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information-theoretic postulates." This is not true, and not supported by the cited references Barnum et al. (2000) and Wilce (2017) (I couldn't access
Cassinelli and Lahti (2017)). Also the claim "Because Gleason's theorem yields the set of all quantum states, pure and mixed, it can be taken as an argument that pure and mixed states should be treated on the same conceptual footing, rather than viewing pure states as more fundamental conceptions" is strange. Gleason's theorem has hardly anything to do with this discussion, and the reference cited to support this merely notes that "Gleasonâs theorem might be interpreted as telling us exactly why it is the most general such state".
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considered to be too strong should be stated in the article. Also, it should be explained why
Gleason's assumptions are better than von Neumann's (it is because von Neumann assumed linearity of expectation values for non-commuting observables, which is an unmotivated assumption of a technical flavour, whereas Gleason assumed non-contextuality, which is both well-motivated and physically meaningful).
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directly about them are not physically meaningful. Instead, one needs to phrase assumptions in terms of projective measurements. That's why
Gleason's theorem is much more important. Now, the problem is that this is my personal opinion, and I'm not aware of a source that says the same thing, so I don't know how to include this information in the article.
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first case the article emphasises that they are non-contextual, and in the second case it doesn't mention it (the fact that
Gleason allowed the weight to be different than one is a mathematical detail of no consequence, since you can always renormalize such frame function to have weight 1. I wouldn't even mention that on the article.).
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it was
Gleason's and is still used in various places. "Bivalent probability measure" was, I think, the jargon in the article before I came along, and probably copied over from whatever source was originally used c. 2006. It can probably be changed to something more illuminating. The bit about "some versions of
1117:. This sentence here is a mess. What are quantum events? With which description of measurement outcomes? I'm not an expert in quantum logic, but as far as I know it just treats projectors (on measurement outcomes) as logical propositions. The mess becomes worse in the following paragraph: the eigenvalues
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Looking at that passage in
Wallace's chapter, I think it's more about what he calls "inferential conceptions" of physical theories ("on the inferential conception there is even less reason to deny that a mixed state is a legitimate state of a system", etc.). Since he only touches on Gleason's theorem
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Thanks for your comments! They look to be very useful, and I will work to address them. The two different statements in the "Overview" section (which I agree ought to be retitled somehow) came about because when I first found the article, it had the quantum-logic version of the statement, written in
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hidden variables is a good one â the body of the article goes into detail on this, but perhaps the qualification should be worked into the introduction as well. "Frame function" can probably be replaced with "probability measure" here and there, though we should I think mention the terminology since
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I like that idea. I guess I had been thinking of the KochenâSpecker qubit model (and the BellâMermin one from section D.2) as demonstrations that hidden variables work for reproducing the quantum statistics for orthonormal measurements in dimension 2, whereas what
Kadison seems to have been talking
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I'm glad you appreciated my comments. I don't think it makes sense to call Bub and
Pitowsky quantum Bayesians. They support a Bayesian (subjectivist) approach to quantum logic, which confusingly enough is not the same as Fuchs' quantum Bayesianism. Furthermore, Pitowsky died in 2010, the year when
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I'm not sure historical accuracy is such a good idea, as the primary source is often rather obscure. But if you want to describe what
Gleason said, I think you can use the same definition of frame function in both parts and add a remark in Gleason's part to the effect that he actually allowed the
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I think it is fine to use the name "frame function", but you should define where it first appears. Currently the article introduces them without using the name, and afterwards introduces them again and gives the name. The reader doesn't know that they are the same thing, especially because in the
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Another way of phrasing the theorem uses the terminology of quantum logic, which makes heavy use of lattice theory. Quantum logic treats quantum events (or measurement outcomes) as logical propositions and studies the relationships and structures formed by these events, with specific emphasis on
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I would suggest using the Kochen-Specker model instead (you can find it e.g. in section D.3 of arXiv:0706.2661). It is much more relevant to physics, as although it's explicitly not Born-like it can be used to reproduce the Born rule by selecting the appropriate probability distribution over the
974:
Thanks again! After so long a time when I was the only one who seemed to be taking an interest in the page, getting comments with this level of detail is really quite refreshing. I just rewrote the sentence that you rightfully pointed out as being vague. Personally, I like the mix of history and
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A general question: is it correct that
Gleason's theorem is a theorem which can be stated completely in terms of mathematical notions such as, say, Hilbert space etc. and that (quantum) physics uses an interpretation of these mathematical notions? If so (which I suspect), it might be worthwhile
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I'm an expert on the subject. I'll add some comments here in the hope that they will be useful. If you so wish I can also provide a more in-depth review. First of all, the article claims that Gleason's theorem rules out local hidden variables. This is not true. It rules out noncontextual hidden
