Talk:Gleason's theorem/Archives/2021

Early and orphan comments

The term "Gleason's Theorem is pretty common among philosophers -- I put up a stub to see who would come out of the woodwork -- what I'd like to do is sketch a short proof, and give a decent bibliography. I put a pointer in the Bell article to see who might come this way. I've contributed before, but never started an article.

--Drewarrowood 08:08, 12 September 2006 (UTC)

I second "no deletion" and the need for improvement. The 1957 paper by A.M. Gleason ("Measures on the Closed Subspaces of a Hilbert Space", Journal of Mathematics and Mechanics 6: 885-93) is a classic paper on the foundations of quantum mechanics. It contains the first and (IMHO) still best derivation of the general quantum-mechanical rule for calculating the probabilities of measurement outcomes. (To understand its importance one has to bear in mind that probabilities of measurement outcomes are the only interference between quantum theory and experiments.)

--Ujm 08:27, 10 September 2006 (UTC)

What a foul trick, trying to rope a poor philosopher into doing this article. Well, I am such a person - although I didn't come here from Bell, I was just wondering what Wikipedia's coverage of the subject is like, and was disappointed to see it was merely a statement of the theorem.

I shall expand the article a bit, but I have no interest in sketching the proof...it is hideously complicated in its original form (i.e. Gleason's original paper) and even the elementary versions of it extend over several pages and are not easily summarised. Someone more used to identifying "key moves" in proofs and so forth is welcome to add a "proof section". The constructive proof can be found here, should anyone be interested. Maybe one day I'll do it, but not today.

But since, as Drewarrowood so slickly put it, the theorem is mostly used by philosophers, the focus of the article should probably be more on what the theorem actually says, why it is important, and what it is used for. So, I shall put in some blab about quantum logic, and how the theorem is a key ingredient in the derivation of the quantum formalism from logical structures (and how this works). Then, a brief bit about the philosophical implications. We really do not need to delete this article! It is of seminal importance to a serious field which is already not covered properly here: unfortunately questions of the interpretation of QM tend to be plagued with crankery, New Age flapdoodle, and positional soapboxing for various outlooks (many-worlds vs. Copenhagen, etc.).

Right. Now let me get cracking. Byrgenwulf 14:23, 23 September 2006 (UTC)

Ha! I just looked at who was commenting here...Herr Mohrhoff: you may remember my comment on your Koantum blog about Nietzsche...I never did get around to replying to you, since I have been caught up in the most awful fight here on Wikipedia. Anyway, feedback on my efforts here would be welcome: make sure I don't wander too far off into perspectivist diatribes! Byrgenwulf 14:30, 23 September 2006 (UTC)

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"Gleason's theorem" has 10,400 hits on Google. So the article needs improvement at worst -- but certainly not deletion!!!

Yours truly, Ludvikus 15:10, 5 September 2006 (UTC)

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However, there are more than on Gleason mathematician that have lived. And there does not appear to be a common reference to any "Gleason's theorem", or Gleason Theorem in my search of MacTutor and MathWorld. So the Author herein needs to justify his usage, or I shall be fored to agree with the Wikipedia Editor who recommended Deletion. So far, I'm Neutral on Deletion.

References:

• MacTutor - http://www-history.mcs.st-and.ac.uk/history/Indexes/G.html

• MathWorld - http://mathworld.wolfram.com/

Yours truly, Ludvikus 15:31, 5 September 2006 (UTC)

Great job!

I'd just like to thank the two editors concerned for turning, in five hours, a small stub into an article that I enjoyed reading (and will watch).

I'm more used to this taking a number of days, and intermediate steps, on Wikipedia, but this is a pretty motivating counterexample :)

RandomP 20:39, 23 September 2006 (UTC)

Uniqueness

For a Hilbert space of dimension 3 or greater, the only possible measure of the probability of the state associated with a particular linear subspace a of the Hilbert space will have the form Tr(P(a) W), where Tr is a trace class operator of the matrix product of the projection operator P(a) and the density matrix for the system W.

