Hi Michel,
Thank you for your excellent essay, which advances the discussion of the topics of quantum non-locality and contextuality, and thanks for your response with the references, which I will follow up shortly. Your work is highly relevant to my own and will be one of the most important of the competition for me. On Friday, July 26, I gave your essay a very high rating.
Sincerely,
Charles Card
Greetings Michel,
I have just given you a well deserved boost, after reading (or reviewing) your wonderful paper. As I understand it; the Dessins are contextual maps, showing the connections involved and the object-observer relationships. You state that the Fano plane is the smallest projective plane possible. I guess that means the octonions are the smallest irreducible representation of object and observer context. Anyhow; it was a very well written and fun introduction to a subject which could have been a lot less engaging. Your enthusiasm for your subject is infectious Michel, and you definitely portray the child-like playful explorer well.
Have Fun!
Jonathan
Dear Jonathan,
My best acknowledgements for the boost. I appreciate that the technicalities do not discourage you. I hope that the infection will propagate!
You are right that the octonions lurk around the examples I selected. Not all dessins are feature quantum contextuality. The simplest case given here is Mermin's square and can be seen as an archetype (in the language you use in your essay). The next case is Mermin's pentagram, there are 12096 of them with three qubits and 12096 is also the size of the automorphism group of G2(2) (related to the octonions as John Baez explains in his famous note). This is discussed in our recent papers on ArXiv.
My kind regards,
Michel
Thank you Michel!
I am pleased to help your wonderful essay rise higher. I expect that many more wonderful insights await, in the collection of your papers (or where you are an author) I have downloaded from arXiv. The overlooked importance of something small like the fundamental nature of the (0, 1, /infty) triple is seldom made known. I noticed you commented to this effect on Akinbo's essay site as well as in your own.
People are unaware that in an ab initio formulation, if we actually start at the very beginning and move forward from first principles, one can only know there is an extent; we can call it 1 but it could also be infinite as there is only nothing to compare it to. That is the rule for constructive geometers. In Ian Durham's essay; he makes the point that even knowing something is a unit, we still don't know how 'big' it is (e.g. - a bit or a trit). Perhaps ternary digits are more useful, after all.
But I like imagining that 1 is a nice balance point between 0 and infinity.
All the Best,
Jonathan
Dear Michel,
Thank you so much for your essay. As far as I understand, in your essay, the conventional logic is used. When we use the different type logic, for example, topos http://arxiv.org/abs/quant-ph/0703060 , how to relate your approach? Also, the problem seems to be completely solved.
Best wishes,
Yutaka
Dear Yukata,
My essay don't refer to logic at all (although Grothendieck's topos may be hidden in it at some level).
Should I remind that Grothendieck introduced the concept of a topos.
It seems that you did not understand what my essay is about, but still scored it low, this is unfair.
http://en.wikipedia.org/wiki/Topos
I don't uderstand your sentence "the problem seems to be completely solved"!
Michel
Dear Jonathan,
As far as dessins d'enfants are concerned, the members of the triple (0, 1, /infty) have well defined meaning. Sorry that I just copy my earlier post:
The Belyi theorem (see the step 3 in my Sec. 2 giving the definition of a child's drawing) and the property that the child's drawing D itself is the preimage of the segment [0,1], that is D=f^-1([0,1]), where the Belyi function f corresponding to D is a rational function. All black vertices of D are the roots of the equation f(x)=0, the multiplicity of each root being equal to the degree of the corresponding vertex. Similarly, all white vertices are the roots of the quation f(x)=1. Inside each face, there exits a single pole, that is a root of the equation f(x)=\infty. Besides 0, 1 and \infty, there are no other critical value of f.
