It from qubit: how to draw quantum contextuality by Michel Dr Planat

Mr Planat,

As I have carefully explained in my essay BITTERS, everything in the real Universe is unique, once.

Although I do not doubt that Wheeler's yes/no binary code and Bell's parameters and Mermin's failing emerging EPR realty criterion and your ability to draw quantum contextuality out of nowhere could be important abstractly, they do not appear to me to be unique.

Michel, it is good to see some new ideas from information theory being put to use in this contest. This is a mathematically very sophisticated and I was not familiar with the relationships around dessins d'enfants so it is very enlightening. I wonder how many times I will have to read it to fully appreciate it.

I included the Kochen-Specker Theorem in my essay last year so I have touched on some corners of these ideas before. It is remarkable how many concepts converge in the theory of qubits

I recognised your name as author of http://arxiv.org/abs/1005.1997 which I looked at when I was looking at qubits http://arxiv.org/abs/1005.1997 . I should look at some others.

Dear Michel,

I've read your essay already when it appeared on the arXiv, and have since been waiting for a chance to comment on it. Having done some work on quantum contextuality myself, I was naturally very curious about your ideas, and though I'll need a bit more time to digest the mathematics, I must say I'm very intrigued.

From the title, I at first assumed you were going to consider the observation that the Lovasz theta function, a measure for a graph's Shannon capacity, gives the quantum violation of contextuality inequalities in at least some cases (something realized first, I think, by Cabello and co-workers). But while you also talk about the Shannon capacity, to me at least the relationship seems not obvious.

First of all, I think your observation that "Wheeler's observer-participancy is contextual" is spot on: in fact, this is essentially the lesson of Bohr's complementarity---we cannot describe all our observations within one single classical picture, just as, in contextuality, we cannot give a single probability distribution (or truth-valuation) for all observables.

However, I think I've missed the main point, probably, which is the precise connection between the dessins and contextuality. Usual proofs of the Kochen-Specker theorem rely on finding a set of vectors in Hilbert space such that the associated graph is not 2-colorable, that is, one cannot find a truth valuation. What do the dessins tell you about this? (Apologies if it's obvious and I just haven't been paying attention.)

Also, on an unrelated note, since I know you've done some work on the black hole/qubit correspondence: in your opinion, does the correspondence tell us something 'deep' about nature, or is it just a mathematical curiosity? I used to be pretty skeptical, thinking that it's probably just based on the coincidence that the group SL(2,C) crops up in both context, but now that entanglement is wormholes ;-), why not have qubits be black holes?

Cheers, and my best wishes for the contest,

Jochen

Dear Michel Planat

I just read your superb essay. I will try to comment in greater detail later today after I have read your essay a second or third time. I have been concerned with the role of Cayley numbers, the projective Fano plane, Freudenthal cubic equation or determinant in quantum gravity. My essay I conjecture some role for octonions in quantum field theory or quantum gravity and its implication for nonlocality. This is in the second part of the essay after I illustrate a formal incompleteness of any causal scheme. Your essay hits on these issues within the context of nonlocality and the Bell-CHSH inequality.

What do you think of algebraic curves over [0, 1, ∞] and the Langlands program? This seems to suggest there are generalized Tanyama-Shimura theorems for curves on general surfaces such as K3xK3 that obey certain conditions or constraints, such as given by the Fano plane.

Cheers LC

If you have comments about my essay of course feel free to comment there. My take on logic is the modal logic of causality, which is used to argue that potentially the associative property is violated in some subtle manner with vacuum physics. I generally think that quantum mechanics is on the Cayley numbers 1, 2, 4, 8 complex or #2. QM may have states that are generated by quaternionic operators (standard physics actually) and further with uncertainty fluctuations of event horizons the ordering ambiguity of operators in an S-matrix channel is a nonassociative condition. My arguments tend to be rather physical at this point, having departed from the more metaphysical issue with modal logic.

It is curious that Grothendieck would comment on this dessins d'enfant, for this does have the appearance of category theory of sorts, which was his area of mastery. In effect what appears possible is that a set of curves defined on [0, 1, ∞] with projective properties, such as with the projective Fano plane, are those which construct modular forms corresponding to curves or spaces of curves (orbits) on spaces of dimension 3, 4, 6 and 10, which are the Cayley numbers plus 2. The "plus 2" comes about because the "trivial case" with 0 is for Ricci flat spaces in two dimensions, T^2 torus, with elliptic curves that define points on them. This is of course the Tanyama-Shimura conjecture --- now proof by Wiles.