Is these only one such Tr?

• If so, this should say the trace class operation
• If not, it should say the only possible measures. Septentrionalis 22:35, 23 September 2006 (UTC)

I think I've corrected it now - the trace on a Hilbert space (more precisely, on the endomorphisms of a Hilbert space, as a partially-defined map) is unique, and usually referred to as "the trace" rather than "the trace class operation"; its domain is the set of trace class operators.

I've also replaced "matrix product" by "operator product", though "composition" might be more consistent with modern terminology; however, "density matrix" is traditional, and "matrix product" might be the right choice of terms if we want to keep this in a matrix mechanics-oriented view.

RandomP 23:19, 23 September 2006 (UTC)

Thanks: "trace" is correct, I think. I must confess I didn't check the statement of the theorem, I just left it as I found it...I think what it was trying before is that Tr is a (specific) operator that falls into the "trace class", as opposed to, say, the "inner product class"...

I'm also going to reword the theorem a tad, because it uses P in a different sense to how I used it later on (not having read the statement of the theorem given here, I didn't notice it). Nothing like a night's sleep to highlight all the slip-ups of the day before. Byrgenwulf 10:30, 24 September 2006 (UTC)

I also shifted the position of the W, since it could previously have been read to mean that the system is called W, when it is, in fact, the label for the density matrix. Byrgenwulf 10:36, 24 September 2006 (UTC)

(Mildly) Off-topic: wikitex error

In the Application paragraph, we find the following wiki text:

We let A represent an observable with finitely many potential outcomes: the eigenvalues of the Hermitian operator A, i.e. $\alpha _{1},\alpha _{2},\alpha _{3},...,\alpha _{n}$. An "event", then...

At least with my settings, there's a spurious "-" inserted after $\alpha _{n}$ in the HTML. Does this happen to anyone else?

RandomP 21:13, 24 September 2006 (UTC)

The other Gleason's Theorem

There is another candidate for this title, on the weight enumerators of binary self-dual codes (Pless, Vera (1982). Introduction to the theory of error-correcting codes. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons. p. 116. ISBN 0-471-08684-3.). I presume they are not the same? Richard Pinch (talk) 08:06, 11 July 2008 (UTC)

It is of course the same Gleason, but it is not the same theorem. However, a quick look on Google gives < 700 results for the theorem on weight enumerators and > 45,000 for the theorem on probability measures. Thus while the self-dual codes one seems to be important in its field, and hence worth including, it should probably go on another, disambiguated, page -- point being that if a Bayesian asked a random person about Gleason's theorem, and wasn't met with a blank stare, he would in all likelihood expect his subject to start talking about Hilbert space. Unless information theory people are less likely to put up webpages on their subject than physicists, but that hypothesis is doubtful.--82.24.120.123 (talk) 16:05, 22 July 2008 (UTC)

possible missing qualification in definition of state.

Shouldn't there be some sort of maximality constraint on the x1...xn in clause 2 in the definition of state? Otherwise the sum of probabilities for x1, x2 would bave to be 1 (by clasue 2), and also the sum of probabilities for x3, ..., xn would have to be 1 (by clause 2), giving the sum over x1...xn as 2, (contrary to clause 2).

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HendrikBoom —Preceding unsigned comment added by 69.165.131.134 (talk) 20:02, 25 July 2010 (UTC)

Theorem was not expressed meaningfully

For a Hilbert space of dimension 3 or greater, the only possible measure of the probability of the state associated with a particular linear subspace a of the Hilbert Space will have the form Tr(P(a) W), where Tr is a trace class operator of the matrix product of the projection operator P(a) and the density matrix for the system W.

This is not expressed right.

(1) The system W and the density matrix for the system W are represented by the same symbol W. The matrix W is not defined or quantified over.

(2) The set on which the probability measure is to be defined is not specified. From context, presumably it is the set of "states", which is identified with the lattice of (closed?) linear subspaces, which in turn is identified with the lattice of orthogonal projections (onto closed subspaces).

