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)

"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)

- 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. 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).

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&#39;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&#39;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: 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.
 * 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.

XOR&#39;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&#39;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&#39;easter (talk) 16:42, 5 January 2020 (UTC)

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&#39;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&#39;easter (talk) 17:34, 17 February 2020 (UTC)
 * That was a good solution, thanks.Tercer (talk) 18:51, 17 February 2020 (UTC)