Essay Abstract

Wheeler's {\it observer-participancy} and the related {\it it from bit} credo refer to quantum non-locality and contextuality. The mystery of these concepts slightly starts unveiling if one encodes the (in)compatibilities between qubit observables in the relevant finite geometries. The main objective of this treatise is to outline another conceptual step forward by employing Grothendieck's {\it dessins d'enfants} to reveal the topological and (non)algebraic machinery underlying the measurement acts and their information content.

Author Bio

Michel Planat is a senior scientist at FEMTO-ST/CNRS, Besançon, France. His present main interest is in fundamental problems of quantum information and their relationship to mathematics. He wrote about 110 refereed papers or book chapters.

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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.

    Joe,

    Thank you for reading it. I also red yours. Yes, everything in the real world is unique and quantum mechanics tell us 'no-cloning'. But in my opinion physics can just explain the how, not the fine details of the existence. A step in the direction of explaining contexts is my approach through "dessins d'enfants' that drive the compatibilities of observables.

    Michel

    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

    Dear Philip,

    I am glad that you learned something. Apart from the Kochen-Specker theorem we have a few common interests: symmetries, the black-hole qubit analogy and number theory. May be you can have a look at my papers (e.g.in google scholar) containing the title 'Dedekind psi function'. Thank you again.

    Michel

      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 Jochen,

      Thank you for your post. I am familiar with your interesting papers on contextuality as well.

      Concerning your main remark, I still do not fully understand the precise connection between Grothendieck's dessins and the finite geometries underlying the compatibility observables - Mermin's pentagram is one of the simplest objects displaying contextuality but there is more to come.

      I agree that the black-hole/qubit correspondance is a toy model, I like it due to its link to string theory. I suggest you have a look at the work of my colleagues Peter Levay and Metod Saniga on this topic

      http://xxx.lanl.gov/abs/0808.3849

      An important object is the split Cayley hexagon that has 12096 automorphisms as the number of three-qubit pentagrams.

      My best regards,

      Michel

      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

      Dear Lawrence,

      I am delighted to read your opinion because I have liked your own essay and also thought at a connection between 'dessins d'enfants' and quantum logic. I intend to write you later on your topic. Now I am waiting for your extra remarks.

      Unfortunately, at this stage, I am not able to say anything relevant about the connection between algebraic curves and the Langland's program, this is a difficult mathematical question and I am more focused on the occurence of algebraic curves in the realm of quantum physics. Of course Grothendieck had these questions in mind.

      Cheers,

      Michel

        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

        You comments are very relevant and I should think more on them.

        In my comments on your essay, I mention that G2 and thus the octonions relate to the contextuality for three qubits http://xxx.lanl.gov/abs/1212.2729

        Yes, you may know this webpage

        http://www.math.jussieu.fr/~leila/grothendieckcircle/biographic.php

          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 Hoang

          Thank you, I will look at your essay.

          Also Poincaré wrote in 1905 in "Science and hypothesis"

          The fundamental propositions of geometry, for instance, Euclid's

          postulate, are only conventions, and it is quite as unreasonable

          to ask if they are true or false as to ask if the metric system is true or false. Only, these conventions are convenient, and there are certain experiments which

          prove it to us.

          Michel

          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