This should be clarified, since an elementary definition of state is a unit vector in H (called a state vector), or the 1-dimensional linear subspace it spans.

(3) There must be some additional hypothesis or qualification on the probability measure that relates it to the Hilbert space.

Maybe it's compatible with the lattice of subspaces in some sense? For example, monotone with respect to the lattice and sums to 1 with the complement?

Then this is not, strictly speaking, a measure (meaning something defined on a sigma-algebra of subsets of something), but rather a function on the orthocomplemented lattice of closed subspaces of H, that slightly generalizes a measure. I would have to think this through.

(4) Presumably P(a) means the orthogonal projection onto the subspace a, but this should be stated.

(5) As pointed out by other commenters, Tr is not an operator. It is the (unique) trace defined on the set of trace-class operators on the Hilbert space.

P(a) will be a bounded operator, never trace-class unless it has finite-dimensional image. So W has to be trace class to assure that P(a) W is trace-class and the trace is allowable.

Since W is a hanging (unquantified) variable, we have to quantify it. I might guess that we should say "every measure (of a certain type) on the space of states can be represented in the form a → Tr(P(a)W) for some trace-class self-adjoint operator W on H".

Would we then interpret W as an observable?

Requiring that observables are trace-class is pretty strong. Many of the most important observables are not even bounded operators. But this difficulty is generally prevalent in quantum mechanics, so maybe it's not the point.

There's a better statement of Gleason's theorem at quantum logic. But this version's a mess.

178.38.81.33 (talk) 15:41, 23 April 2015 (UTC)

Simple solution -- I just copied the theorem from there. It gets everything right. Note that the previous version was so unclear that I even mixed up observables and states.

Difficulties in "Application" section

I also did some fixing-up of the Application section. Here Gleason's theorem is stated somewhat more correctly, but serious problems remain.

The most conspicuous one is that an observable A is introduced and induces a finite sublattice. But then suddenly it is forgotten, and P is defined on the full lattice. Yet similar notation is used. Atoms are defined only through A and the operator has to have distinct eigenvalues by assumption. So the Hilbert space is finite dimensional.

Very confusing. The paragraph on A is irrelevant and should be removed.

Also, the identity involving P(y) is treated incorrectly, being introduced as an observation instead of a definition or requirement.

Finally, it is not obvious how this statement relates to the one in the introduction, mostly because the notation has not been brought in line. (Nor was it in line before I fixed the introduction.)

No more energy for this at the moment. But the section must be rewritten. 178.38.81.33 (talk) 17:55, 23 April 2015 (UTC)

Upshot: all the needed definitions are at quantum logic. There, all the questions in this section and the previous one are answered.178.38.81.33 (talk) 18:20, 23 April 2015 (UTC)

Link to atom

From the article, as I found it:

The events $x_{i}$ generate a sublattice of the Hilbert space which is a finite Boolean algebra, and if n is the dimension of the Hilbert space, then each event is an atom.

Clicking on the word "atom" gives interesting insight into the conceptual basis of quantum mechanics. ;-)178.38.81.33 (talk) 16:52, 23 April 2015 (UTC)

Confusing wording in the introduction

The last sentences of the introduction were:

This implies that the Standard Quantum Logic can be viewed as a manifold of interlocking perspectives that cannot be embedded into a single perspective. Hence, the perspectives cannot be viewed as perspectives on one real world. So, even considering one world as a methodological principle breaks down in the quantum micro-domain.