In experiments you will have 0 or 1 as the result of the experiment (in the single or multiple qubit context) but the unobserved \infty is needed in the explanation. The way the black points (bit 0) and white points (bit 1) ly on the dessin (a graph on the oriented surface such as the sphere S2, or a Riemann surface with holes) is such that sigma(0)*sigma(1)*sigma(infty)=id, where
sigma (0) is the permutation group attached to the black points 0 (how the edges incident on the black points rotate) and sigma (1) is the permutation group attached to the white point 1 (how the edges incident on the white points rotate).
It is still binary logic but in a more clever way (may be this has to do with Grothendieck's topos, I have not thought about this aspect).
Thanks again for your interest.
I intend to write you again about the Hopf fibrations.
My kind regards.
Michel
Dear Michel,
For completeness, I'm posting the following (slightly modified) response from my FQXi blog. Sorry to see that you are also the victim of some "unfair" voting. Regrettably, I scored many "ones" without the accompanying "critiques (the very reason that I entered, as spelt out in my essay).
......
Dear Michel,
Many thanks for continuing the discussion. I hope we will get to do more of it in the future.
As for my acceptance (or otherwise) of COUNTERFACTUAL REASONING, let me offer the following proposition:
Perhaps the related problems are due to COUNTERFACTUAL TESTING!?
For example: In my Essay, referring to the CHSH Inequality -- page 7, equations (21)-(22) -- you will see that the inequality is based on a TRUISM (21). But we do NOT test the truism; rather, we test the best approximation that we can (22).
So, by this view, it is not counterfactual reasoning that's at fault. It's the failure, even the impossibility, of testing it.
Hence the question: Why should that impossibility be regarded as a valid strike against a rational local-realism?
Especially when QM fails to deliver in the same impossible context?
WHILE both theories deliver the same experimental outcomes!
Thus the need for further discussions continues.
With best regards; Gordon
Dear Michel,
yes, I read your essay but was on the vacation before I had the chance to write you. I like your geometric model very much (I rated your essay long ago with maximum score).
Now after a second reading I have some questions:
- You used the dessin d'enfants to visualize the contextuality. I understood the Mermin square but how did I see it in dessin d'enfant (Fig. 3b). Is it the number of half-edges (odd number) which produces the contradiction?
- Why is the transitive action so important? In case of a non-trivial orbit, you can check every point seperately.
One remark about the triple 0,1,infty: In the projective geometry, you always have the invariance w.r.t. the inversion operation. In the context of your model it means you have the operator and its inverse operator. Then 0 is related to infty and 1 is related to itself via inversion.
Thanks in advance for the answers.
Torsten
PS: Did you saw my post (June, 17) above?
Dear Gordon,
Thank you for your continuing interest. I really appreciate your feedback at this time of the competition. We can certainly learn more from each other after the end of the contest. I will rate your essay highly, as it deserves, I would like to see you in the finalists, hopefully I will be too.
Concerning counterfactuality, as soon as a good theory of quantum observability is written, one will be able to check it as others assumpions in science. I claim that Grothendieck's approach with dessins d'enfants is an excellent starting point because it has all attributes of an archetype (read Dickau's essay) or a monad (read Ojo's essay) and other good ontological properties which I don't list here. Topos theory is not too far.
There are important essays here that pushed me to see the dessins d'enfants as "explicate imprints" of a more general (possibly spatio-temporal) algebraic geometry. I have in mind the Hopf fibrations as an excellent tool. For example you can lift S2 (the Riemann sphere) to S3 (the 3-sphere, i.e. the space of a single qubit (Jackson's intelligent qubit?), also the conformally compactified Minkowski space (see Matlock' essay and in relation to Bell's theorem Joy Christian 'realistic' approach).
Local/nonlocal arguments are insufficient, I think, mathematics should help in revealing the hidden machinary of the physical and ontological universe. May be this is Einstein's dream, not contradicting Wheeler, at the end of the day because we are, more or less, their children in knowledge.
Yes our discussion should live.
All the best,
Michel
Dear Torsten,
Thanks for your careful reading.