I guess that Grothendieck either completely disappeared or maybe he is dead by now. He was a brilliant but rather odd man.

Cheers LC

Just stumbling across this, the octonions are very relevant for three-qubit geometry in general: the state space is a 15-dimensional sphere, an S^8 with fiber S^7, i.e. the last Hopf fibration; likewise, two qubits are related to the second Hopf fibration, S^3 over base S^4, and of course the Bloch sphere is just S^2 fibered with the global phase S^1. What's intriguing is that the requisite maps can be used to characterize the entanglement in the state, as discussed in this paper by Barnevig and Chen.So in this sense, it's sort of natural to consider a 3-qubit state a 'octonionic spinor' (o_1,o_2), parametrizing the S^15 via the normalization |o_1|^2 |o_2|^2 = 1.

Not sure if it's anything deep, but it's always struck me as a curious and perhaps interesting observation.

I discuss the connnection to twistor theory on my page , which is a bi-spinor theory.

Thanks for the information on G_2. I discuss some of this on my page today. Duff has worked out connections between qubits, (2, 4, and 8)-qubits with C, H and O. I will read these papers on the E8 automorphism and comment later.

Cheers LC

Dear Michel,

great essay. I like to see abstract methods like Dessins d'enfants. Additionally I wil also read your paper mentioned above. I also worked in quantum information theory around 2003 to 2006. There I remembered on a discussion withthe group in Karlsruhe (Prof Beth) about the uniqueness of te Hamiltonian representation of qubit operations. One member of the group thought about a decomposition of three or higher qubit operation using only 2-qubit interaction Hamiltonians. I have the feeling that this problem is connecetd with your three qubit problem above.

Then I was able to prove a No-Go theorem (using ideas about the non-parallelizability of spheres). We were not able to publish the paper. Everyboday told us that it is not interesting or trivial.

Here is the link:

http://arxiv.org/abs/quant-ph/0508029

Maybe you can use it....

I will also thought about the G_2.

More later

Best

Torsten

Michel,

I reread your paper again this last Sunday. The desin d'enfant leads at the end to Mermin's pentagons. These are of course an aspect of the Kochen-Specker theorem. This is of course the main theorem on contextuality in QM. In my paper I discuss the quantum homotopies of associators at various dimensions, which are pentagonal systems. I copy this post on my essay blog page, so you can respond to this there as well.

I notice you have considerable interest in the G_2 group, which is the automorphism of the E8 group. The F_4 group is a centralizer in E8, whereby G_2 action keep it fixed; the elements of F_4 and G_2 commute.

The Kochen-Specker theorem is connected with the F_4 group, or the 24 cell. The 117 projectors with the original KS theorem in 3-dim Hilbert space is simplified by considering a four dimensional Hilbert space, or a system of 4 qubits. This involves only 18 projector operators. The space 24-cells is a system of root vectors for the F_4 group. Each root vector is paired with its negative to define a line through the origin in 4d space. These 24 lines are the 24 rays of Peres. The root vectors are

1 (2,0,0,0) 2 (0,2,0,0) 3 (0,0,2,0) 4 (0,0,0,2)

5 (1,1,1,1) 6 (1,1,-1,-1) 7 (1,-1,1,-1) 8 (1,-1,-1,1)

9 (-1,1,1,1) 10 (1,-1,1,1) 11 (1,1,-1,1) 12 (1,1,1,-1)

13 (1,1,0,0) 14 (1,-1,0,0) 15 (0,0,1,1) 16 (0,0,1,-1)

17 (0,1,0,1) 18 (0,1,0,-1) 19 (1,0,1,0) 20 (1,0,-1,0)

21 (1,0,0,-1) 22 (1,0,0,1) 23 (0,1,-1,0) 24 (0,1,1,0)

(I hope this table works out here) Consider these as 24 quantum states |ψ_i>, properly normalized, in a 4 dimensionl Hilbert Space e.g. it might be a system of two qubits. For each state we can define a projection operator

P_i = |ψ_i)(ψ_i| --- I have to use parentheses because carrot signs fail in this blog.

P_i are are Hermitian operators with three eigenvlaues of 0 and one of 1. They can be considered as observables and we could set up an experimental system where we prepare states and measure these observables to check that they comply with the rules of quantum mechanics. There are sets of 4 operators which commute because the 4 rays they are based on are mutually orthogonal. An example would be the four operators P_1, P_2, P_3, P4.