This was confusing to me. Is 'perspectives' a technical word? Does 'manifold' have the usual meaning in math/physics? If so, the corresponding wikipedia articles should be linked. But looking at the body of the article, there is no further mention of manifolds or perspectives. It seems like this section is not meaningful (or at best very unclear) so I have removed it. If I am incorrect in doing so, I would be interested in what the intended meaning was. — Preceding unsigned comment added by Doublefelix921 (talk • contribs) 11:10, 7 May 2017 (UTC)

Try reading: http://alpha.math.uga.edu/~davide/The_Mathematical_Foundations_of_Quantum_Mechanics.pdf — Preceding unsigned comment added by David edwards (talk • contribs) 13:40, 8 May 2020 (UTC)

Merely restoring a confusing passage does not make it clear. Moreover, the text in question is a verbatim copy of the source, which is not how Wikipedia works. XOR'easter (talk) 14:41, 8 May 2020 (UTC)

I should add that this article recently went through an extensive review process that concluded with giving it the second-highest level of community approval that Wikipedia can grant. Under such circumstances, it is best to propose major changes before making them. Cheers, XOR'easter (talk) 00:49, 9 May 2020 (UTC)

Reversion

I've removed the following text:

Gleason's theorem is also valid in real Hilbert space and can be extended to quaternionic Hilbert spaces as proved by Varadarajan in . These are in fact the only three possibilities admitted by Solèr's theorem when formulating quantum mechanics over a lattice of orthogonal projectors in a Hilbert space. This generalization contained however a gap due to the notion of the trace of an operator defined in a quaternionic Hilbert space. The complete generalization, using the notion of real trace has been obtained recently .

And the following two references:

• Moretti, V. and Oppio, M. The correct formulation of Gleason's theorem in quaternionic Hilbert spaces, Annales Henri Poincaré 14 (2018) in print arXiv:1803.06882
• Varadarajan, Veeravalli S., Geometry of quantum theory. Springer Science+Business Media, 1968, 2nd edition 2007.

The 2018 paper looks fine, but it has yet to attract any substantial scholarly evaluation (it was first posted this year, updated to an arXiv v2 this month, and is still "in press" at a journal), so it shouldn't be used in this way.

References

XOR'easter (talk) 16:08, 27 September 2018 (UTC)

Hm. I think it is a pity that these - obviously not *very* notable - generalisations and variations of Gleason's theorem are not even referenced any more. Wikipedia is really good at helping one find complete lists of relevant articles even if one does not need to cite, let alone read, most of them ... The journal where Moretti and Oppio got pulbished is a very very serious mathematics journal. Not some predatory junk publisher. So the paper has almost surely been very carefully reviewed by very knowledgable persons. Richard Gill (talk) 06:47, 5 January 2020 (UTC)

They actually are referenced; see footnotes 3 and 6. I think the way they are used now is fine. XOR'easter (talk) 16:36, 5 January 2020 (UTC)

Completely Broken

The overview seems broken in the definition of the quantum probability function and should be cleaned up!!

Problem 1: A quantum probability function is described as a function on the atoms and then, in the very next sentence the function is applied on element 0. However, 0 is not an atom, so the function is not defined. Moreover, the non-negativity condition is formulated for all elements of the lattice L. Again, the function is not defined on all elements of the lattice (according to the article).

Problem 2: Only the sum over all orthogonal atoms is 1. The stated condition misses the crucial word "all". One can argue that this is sufficient, since earlier n is mentioned as the dimension. However, this is in a different paragraph higher up so the connection is not fully clear and should (and could) be made clearer by a more precise formulation. Moreover, this criterion holds only when n is finite.

Problem 3: $P(y)$ is, according to the text, not defined for non-atomic y. — Preceding unsigned comment added by 217.95.163.80 (talk) 19:18, 9 September 2019 (UTC)

Cooke, Keane and Moran (1985)

We are missing the fact that there is now also a pretty elementary proof of Gleasons' theorem due to Keane and others. https://www.researchgate.net/publication/231919294_An_elementary_proof_of_Gleason's_theorem Richard Gill (talk) 06:34, 5 January 2020 (UTC)

Huh. I had thought that was already in the reference list at least. Thanks for pointing out the omission; I've added it now. XOR'easter (talk) 16:42, 5 January 2020 (UTC)

GA Review

This review is transcluded from Talk:Gleason's theorem/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: Jakob.scholbach (talk · contribs) 20:18, 6 January 2020 (UTC)

Thanks, XOR'easter for the nomination -- I will be doing a review as soon as I can; other reviewers are welcome! Jakob.scholbach (talk) 20:18, 6 January 2020 (UTC)

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 variables. See for example Kochen–Specker theorem 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 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).

