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

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

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