Quantum mechanics tells us if we measure these commuting observables in any order we will end up with a state which is a common eigenvector i.e. one of the first four rays. The values of the observables will always be given by 1,0,0,0 in some order. This can be checked experimentally. There exist 36 sets of 4 different rays that are mutually orthogonal, but we just need 9 of them as follows:

{P2, P4, P19, P20}

{P10, P11, P21, P24}

{P7, P8, P13, P15}

{P2, P3, P21, P22}

{P6, P8, P17, P19}

{P11, P12, P14, P15}

{P6, P7, P22, P24}

{P3, P4, P13, P14}

{P10, P12, P17, P20}

At this point you need to check two things, firstly that each of these sets of 4 observables are mutually commuting because the rays are othogonal, secondly that there are 18 observables each of which appears in exactly two sets.

Now assume there is some hidden variable theory which explains this system and which reproduces all the predictions of quantum mechanics. At any given moment the system is in a definite state, and values for each of the 18 operators are determined. The values must be 0 or 1. with the rules they are equal to 1 for exactly one observable in each of the 9 sets, the other three values in each set will be 0. Consequently, there must be nine values set to one overall. This leads to a contradiction, for each observable appears twice so which ever observables have the value of 1 there will always be an even number of ones in total, and 9 is not even.

To add another ingredient into this mix I reference , which illustrates how the Kochen-Specker result is an aspect of the 24-cell. The 24-cell has a number of representations. The full representation is the F_4 group with 1154 Hurwitz quaternions. The other is the B_4, which is the 16 cell Plus an 8-cell, and the other is D_4 which is three 8-cells. The more general automorphism is then F_4. The quotient between the 52 dimensional F_4 and the 36 dimensional so(9) ~ B_4 defines the short exact sequence

F_4/B_4:1 --> spin(9) --> F_{52\16} --> {\cal O}P^2 --> 1,

where F_{52\16} means F_4 restricted to 36 dimensions, which are the kernel of the map to the 16 dimensional Moufang or Cayley plane OP^2. The occurrence of 36 and 9 is no accident, and this is equivalent to the structure used to prove the KS theorem.

F_4 is the isometry group of the projective plane over the octonions. There are extensions to this where the bi-ocotonions CxO have the isometry group E_6, HxO has E_7 and OxO has E_8. This forms the basis of the "magic square." F_4 plays a prominent role in the bi-octonions, which is J^3(O) or the Jordan algebra as the automorphism which preserves the determinant of the Jordan matrix

The exceptional group G_2 is the automorphism on O, or equivalently that F_4xG_2 defines a centralizer on E_8. The fibration G_2 --> S^7 is completed with SO(8), where the three O's satisfy the triality condition in SO(8). The G_2 fixes a vector basis in S^7 according to the triality condition on vectors V \in J^3(O) and spinors θ in O, t:Vxθ_1xθ_2 --> R. The triality group is spin(8) and a subgroup spin(7) will fix a vector in V and a spinor in θ_1. To fix a vector in spin(7) the transitive action of spin(7) on the 7-sphere with spin(7)/G_2 = S^7 with dimensions

dim(G_2) = dim(spin(7)) - dim(S^7) = 21 - 7 = 14.

The G_2 group in a sense fixes a frame on the octonions, and has features similar to a gauge group. The double covering so(O) ~= so(8) and the inclusion g_2 \subset spin(8) determines the homomorphism g_2 hook--> spin(8) --> so(O). The 1-1 inclusion of g_2 in so(O) maps a 14 dimensional group into a 28 dimensional group. This construction is remarkably similar to the moduli space construction of Duff et al. .

Cheers LC

Dear Michel,

I read your polished essay rather too quickly soon realizing its technicalities were beyond my understanding. As an artist I was fascinated by the concept of Dessin d'Enfent, but it soon became clear it was some sort of variant of network theory (?) - perhaps a causality map (?). It needs more study.

More importantly I feel that you base your paper on 'standard' quantum philosophy - that probability is at the base of everything, and that knowing Nature is observer-related. I and many other sans-culotte feel that these are derivative phenomena - that there is an absolute universe that explains all these phenomena without the 'weirdness' that has become the hallmark of the field. It is a long discussion, but my incomplete and qualitative Beautiful Universe Theory will explain why I have responded as I did to your paper.

With best wishes

Vladimir

Rdposted from my area

Michel,

I don't have as much time this morning to expand on this, so I will just make this rather brief for now. I will try to expand on this later today or tomorrow.