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I'm a bit worried about the first paragraph of the Generalizations section. It talks about Busch's theorem as if it is the best thing since sliced bread, but this is not really the case. The theorem didn't have much impact, because (I assume) POVMs are not fundamental, so assumptions that talk
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From the single premise that the âexperimental propositionsâ associated with a physical system are encoded by projections in the way indicated above, one can reconstruct the rest of the formal apparatus of quantum mechanics. The first step, of course, is Gleason's theorem, which tells us that
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In many mathematical statements are of the form "all examples of an (apparently more flexible) notion arise by applying some construction to a simpler (seemingly more restrictive) notion". Is this the case here, too? I.e., I guess it is semi-obvious (??) that functions of the form <x, Wx:
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for a proper take on the relationship between nonlocality and contextuality. What is taken to rule out local hidden-variables is Bell's theorem, first because it deals directly with locality, and secondly because its assumptions are much weaker. The fact that Gleason's assumptions are widely
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It's true that in this 2001 paper you link by Caves, Fuchs, and Schack (who are QBists without any doubt), they do use Gleason's theorem to get Born's rule, but that's not the approach they favour anymore. After they invented SIC-POVMs they decided to use them to postulate the Born rule as
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quantum Bayesianism was named and defined. Ironically enough, the paper you cite from Bub explains the difference between his and Fuchs' approach. Now, the quantum logic people indeed use Gleason's theorem as a foundational result, which is probably what you should mention in the article.
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The referencing was like that when I found the article, and at the time, the path of least resistance was to add to it rather than to reformat it all. (I probably didn't think that I'd be adding too many references when I started fixing the page up, and I might not have known about the
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That's great. I also edited the quantum logic section, I don't see any remaining issues with it. The formatting of the references is a bit bizarre, though. You have to look at the author-year, and the look for the actual reference in the list below. Is there any reason to be like this?
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a way that I found hard to follow. I tried to clean it up and also include a statement phrased in the language I was more familiar with. It's entirely possible that the repetition is more confusing than I had hoped. Anyway, thanks again, and I look forward to your feedback on the rest.
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Furthermore, the "Implications" section contains some dubious claims. It says, for example, that "Alternatively, such approaches as relational quantum mechanics and some versions of quantum Bayesianism employ Gleason's theorem as an essential step in deriving the quantum formalism from
1547:, would you be willing to finish the reviewing process? You state above that you have read the article word by word, so I am very confident that you will be able to give a meaningful review without much additional effort from your side. I would be much relieved, thank you!
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Yes, the "some varieties of quantum Bayesianism" phrasing made sense in my brain, but that's no guarantee it would make sense anywhere else! :-) I got curious and dug through the history for where the "relational quantum mechanics" part came in. Turns out that it was added
559:"given that each quantum system is associated with a Hilbert space" -- the ignorant reader (like me) had to guess (?) that a quantum system is just a physics-lingo for a Hilbert space. Is that correct? If yes, such a statement belongs further up in the section, I think.
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I could imagine that the proof outline could be fleshed out a bit more. I (as a layman in this area) do get a reasonable overview, but I figure it makes sense to many readers of this article to get more information. For example, how does the reduction step from d:
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Fair points. In that paragraph, I was trying to stick with Gleason's terminology, as part of summarizing his original argument. I still think that's a reasonable thing to do at that point in the article, but I will try to tie it better with the previous section.
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proof outline, since the particular method that Gleason used is "part of history", and other people replaced parts of his argument in the following years. But I can also see merit in dividing them up more cleanly. Regarding Kadison's counterexample,
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First off: I am by no means an expert on this topic; I am a mathematician who knows what a Hilbert space and a probability measure is, but otherwise the topics in this article are new to me. This will also no doubt be reflected in my comments below.
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Trassinelli's reference is a weird one. He is advocating a marriage between quantum logic and RQM. Which is fine, but it is not how RQM is usually understood, and Gleason's theorem again plays its role in the quantum logic part of the argument.
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Indeed. I had interpreted a lack of a "done" marking as the issue not being addressed, but this is not case, the unmarked issues have also been addressed. I would like to hear your thoughts about the points 2, 3, and 5 that I raised, though.
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In dimension two a frame function can be defined arbitrarily on a closed quadrant of the unit circle in the real case, and similarly in the complex case. In higher dimensions the orthonormal sets are intertwined and there is more to be
612:"For simplicity, we can assume" -- maybe rephrase as "For the purposes of this overview, the Hilbert space is assumed to be finite-dimensional in the sequel."Â ? Also, it would seem to make sense how this assumption is a simplification?