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 measure"?

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 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 "Gleason’s theorem might be interpreted as telling us exactly why it is the most general such state". Tercer (talk) 12:01, 11 January 2020 (UTC)

Thanks for your comments. The point about noncontextual hidden variables is a good one — 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 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 Quantum Bayesianism" is true — Bub and Pitowsky make much of it [1][2][3], as did Carl Caves and coauthors back in the day [4]. The "relational quantum mechanics" part is supportable in principle [5], but I wouldn't mind taking it out. Wilce (2017) does talk about Gleason and reconstruction: From the single premise that the “experimental propositions” 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 probability measures on $L(H)$ correspond to density operators. [...] The point to bear in mind is that, once the quantum-logical skeleton $L(H)$ is in place, the remaining statistical and dynamical apparatus of quantum mechanics is essentially fixed. In this sense, then, quantum mechanics—or, at any rate, its mathematical framework—reduces to quantum logic and its attendant probability theory. Cassinelli and Lahti (2017) is explicitly about reconstructing quantum theory by way of Solèr and Gleason. I think there's a problem of labels — e.g., where's the line between quantum information and quantum logic? — so I'll try revising that passage of the article.

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 parenthetically, I think it's better to take that line out rather than rewrite it. XOR'easter (talk) 18:57, 11 January 2020 (UTC)

I've removed "bivalent", defined "frame function" and made assorted other modifications. XOR'easter (talk) 22:23, 11 January 2020 (UTC)

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 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.). Tercer (talk) 13:39, 12 January 2020 (UTC)

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. XOR'easter (talk) 22:10, 12 January 2020 (UTC)

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 function to sum to any non-negative real number, but that one can assume wlog that it sums to 1. Tercer (talk) 22:35, 12 January 2020 (UTC)

That's much like what I was considering doing. I'll go off and try to find a decent phrasing now. XOR'easter (talk) 22:37, 12 January 2020 (UTC)

Great, I think the result was quite decent. Tercer (talk) 23:18, 12 January 2020 (UTC)

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 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.

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 fundamental. This is the approach used in this 2010 paper, the one that named and defined quantum Bayesianism, does not even mention Gleason's theorem.

The reconstructions you mention by Wilce, Cassinelli, and Lathi are explicitly quantum logic reconstructions, that indeed do use Gleason's theorem.

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. Tercer (talk) 15:00, 12 January 2020 (UTC)

I've always heard "quantum Bayesianism" defined more broadly than "QBism"; the former includes Bub and Pitowsky, while the latter is what Fuchs, Mermin and Schack advocate. See for example Duwell (2010). To avoid the ambiguity, I've rewritten that passage. XOR'easter (talk) 15:44, 12 January 2020 (UTC)

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. Tercer (talk) 17:00, 12 January 2020 (UTC)

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 in September 2006. I probably should have cut it out or changed it to talk about quantum logic instead, but it slipped through the net. XOR'easter (talk) 22:10, 12 January 2020 (UTC)

Review by Jakob.scholbach (talk)

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.