The three-qubit entanglement corresponds to a BPS black hole. The four qubit entanglement is the case of an extremal black hole. I think there is an underlying relationship between functions of the form (ψ|ψ) = F(ψψψ), an elliptic curve with the cubic form corresponding to the 3-qubit, and the "bounding" Jacobian curve that defines a quartic for G(ψψψψ). This I think is some sort of cohomology.

The G2 I think defines a frame bundle on the E8 which defines the F4 condition for 18 rays in the spacetime version of Kochen-Specker.

As I said I should have more time later to discuss this in greater depth.

Cheers LC

Dear Michael,

Like Philips I was not familiar with the relationships around dessins d'enfants so also for me it is very enlightening. And I need time to understand it. This is one of the advantages of participation in the contest. I do not feel competent to comment all essays and not all of them are worth commenting.

Nice to learn something new and interesting.

My essay is much simpler and short.

Best regards

Thank you Michael,

I will keep checking Arxiv.

Regards

Cher Michel Planat,

Thank you for an interesting suggestion. Are you saying that this formalism will help us understand the difference between quantum and classical bounds of the Bell inequality, and if yes, then how?

Best regards,

Alexei Grinbaum

Dear Michel,

I returned from vacation, and left a reply to your comment on my essay's page.

You presented beautiful and surprising connections between dessins d'enfants and quantum observables, building on your 2004 conjecture suggesting a connection between the existence of mutually unbiased bases and the existence of projective planes. I understand from your reply to Jochen Szangolies's comment, that you "still do not fully understand the precise connection between Grothendieck's dessins and the finite geometries underlying the compatibility observables". With this in mind, do you have a geometric/topological interpretation of the Riemann surfaces arising from Grothendieck's dessins d'enfants? Are there possible configurations of quantum observables corresponding to higher dimensional varieties?

I look forward to see your forthcomming papers on this subject.

You may be interested in Florin Moldoveanu's approach to quantum mechanics, using the Grothendieck group.

Best regards,

Cristi Stoica

Dear Dr. Michel,

I have down loaded your essay and soon post my comments on it. Meanwhile, please, go through my essay and post your comments.

Regards and good luck in the contest.

Sreenath BN.

http://fqxi.org/community/forum/topic/1827

Dear

Thank you for presenting your nice essay. I saw the abstract and will post my comments soon.

So you can produce material from your thinking. . . .

I am requesting you to go through my essay also. And I take this opportunity to say, to come to reality and base your arguments on experimental results.

I failed mainly because I worked against the main stream. The main stream community people want magic from science instead of realty especially in the subject of cosmology. We all know well that cosmology is a subject where speculations rule.

Hope to get your comments even directly to my mail ID also. . . .

Best

=snp

snp.gupta@gmail.com

http://vaksdynamicuniversemodel.blogspot.com/

Pdf download:

http://fqxi.org/community/forum/topic/essay-download/1607/__details/Gupta_Vak_FQXi_TABLE_REF_Fi.pdf

Part of abstract:

- -Material objects are more fundamental- - is being proposed in this paper; It is well known that there is no mental experiment, which produced material. . . Similarly creation of matter from empty space as required in Steady State theory or in Bigbang is another such problem in the Cosmological counterpart. . . . In this paper we will see about CMB, how it is generated from stars and Galaxies around us. And here we show that NO Microwave background radiation was detected till now after excluding radiation from Stars and Galaxies. . . .

Some complements from FQXi community. . . . .

A

Anton Lorenz Vrba wrote on May. 4, 2013 @ 13:43 GMT

....... I do love your last two sentences - that is why I am coming back.

Author Satyavarapu Naga Parameswara Gupta replied on May. 6, 2013 @ 09:24 GMT

. . . . We should use our minds to down to earth realistic thinking. There is no point in wasting our brains in total imagination which are never realities. It is something like showing, mixing of cartoon characters with normal people in movies or people entering into Game-space in virtual reality games or Firing antimatter into a black hole!!!. It is sheer a madness of such concepts going on in many fields like science, mathematics, computer IT etc. . . .

B.

Francis V wrote on May. 11, 2013 @ 02:05 GMT

Well-presented argument about the absence of any explosion for a relic frequency to occur and the detail on collection of temperature data......

C

Robert Bennett wrote on May. 14, 2013 @ 18:26 GMT

"Material objects are more fundamental"..... in other words "IT from Bit" is true.