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Yeah, that's a funny conceptual twist. One can anyway do both: use equation (15) with some fixed λ as the counterexample, and remark that it can be used to build a nice hidden-variable model for a qubit, the Kochen-Specker model.
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is explicitly about reconstructing quantum theory by way of SolĂšr and Gleason. I think there's a problem of labels â e.g., where's the line between quantum information and quantum logic? â so I'll try revising that passage of the
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I think the second paragraph in the subsection "Conceptual background", about pure states and mixed states, is going into too much detail about something that's not relevant for Gleason's theorem. I would remove it
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The lattice of subspaces is just the set of subspaces endowed with the containment relation, right? If so, it might be an idea to say "Each event is a subspace of H" instead of this somewhat artificial "atom of the
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had. (At least, the problematic text isn't there any longer.) The remaining question that I see is how much more detail to include in the "Outline" section, like perhaps saying more about reducing the problem to
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Also, I find the article unnecessarily heavy in jargon. Why use "frame function", instead of simply "probability function"? Why use "bivalent probability measures", instead of simply "deterministic probability
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where all measurement outcomes already. Is it a union of measurement outcomes? But isn't the formula trivial in this case? Now comes the statement of Gleason's theorem, and it finally makes it clear that the
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The section title "Overview" is not very descriptive for the contents of the section. What about adding some subsections, and retitling it to something like "Statement (and interpretations) of the theorem"?
1175:. Then it is stated that events are atoms of the lattice. Now the function P is defined as summing to one when applied to "orthogonal atoms". I don't know that it means for atoms to be orthogonal. The page
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Is it possible and sensible to describe Kadison's counterexample in d=2? This could even illustrate some of the concepts earlier in the article, thus highlighting a bit further how the theorem is notable.
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The mapping u â âšÏu,uâ© is continuous on the unit sphere of the Hilbert space for any density operator Ï. Since this unit sphere is connected, no continuous probability measure on it can be deterministic.
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about is the possibility of statistics that don't look at all like the Born rule but are still consistent with the frame-function assumptions (because in dimension 2, there's no "intertwining").
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I am sorry -- I have been incredibly slow in responding to the recent edits here, and it looks like I will not have the time needed to continue a thorough review in a reasonable time frame.
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is in place, the remaining statistical and dynamical apparatus of quantum mechanics is essentially fixed. In this sense, then, quantum mechanicsâor, at any rate, its mathematical frameworkâ
1490:, Gleason just did the obvious thing, he showed that if a frame function in a higher dimension is regular when restricted to every three-dimensional subspace, then it is just regular.
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I am not convinced that mixing historical comments with the proof outline fits so well. You could consider putting the proof to the first section containing the statement.
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From this and the above review, it sounds like the "quantum logic" subsection is in bad shape. Looking into it, that stretch of text was apparently based closely on
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I don't think this is true, none of the quantum information derivations I know (e.g. arXiv:quant-ph/0101012, arXiv:0911.0695, arXiv:1011.6451) use Gleason's theorem.
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Gleason's theorem is of particular importance ... for the effort in quantum information theory to re-derive quantum mechanics from information-theoretic principles.
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1509:'s review, it seems to me that the bullet points not yet marked "done" have mostly been addressed by the trimming and refactoring of the quantum-logic material.
636:"Equivalently, we can say that ..." -- is that really equivalent? After all you need to choose a basis to begin with, no? Also "we can say that" is redundant.
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1279:; as a result, it doesn't really have an encyclopedic tone. I think it could be streamlined and clarified. I'll make that the first thing on my list to fix.
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The reconstructions you mention by Wilce, Cassinelli, and Lathi are explicitly quantum logic reconstructions, that indeed do use Gleason's theorem.
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Is such a complicated argument really needed? I would just mention that the probability rule must be the Born rule, and that's not deterministic.
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I don't see the point in the assertion that the structure of a quantum state space (again ?? = Hilbert space) follows from Gleason's theorem.
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584:"particular mathematical entities" -- do you mean probability measures here? The vagueness of the phrasing does not seem to help (me) here.
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I've shortened and rearranged the quantum-logic material, and I added more details about how to construct a counterexample in dimension 2.
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It seems that the Hilbert spaces in the article are either complex or real, correct? If so, Hermitian should probably be rephrased somehow?
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I'm not the actual reviewer, but I went through the article word-by-word, and I hope the resulting comments will be useful to improve it.