Overview

• 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 separating a bit more the mathematical basis of the story and the physical interpretation.
• Per WP:MOS (MOS:WE) I think it is advisable not to address the reader directly. This happens in many spots, e.g. "Consider a quantum system..."
• "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.  Done Moved during reorganization XOR'easter (talk) 17:24, 23 January 2020 (UTC)
• "particular mathematical entities" -- do you mean probability measures here? The vagueness of the phrasing does not seem to help (me) here.  Done Rewritten to be more precise XOR'easter (talk) 17:24, 23 January 2020 (UTC)
• 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.
• "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." ? Also, it would seem to make sense how this assumption is a simplification?  Done XOR'easter (talk) 17:24, 23 January 2020 (UTC)
• "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.  Done Passage revised XOR'easter (talk) 17:24, 23 January 2020 (UTC)
• "A density operator is a positive-semidefinite operator" -- on what?  Done XOR'easter (talk) 17:24, 23 January 2020 (UTC)
• Is "quantum-mechanical observable" a synonym for "measurement"?  Done XOR'easter (talk) 17:24, 23 January 2020 (UTC)
• Why does the statement of G's theorem appear twice, however in a somewhat different form?
• Further down: is the section with "Let H denote the Hilbert space..." related to the quantum-logic interpretation of the theorem?  Done The subsectioning should make this clear now XOR'easter (talk) 17:24, 23 January 2020 (UTC)
• 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 lattice"?
• Is a "proposition" the same as speciyfing a subspace of H which has codimension 1?
• How do the x_i generate a sublattice? Just the intersections of these subspaces?
• Previously, the probability measure P was defined on H; now it seems to be defined on the set (or lattice) of all subspaces of H. How does this relate to the setup above?
• In the indented statement of G's theorem all jargon that is not crucial should be eliminated: can the mention of L be eliminated / what about the bold-face / italic x? Also the notation for the Hilbert space H should be repeated here, to make the statement as self-contained as possible.
• 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> give a probability function on H (also in dim = 1, 2), and the content of the theorem is that the converse holds, too?  Done XOR'easter (talk) 17:24, 23 January 2020 (UTC)
• It seems that the Hilbert spaces in the article are either complex or real, correct? If so, Hermitian should probably be rephrased somehow?
• 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"?  Done XOR'easter (talk) 17:24, 23 January 2020 (UTC)

OK, I will review the remaining sections as soon as I can. Jakob.scholbach (talk) 20:22, 19 January 2020 (UTC)

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 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. XOR'easter (talk) 20:51, 19 January 2020 (UTC)

OK, I've retitled, reorganized and in parts rephrased the "Overview" section. Hopefully this takes it in a good direction at least. XOR'easter (talk) 19:00, 20 January 2020 (UTC)

Can you please very briefly indicate which comments of the above have been addressed? This would simplify the exchange. Jakob.scholbach (talk) 19:08, 22 January 2020 (UTC)

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. XOR'easter (talk) 21:33, 22 January 2020 (UTC)

History

• "in a theory founded on Hilbert space" sounds both vague and a bit ungrammatical to me.  Done Rephrased. XOR'easter (talk) 17:24, 23 January 2020 (UTC)
• 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.
• 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.
• 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>3 to d=3 work? Jakob.scholbach (talk) 19:08, 22 January 2020 (UTC)

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 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, Gleason's article doesn't mention him specifically; he just makes the observation, 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 said. The article by Chernoff that mentions Kadison doesn't go into detail about what his counterexample was. However, other papers give counterexamples for $d=2$, and we could provide one of those (e.g., the one in Wright and Weigert (2018). XOR'easter (talk) 21:33, 22 January 2020 (UTC)

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 hidden variables. It was all the rage in the past few years as people were studying the limits of psi-epistemic models. Tercer (talk) 09:49, 23 January 2020 (UTC)

I like that idea. I guess I had been thinking of the Kochen–Specker qubit model (and the Bell–Mermin 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 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"). XOR'easter (talk) 13:31, 23 January 2020 (UTC)

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. Tercer (talk) 14:14, 23 January 2020 (UTC)

Pseudo-review by Tercer

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.

