a "Bell-local" description, in the sense that it does not provide an expression for the correlation function of the type given in equations 3 and 4.
More generally, whenever anybody claims that his interpretation of quantum mechanics is local, in the face of Bell's theorem, he must be using the word "local" in a different sense (e.g., signal locality). It should be sufficient, for the present purposes, to treat this as the term "realism" was treated, i.e., by quoting some examples, and pointing out that the term "local" used in those references has not been defined mathematically, or has been defined differently (like Jarrett did for parameter independence). It should then simply be pointed out that introducing a new meaning for the word "local," different from the one used by Bell (and Einstein), is inappropriate and misleading, especially if it is done within discussions of Bell's theorem itself.
31. The ending section 11, on the relationship with relativity, is, as noted above, also without a sharp conclusion. It does describe fairly well Bell's long-held attitude, but it should be remembered that he retreated from this position in his 1987 publication "Are there quantum jumps?" which ends with: "It takes away the ground of my fear that any exact formulation of quantum mechanics must conflict with fundamental Lorentz invariance."
32. What to do then with the closing sections? Regarding section 10, I would shorten sections 10.7 and 10.8 to mere paragraphs, and include them in section 10.5 about controversies regarding the locality condition. After all, it is clear that the proponents of these views are not using the word "local" in the same sense that Bell used it. One has to make a clean separation between the mathematical discussion, within which Bell's theorem is just that: a proven theorem, and the question "which notion of locality is the most relevant to physical reality?" Regarding the latter, of course there is much ambiguity. In fact, the experiments refute Einstein's and Bell's initial expectations regarding locality and natural phenomena! Which, if any, of the other notions of locality which have been brought up by various authors is appropriate for the description of natural phenomena (physical reality) is thus an open question, which could well be reiterated in the closing section 11.
33. I would consider adding a subsection to section 10 with a title such as "Omitting mention of the causal arrow of time." The point is that it is often argued that Bell's theorem shows that quantum mechanics in some sense "contradicts" the theory of relativity, whereas both the theory of relativity and the measurement problem of quantum mechanics could encourage us to adopt a time-reversal symmetric point of view (see, e.g., recent discussions of time-reversal symmetric quantum mechanics in Physics Today), and it is only by assuming that this symmetry is broken (the causal arrow of time) that a proof of Bell's theorem can be carried through. Despite the fact that Bell was careful enough in talking about "local causality" rather than "locality," I think that he is to be identified as a proponent of this "misunderstanding," as he discussed Lorentz invariance as if were impossible to include "improper" Lorentz transformations within the theory of relativity.
In the context of such a subsection, I would think it appropriate to quote also my recent article: N. Argaman, "Bell’s theorem and the causal arrow of time," Am. J. Phys. Vol. 78, p. 1007 (2010), as this article presents a toy-model which demonstrates how Bell inequalities may be violated by a mathematical description which is local (in the time-reversal invariant sense associated with spacelike separation) but not causal. Such simplistic toy-models are ideal for resolving controversies of the type discussed in section 10. For example, if counterfactual definiteness were indeed an essential assumption of Bell's theorem, then it would have been easy to construct a relevant "counterfactually indefinite but local" toy-model. I would like to add that there are many other relevant references on the arrow-of-time issue which should be quoted (Costa de Beauregard, Feynman, Price), but in my view this shoud be done already in the first few sections.
34. In accordance with the above, it seems to me that it would be appropriate to shorten (dramatically) the current section 11 on "nonlocality and relativity," and include it within the subsection of section 10 which would clarify the interplay between non-locality, relativity and the arrow of time.
35. I would suggest that the article end instead with a much stronger conclusion, reiterating the main points (including what is included in the mathematical theorem and which issues are still open), and providing an indication of some of the more recent developments which are of closely related interest. I refer here to the various aspects of "quantum entanglement," several of which continue to be actively studied to this day. For example, one may mention the "proof" of non-locality associated with "quantum teleportation," and its rebuttal (see, e.g., arXiv:0806.1679), and/or the issues of Tsirelson's bound, PR boxes, and the recent discussions of "information locality." Further examples include discussions of various quantum information channels, including entanglement-assisted channels.
I have provided above a long and detailed list of remarks. At times, I have been rather harsh, and have surely been repetitive. I hope that the authors still find my remarks constructive, rather than a burden. I look forward to reading the authors' response to this list, and subsequently, to accepting the article.