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I've always heard "quantum Bayesianism" defined more broadly than "QBism"; the former includes Bub and Pitowsky, while the latter is what
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OK, I've retitled, reorganized and in parts rephrased the "Overview" section. Hopefully this takes it in a good direction at least.
556:(MOS:WE) I think it is advisable not to address the reader directly. This happens in many spots, e.g. "Consider a quantum system..."
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Further down: is the section with "Let H denote the Hilbert space..." related to the quantum-logic interpretation of the theorem?
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that mentions Kadison doesn't go into detail about what his counterexample was. However, other papers give counterexamples for
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It's not really possible to illustrate the theorem itself with an imagine, so I think the ones it has are as good as it gets
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doesn't explain it either. Perhaps you mean orthogonal projectors? Then the probability of obtaining measurement outcome
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Ah, so that's what you had in mind! In my head they were synonyms, but I'm glad you agree that they are easy to confuse.
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hidden variables. It was all the rage in the past few years as people were studying the limits of psi-epistemic models.
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Good idea. I will try doing so after the next round of revisions I make, since I might be able to check off a few more.
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Can you please very briefly indicate which comments of the above have been addressed? This would simplify the exchange.
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500:. I probably should have cut it out or changed it to talk about quantum logic instead, but it slipped through the net.
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The second paragraph of the lead feels a bit weird to me. Should we really try to teach quantum mechanics in the lead?
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In the indented statement of G's theorem all jargon that is not crucial should be eliminated: can the mention of
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I'm afraid putting more detail would confuse instead of enlighten. As for the reduction of the problem to
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The massive amount of work in the review was mostly dedicated to address this, now I think it is perfect.
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correspond to density operators. The point to bear in mind is that, once the quantum-logical skeleton
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are introduced, but play no role. Then a (quantum?) event is introduced as the measurement outcome
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That's much like what I was considering doing. I'll go off and try to find a decent phrasing now.
1446:. I think the current level of detail is OK, but perhaps more would be better? Opinions welcome!
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function to sum to any non-negative real number, but that one can assume wlog that it sums to 1.
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for the nomination -- I will be doing a review as soon as I can; other reviewers are welcome!
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It looks like redoing the quantum-logic material has addressed several of the comments that
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separating a bit more the mathematical basis of the story and the physical interpretation.
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Why does the statement of G's theorem appear twice, however in a somewhat different form?
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I've removed "bivalent", defined "frame function" and made assorted other modifications.
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parenthetically, I think it's better to take that line out rather than rewrite it.
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Well, you, I was planning to do it now but there is nothing left to do. Thanks.
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How do the x_i generate a sublattice? Just the intersections of these subspaces?
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should be repeated here, to make the statement as self-contained as possible.
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Thanks! Looks like we've gotten it done (and yes, it was long overdue).
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template for providing page or section information after a footnote.)
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remove the paragraphs I found problematic, and finish the GA review.
661:"A density operator is a positive-semidefinite operator" -- on what?
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doesn't mention him specifically; he just makes the observation,
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Is "quantum-mechanical observable" a synonym for "measurement"?
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is described as the sum of the probability of the atoms under
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The lead should still be improved, but it's good enough for
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OK, I will review the remaining sections as soon as I can.
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In the Quantum Logic subsection, the article states that
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Is a "proposition" the same as speciyfing a subspace of
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Sure, I can do it, it's pretty much finished anyway.
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In the Implications section, the article states that
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are unit vectors representing logical propositions.
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quantum logic and its attendant probability theory.
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222:" is true â Bub and Pitowsky make much of it
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188:KochenâSpecker theorem
1864:It is illustrated by
1816:neutral point of view
1772:broad in its coverage
1485:
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1241:
1239:{\displaystyle x_{i}}
1213:
1211:{\displaystyle x_{i}}
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1168:{\displaystyle x_{i}}
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992:
297:{\displaystyle L(H)}
279:
268:{\displaystyle L(H)}
250:
1877:fair use rationales
1177:Atom (order theory)
1115:quantum measurement
1007:{\displaystyle d=2}
986:article by Chernoff
220:Quantum Bayesianism
1848:No edit wars, etc.
1690:factually accurate
1480:
1436:
1236:
1208:
1165:
1134:
1004:
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1895:suitable captions
1705:reference section
1144:of an observable
1087:Pseudo-review by
977:Gleason's article
498:in September 2006
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43:Copyvio detector
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1891:appropriate use
1813:It follows the
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1577:Jakob.scholbach
1549:Jakob.scholbach
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956:3 to d=3 work?