• In the lead, the article states that Gleason's theorem is of particular importance ... for the effort in quantum information theory to re-derive quantum mechanics from information-theoretic principles. 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.
• The second paragraph of the lead feels a bit weird to me. Should we really try to teach quantum mechanics in the lead?
• 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 entirely.
• In the Quantum Logic subsection, the article states that 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 quantum measurement. 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 $\alpha _{i}$ of an observable A are introduced, but play no role. Then a (quantum?) event is introduced as the measurement outcome $x_{i}$. 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 Atom (order theory) doesn't explain it either. Perhaps you mean orthogonal projectors? Then the probability of obtaining measurement outcome y is described as the sum of the probability of the atoms under y. But what is y? I thought the $x_{i}$ 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 $x_{i}$ are unit vectors representing logical propositions.
• In the Implications section, the article states that The mapping u → ⟨ρu,u⟩ is continuous on the unit sphere of the Hilbert space for any density operator ρ. Since this unit sphere is connected, no continuous probability measure on it can be deterministic. 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.
• 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 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. Tercer (talk) 15:40, 5 February 2020 (UTC)

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 section 3.1 of Pitowsky (2006); 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. XOR'easter (talk) 13:33, 6 February 2020 (UTC)

I've shortened and rearranged the quantum-logic material, and I added more details about how to construct a counterexample in dimension 2. XOR'easter (talk) 19:11, 15 February 2020 (UTC)

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? Tercer (talk) 20:25, 15 February 2020 (UTC)

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 {{rp}} template for providing page or section information after a footnote.) XOR'easter (talk) 20:34, 15 February 2020 (UTC)

I'll help with the reformatting then. Tercer (talk) 20:45, 15 February 2020 (UTC)

Thanks! Looks like we've gotten it done (and yes, it was long overdue). XOR'easter (talk) 21:32, 15 February 2020 (UTC)

Well, you, I was planning to do it now but there is nothing left to do. Thanks. Tercer (talk) 07:07, 16 February 2020 (UTC)

Section break

It looks like redoing the quantum-logic material has addressed several of the comments that Jakob.scholbach 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 $\mathbb {R} ^{3}$. I think the current level of detail is OK, but perhaps more would be better? Opinions welcome! XOR'easter (talk) 21:51, 15 February 2020 (UTC)

I'm afraid putting more detail would confuse instead of enlighten. As for the reduction of the problem to $\mathbb {R} ^{3}$, 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. Tercer (talk) 07:06, 16 February 2020 (UTC)

Where do we stand? Looking back over Jakob.scholbach'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. XOR'easter (talk) 14:53, 23 February 2020 (UTC)

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. Tercer (talk) 17:26, 23 February 2020 (UTC)

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. Tercer, 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! Jakob.scholbach (talk) 20:20, 24 February 2020 (UTC)

Sure, I can do it, it's pretty much finished anyway. Tercer (talk) 20:37, 24 February 2020 (UTC)

Thanks, I appreciate it. Jakob.scholbach (talk) 07:51, 25 February 2020 (UTC)

Final review

I decided to WP:BOLDly remove the paragraphs I found problematic, and finish the GA review.

GA review (see here for what the criteria are, and here for what they are not)

1. It is reasonably well written.

   a (prose, spelling, and grammar): b (MoS for lead, layout, word choice, fiction, and lists):

   I'm not really happy with the lead

2. It is factually accurate and verifiable.

   a (reference section): b (citations to reliable sources): c (OR): d (copyvio and plagiarism):

   The massive amount of work in the review was mostly dedicated to address this, now I think it is perfect.

3. It is broad in its coverage.

   a (major aspects): b (focused):

4. It follows the neutral point of view policy.

   Fair representation without bias:

5. It is stable.

   No edit wars, etc.:

6. It is illustrated by images and other media, where possible and appropriate.

   a (images are tagged and non-free content have fair use rationales): b (appropriate use with suitable captions):

   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

7. Overall:

   Pass/Fail:

   The lead should still be improved, but it's good enough for WP:GOOD.

Spekkens (2005) and Spekkens (2014)

Both Spekkens (2005) and Spekkens (2014) argue that theorems analogous to Gleason and Kochen–Specker apply to a single qubit. The latter adopts the definition of "noncontextual operational model" proposed by the former, and then argues that unsharp measurements in quantum theory must be represented by outcome-indeterministic response functions. In the discussion section, he criticizes some earlier claims to rule out a noncontextual model of quantum theory using a proof that appeals explicitly to unsharp measurements, because they assumed outcome determinism for unsharp measurements ("ODUM"). But then he pulls out a "Nevertheless": it is possible to construct a no-go theorem for noncontextuality for a single qubit without the assumption of ODUM, two examples being given in Spekkens (2005), which he recapitulates. One is based on a finite set of measurements and so is reminiscent of Kochen–Specker, and the other is based on the Gleason-like theorem for POVMs.