1) It is not straight to see the contradiction in the dessin of Fig. 3b, I failed to see it in general (for other contexts). Also there is not a single dessin leading to Mermin's square but many, why is it so? More work is necessary. This non-bijection is general for most geometries I have tried to reconstruct from the n-simplices to projective configurations such as Desargues, Cremona-Richmund (i.e. the doily W(2) of two-qubit commutatitivity) and others.
2) You are right that transitive action may not be a necessary condition. The geometry is constructed by having recourse to the stabilizer of each point in the permutation group relevant to the dessin.
3) Last remark, the geometry is of the projective type not the dessin. Here you have to refer to the theory that is well explained, for example in Lando and Zvonkin (my ref. [6]).
Torsten, please check that you vote was recorded.
Thanks Michel for your message on my blog. My arguments are from a philosophical and classical perspective. It is possible that when viewed from your perspective we may well be saying something similar. I am not expert on the math involved in quantum theory.
Following additional insights gained from interacting with FQXi community members, perhaps you will find the the judgement in the case of Atomistic Enterprises Inc. vs. Plato & Ors, delivered on Jul. 28, 2013 @ 11:39 GMT easier to understand my thinking.
Best regards,
Akinbo
*I have already rated your essay so you may do likewise.
Greetings Michel,
I enjoyed the last comments left on my essay space, and I eagerly await the next chapter on Hopf fibrations - which are already a subject of interest.
It appears the 'infection' has spread, but Dr. Planat is in!
Have Fun,
Jonathan
Hi Michel,
So it appears that the FQXI database has been reset and my comments have disappeared... I will add them back in. As I said before:
Your very interesting essay, taking us right to the foundations, raises some questions. You wrote:
> This implies the non-existence of the Belyi map and puts three-qubit contextuality on a qualitatively different footing when compared with the two-qubit case.
It seems that your quest to model 3-qubit contextuality has an unhappy ending in your essay. Do you thus conclude that (some? all?) mechanisms that explain contextuality for 2-qubit systems will not explain it for 3-qubit systems?
What avenues are you now taking to explore 3-qubit contextuality? (Yes, yes I know I should read your recent papers... after the contest, please...) I am hoping there is another chapter to come in your story... with finally, a happy ending. I do like the idea that you mention of lifting the Riemann sphere to S3 via the inverse Hopf Fibration. I will put some more comments about this on my blog.
> To find the corresponding Belyi map seems to be a challenging math problem.
My thought was there might be a possibility of easier results if we restrict white and black points to lie at k-rational points:
Consider Q(phi), the algebraic extension of the rationals by the golden ratio. These numbers are able to provide cartesian coordinates for all the vertices of Platonic solids, and many other polytopes besides. These could be considered as k-rational points within larger fields, but also as a field on their own. (For such fields, Falting's Theorem should apply and might possibly be useful.) But I was mainly wondering if anyone had looked at the techniques you describe with such k-rational points and fields in mind.
Anyway, if they prove useful, I imagined that such diagrams (living in S3?) might be called "Dessins d'Or"... and eventually lead to a happy ending. Next year's essay, perhaps?
Hugh
I also saved a copy of your response, so I add it here:
--------------------------------------
Dear Hugh,
"different footing"
it is a matter of perspective, the pentagram possesses the same graph than the Desargues configuration that can be drawn in several ways. The non-bijection between drawings and geometries here (and elswhere) is something I am currently working at.
"avenues"
yes, our recent papers pointing out G2(2) and octonions (several comments in this blog) and more to come, including (with you?) the lift to S3.
"k-rational points"
excellent, we are preciselt talking about algebraic curves on the Riemann sphere (S2 say), after the lift we should keep the algebraic property.
"Dessins d'Or"
a lift to Orland circles, or Urland knots.
My kind regards,
Michel
Hi Michel,
Your very interesting essay, taking us right to the foundations, raises some questions. You wrote:
> This implies the non-existence of the Belyi map and puts three-qubit contextuality on a qualitatively different footing when compared with the two-qubit case.