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842:Jakob.scholbach
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755:was defined on
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53:External links
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1781:major aspects
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1601:I decided to
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1408:Section break
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1600:
1597:Final review
1504:
1411:
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1184:
1180:
1145:
1093:
930:
839:
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241:Wilce (2017)
214:
164:
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143:
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135:
131:
117:Article talk
116:
112:
93:
90:
81:Instructions
1661:word choice
934:Rephrased.
104:visual edit
1751:plagiarism
1695:verifiable
1511:XOR'easter
1448:XOR'easter
1363:XOR'easter
1335:XOR'easter
1295:XOR'easter
1281:XOR'easter
1050:XOR'easter
1020:XOR'easter
936:XOR'easter
900:XOR'easter
872:XOR'easter
858:XOR'easter
827:XOR'easter
799:XOR'easter
725:XOR'easter
697:XOR'easter
673:XOR'easter
649:XOR'easter
624:XOR'easter
597:XOR'easter
572:XOR'easter
526:Review by
502:XOR'easter
469:XOR'easter
393:XOR'easter
364:XOR'easter
334:XOR'easter
320:XOR'easter
306:reduces to
168:XOR'easter
48:Authorship
34:GA toolbox
1921:Pass/Fail
1603:WP:BOLDly
1109:entirely.
738:lattice"?
195:measure"?
144:Reviewer:
71:Templates
62:Reviewing
27:GA Review
543:Overview
314:article.
166:Thanks,
157:contribs
76:Criteria
1933:WP:GOOD
1915:Overall
1795:focused
1747:copyvio
1665:fiction
918:History
232:. The "
127:history
108:history
94:Article
1867:images
1842:stable
1840:It is
1818:policy
1770:It is
1688:It is
1667:, and
1657:layout
1628:It is
1613:review
1563:Tercer
1545:Tercer
1527:Tercer
1492:Tercer
1377:Tercer
1349:Tercer
1310:Tercer
1259:Tercer
1089:Tercer
1065:Tercer
1035:Tercer
554:WP:MOS
483:Tercer
447:Tercer
407:Tercer
379:Tercer
349:Tercer
201:Tercer
1893:with
1669:lists
1615:(see
982:said.
136:Watch
16:<
1749:and
1692:and
1653:lead
1651:for
1621:here
1617:here
1581:talk
1567:talk
1553:talk
1531:talk
1515:talk
1496:talk
1452:talk
1381:talk
1367:talk
1353:talk
1339:talk
1314:talk
1299:talk
1285:talk
1263:talk
1069:talk
1054:talk
1039:talk
1024:talk
984:The
962:talk
940:talk
931:Done
904:talk
890:talk
876:talk
862:talk
846:talk
831:talk
823:Done
803:talk
795:Done
729:talk
720:Done
701:talk
693:Done
677:talk
669:Done
653:talk
644:Done
628:talk
620:Done
601:talk
592:Done
576:talk
567:Done
552:Per
532:talk
506:talk
487:talk
473:talk
451:talk
411:talk
397:talk
383:talk
368:talk
353:talk
338:talk
324:talk
205:talk
176:talk
151:talk
123:edit
100:edit
1649:MoS
955:-->
782:-->
1923::
1917::
1899::
1887:b
1881::
1873:a
1850::
1844:.
1826::
1820:.
1799::
1791:b
1785::
1777:a
1774:.
1755::
1743:d
1737::
1733:OR
1729:c
1723::
1715:b
1709::
1701:a
1698:.
1673::
1663:,
1659:,
1655:,
1645:b
1639::
1635:a
1632:.
1611:GA
1583:)
1569:)
1555:)
1533:)
1517:)
1498:)
1454:)
1383:)
1369:)
1355:)
1341:)
1331:}}
1328:rp
1325:{{
1316:)
1301:)
1287:)
1265:)
1126:α
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964:)
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906:)
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1647:(
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1565:(
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1476:3
1471:R
1450:(
1432:3
1427:R
1379:(
1365:(
1351:(
1337:(
1312:(
1297:(
1283:(
1261:(
1232:i
1228:x
1204:i
1200:x
1189:y
1185:y
1181:y
1161:i
1157:x
1146:A
1130:i
1067:(
1052:(
1037:(
1022:(
1002:2
999:=
996:d
960:(
938:(
902:(
888:(
874:(
860:(
844:(
829:(
801:(
785:H
776:H
772:x
768:L
761:H
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753:P
743:H
727:(
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530:(
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289:H
286:(
283:L
263:)
260:H
257:(
254:L
203:(
174:(
154:·
149:(
132:·
129:)
121:(
113:·
110:)
98:(
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