Author Satyavarapu Naga Parameswara Gupta replied on May. 14, 2013 @ 22:53 GMT

1. It is well known that there is no mental experiment, which produced material.

2. John Wheeler did not produce material from information.

3. Information describes material properties. But a mere description of material properties does not produce material.

4. There are Gods, Wizards, and Magicians, allegedly produced material from nowhere. But will that be a scientific experiment?

D

Hoang cao Hai wrote on Jun. 16, 2013 @ 16:22 GMT

It from bit - where are bit come from?

Author Satyavarapu Naga Parameswara Gupta replied on Jun. 17, 2013 @ 06:10 GMT

....And your question is like asking, -- which is first? Egg or Hen?-- in other words Matter is first or Information is first? Is that so? In reality there is no way that Matter comes from information.

Matter is another form of Energy. Matter cannot be created from nothing. Any type of vacuum cannot produce matter. Matter is another form of energy. Energy is having many forms: Mechanical, Electrical, Heat, Magnetic and so on..

E

Antony Ryan wrote on Jun. 23, 2013 @ 22:08 GMT

.....Either way your abstract argument based empirical evidence is strong given that "a mere description of material properties does not produce material". While of course materials do give information.

I think you deserve a place in the final based on this alone. Concise - simple - but undeniable.

Hello, Michel,

Thank you for this interesting essay. As you will see from mine, you are one of the people I critique as overweighting geometry at the expense of energy. The logic of Grothendieck, in my humble opinion, is not the dynamic logic of the universe. I hope we may communicate on this point.

Best regards,

Joseph

Dear Dr. Michel,

Your essay is highly original and intriguing but at the same time it appears as if it is written for the experts in the field but not keeping general audience in the perspective. It is interesting to know how far the different geometric methods, you have followed in this article, are capable of solving other problems prevailing in QM. I congratulate you for producing such an innovative essay.

Sreenath

Dear Dr. Michel,

I appreciate your kind comments. It is good to learn that we share some common basic views regarding the existence of knowledge.

Best regards,

Sreenath

Dear Michal,

Thank you once again for the questions you asked me on my essay. If you visit the FQXi page ,( at the beginning of the page)

http://fqxi.org/community/essay

.............................................

I. GOALS & INTENT

The goals of the Foundational Questions Institute's Essay Contest (the "Contest") are to:

^ Encourage and support rigorous, innovative, and influential thinking about foundational questions in physics and cosmology;

.............................................

They used a word 'innovative', that may mean they want more fundamental thinking and may not be a report on current research prepared for discovery channel viewers...

Best

=snp

Michal

nice essay and interesting ideas, even though I need more knowledge about your math, to understand the technical derivations. I like the one about CHSH.

Best wishes

Mauro

Dear Michel Planat,

Having read your very interesting paper twice, I concluded that you would probably have little interest in mine. But after reading your comment on Stewart Heinrich's essay expressing your interest in the concept of self-awareness and the miraculous efficiency of mathematics for mimicking physical problems, both of which I address in my essay, I decided to invite you to read it and comment. I think it has little connection to your essay yet you may find a new perspective on these two topics.

Thank you for participating in this contest and good luck in the contest.

Best regards,

Edwin Eugene Klingman

Michel,

If given the time and the wits to evaluate over 120 more entries, I have a month to try. My seemingly whimsical title, "It's good to be the king," is serious about our subject.

Jim

Dear Dr. Michel Planat

Thanks and congratulations for your poetic qubits.

I am mere a learner of physics here who have huge interests to know the fundamentals in nature. I think that to resolve the issues like "quantum non-locality and contextuality' in your essay, why not we ask the nature in different ways? I invite you to read my submitted essay for a quite new approach of asking the nature.

With my regards

Dipak

Dear Michel,

Thank you for reading my essay and commenting on it. I have had a chance to review the two papers you referenced there. The Riemann paper discusses the details of a specific partition function, which I find interesting as I base the applicability of the Born probability to my wave function model on the partition function. The other paper, on time perception is also interesting. I had not seen the Poincare discussion of the Continuum, and found that fascinating, as well as your connection. I am somewhat confused as to whether you are proposing the phase locking as the 'mechanism' of time perception or of the 'scaling' of time perception? I can understand how this could relate to scaling, but not perception as I understand it.

Thanks again for reading my essay. I hope it stimulates some ideas for you.

Best,

Edwin Eugene Klingman

Dear Michel,

thank you for stopping by to comment on my essay. I was very much intrigued by your work and remember it from last year. It appears your presentation is more technical this year. I wish your interesting work reserves recognition is deserves among the specialists.

Best of luck :)