One consequence of our analysis is that the restriction of previous no-go theorems for noncontextuality to Hilbert spaces of dimension 3 or greater was an artifact of having a notion of noncontextuality that was limited to sharp measurements. For a qubit, there is only a single measurement context in which any given rank-1 projector can appear, namely, together with its unique rank-1 orthogonal complement. Hence, there is no possibility of a nontrivial variation of the context in which a projector appears and hence no possibility of context-dependence either. When one considers unsharp measurements, on the other hand, there are nontrivial contexts: a given nonprojective POVM may be realized as a convex combination of other measurements in multiple ways, as a post-processing of other measurements in multiple ways, and by reduction of another measurement in multiple ways. However, for unsharp measurements, achieving a noncontextual model is not about assigning outcomes in a context-independent fashion, it is about assigning probabilities of outcomes in a context-independent fashion.

Regardless of whether this is the right position or not (a question that Wikipedia isn't here to settle anyway), I don't see a way to argue that Spekkens is arguing different sides in these two papers.

I tried to get this right with a footnote, putting Spekkens on one side of the "does this apply to a single qubit?" question and Grudka and Kurzyński on the other. But looking at it again, I realized my phrasing could have implied that they were all opposing the claim in the main text, rather than being in opposition to each other. (And not entirely in opposition, at that, since Spekkens agrees with Grudka–Kurzyński that the Cabello–Nakamura proposal for a qubit Kochen–Specker proof appeals to a flawed notion of noncontextuality. But for the purposes of this article, that seems a sideline: Spekkens dismisses one attempt at qubit Kochen–Specker as bad, then holds up another as good, so the overall conclusion is that qubit Kochen–Specker is possible.) XOR'easter (talk) 15:43, 17 February 2020 (UTC)

If only the good professor Gleason were still alive to see all this. EEng 16:11, 17 February 2020 (UTC)

You're right. I thought Spekkens had changed his mind in the intervening years, but alas, he learned nothing. I had focussed on this footnote of Spekkens (2014):

Grudka and Kurzynski [29] have also criticized the notion of noncontextuality used in the Cabello-Nakamura proofs. They argue that in a noncontextual model, one should only assign deterministic values to the projectors that appear in a Naimark extension of the POVM, rather than the POVM elements themselves. It then suffices to note that the projector that extends a given effect varies with the POVM in which that effect appears, and therefore that a noncontextual model does not assign a unique deterministic value to a given effect. In the language of the present article, they argue that a noncontextual and outcome-deterministic value-assignment to projectors on system+ancilla does not imply a non-contextual and outcome-deterministic value-assignment to effects on the system. This attitude is entirely consistent with the view espoused here.

I didn't read it carefully enough, though. Spekkens is merely saying that Grudka and Kurzyński's view are consistent with his, not that he actually agrees with them. Their only point of agreement is that ODUM is nonsense, but Spekkens still thinks that one should assume non-contextuality for POVMs, as he explicitly proves a Busch-like theorem.

I think we shouldn't put controversy in the text and in the note, though; the text already notes that there is a controversy, and gives a reference to support that. I think rather that Spekkens' reference belongs with the whole pile of references supporting Busch's theorem. If he had criticized Grudka and Kurzyński, or at least pointed out explicitly that he disagrees with them, then it would be appropriate to cite them both in opposition.Tercer (talk) 17:12, 17 February 2020 (UTC)

I've moved the footnotes around, to break up the long block and to bring each one in contact with the most pertinent text. XOR'easter (talk) 17:34, 17 February 2020 (UTC)

That was a good solution, thanks.Tercer (talk) 18:51, 17 February 2020 (UTC)