It seems that your quest to model 3-qubit contextuality has an unhappy ending in your essay. Do you thus conclude that (some? all?) mechanisms that explain contextuality for 2-qubit systems will not explain it for 3-qubit systems?
What avenues are you now taking to explore 3-qubit contextuality? (Yes, yes, I know I should read your recent papers... after the contest, please...) I am hoping there is another chapter to come in your story... with finally, a happy ending. I very much like the idea that you mention of lifting the Riemann sphere to S3 via the inverse Hopf Fibration. I will put some more comments about this on my blog.
> To find the corresponding Belyi map seems to be a challenging math problem.
My thought was there might be a possibility of easier results if we restrict white and black points to lie at k-rational points: Consider Q(phi), the algebraic extension of the rationals by the golden ratio. These numbers are able to provide cartesian coordinates for all the vertices of Platonic solids, and many other polytopes besides. These could be considered as k-rational points within larger fields, but also as a field on their own. (For such fields, Falting's Theorem should apply and might possibly be useful.) But I was mainly wondering if anyone had looked at the techniques you describe with such k-rational points and fields in mind.
Anyway, if they prove useful, I imagined that such diagrams (living in S3?) you could call "Dessins d'Or"... and that they could lead you to a happy ending. A topic for next year's essay, perhaps?
Hugh
Dear Hugh,
"different footing"
it is a matter of perspective, the pentagram possesses the same graph than the Desargues configuration that can be drawn in several ways. The non-bijection between drawings and geometries here (and elswhere) is something I am currently working at.
"avenues"
yes, our recent papers pointing out G2(2) and octonions (several comments in this blog) and more to come, including (with you?) the lift to S3.
"k-rational points"
excellent, we are preciselt talking about algebraic curves on the Riemann sphere (S2 say), after the lift we should keep the algebraic property.
"Dessins d'Or"
a lift to Orland circles, or Urland knots.
My kind regards,
Michel
Dear Torsten (a copy is on your blog),
I am trying to better understand your deep essay but it turns out to be quite difficult accounting for my poor knowledge of differential geometry.
I have a naive question. The (first) Hopf fibration S^3 can be seen as the sphere bundle over the Riemann sphere S^2 with fiber S^1. Could you explain what is the sphere bundle S^2 x [0,1] that you associate to the gravitational interaction? May it be considered as some sort of lift from dessins d'enfants on S^2 to S^2 x S^0, and the latter object lives in circles on S^3, right?
I have in mind Matlock's essay as well.
All the best,
Michel
This test is the best that I have read and analyzed in this contest.Explains Dr Planat efficiently and deep map the graph theory applied to information processing, in quantum theory. In this analysis, both logically well argued, as mathematically demonstrated the clear and inevitable connection with the theory of graphs and maps permutations, with information theory.Certainly, from my humble point of view, a clear candidate to receive recognition for this contest.
Rate it all!!
Thanks Dr Planat
Dear Michel,
I rated already your sophisticatedly serious essay on August 1st. Somehow, my comment accompanied with my vote is lost or not posted. I definitely agree that all interpretations must be contextual in its nature. Excellent work!
If I may say, KQID proposes contextuality through KQID Ouroboros Equations of Existence that combines Newton, Maxwell, Planck, Boltzmann, Lorentz,Einstein, Laundauer, Wheeler , Feynman, Ssusskind, Hooft, Wilczek, Bousso and others. The Ouroboros Equations mean each interpretation involves every beginning to every ending. Similarly, everything we do involve the Ouroboros action or totality of any action. Nature is such unbelievable phenomena that we are just now starting to peek into its secret that is shockingly simple in the beginning but infinitely complex in the ending that per KQID every absolute digital time ≤ 10^-1000seconds. Interestingly, the mechanism is also simple. See my essay Child of Qbit in time.
All the Best,
Leo KoGuan