Yes, the entangled qubit interpretation is an approximation to a deeper structure, as the full Freudenthal triple system has more degrees of freedom. For the interpretation to work, even at the 3-qubit level, one is no longer working over the real forms of the FTS, but a complexification thereof. For M-theory compactification on the 7-torus, D=4, N=8 SUGRA has E_7(7) symmetry, while for homogeneous D=4, N=2 SUGRA one recovers E_7(-25). Upon complexification of the two relevant FTS for these cases, one recovers a single FTS with E_7(C) symmetry. The single (complex) variable extension of this FTS is a 57-dimensional structure with nonlinear E_8(C) symmetry.

As of yet, there is no interpretation for the fully complexified FTS with E_7(C) symmetry. It would describe some generalization of the SUGRA systems where the charges are complex valued. Moreover, when the FTS has diagonalized electric and magnetic components, the FTS resembles a 3-qubit system. The complexified FTS, as a bonus, also gives the hermitian symmetric domain that generalizes the non-compact E_7 Riemannian symmetric spaces used in the D=4 SUGRAs.

What are the references for E_{7(7)} and E_{7(-25)} with N = 8 and 4 SUGRA? Are there other compactificiation schemes besides the 7-torus? For instance are there schemes with S^2xK3^2 or some other scheme with CY manifolds?

For K = R, C, H, O we can decompose h(K) into

h^n(K) = RвЉ•h^{n-1}(K)вЉ•K^{n-1}

following Baez in his "Octonions" so we form a type of matrix with diagonals in h_{n-1} and off diagonals in K^{n-1} that comprises the spin factor RвЉ•K^{n-1}. In this isomorphism we may assign K^{n} into a partition K^{n,n+1}. In this way we have one field П€ \in h^{n-1} and another П† \in K^{n-1} forming the diagonals and off diagonals respectively. In this way the projective Fano plane OP^2 and its "Poincare-like dual" as a line in the heavenly sphere OP^1 form a CP^3 with the J^3(O) = h^3(O) construction.

We can then in this way proceed with K'вЉ--K, where for K' = H and K = O we have a manner by which one can describe J^3(O)вЉ--G, of octonion basis with a gauge group action. In this way we can have E_8(C) and E_8(H), where the latter is a quaternionic extension of the FTS. I attach a little note here that illustrates in an elementary manner how quaternions naturally give rise to gauge theory. I have another one of these notes that includes the Dirac field, and how quantum mechanics has in some ways a more natural expression as quaternions. Stephen Adler went into this some.

I was going to write more, but this morning has me rather tightly scheduled.

LC

For SUGRA references on non-compact real forms of E_7 see references [29]-[32] in my essay. The E_7(7) case is a 7-torus compactification while I'm not certain about the other compactification manifold for E_7(-25).

The 56-representations correspond to the FTS Lie algebra gradings of E_7. There are other gradings, however, that have a spin factor-like structure, such as:

e_7(-25) = 1 32 so(2,10) R 32* 1

This grading suggests a (2,10) spacetime signature with 32 spinor. An E_8 grading that generalizes this is:

e_8(-24) = 14 64 so(3,11) R 64* 14

Here one sees a (3,11) spacetime signature with 64 spinor. This grading is manifest in Lisi's E_8 model. Morever, it has been shown by Sezgin there exists a super Yang-Mills theory with (3,11) signature.

    It appears that the E_{7(-25)} and E_{8(24)} could serve as the global groups G_4 and G_3 fior 4, 3 qubits with SO(2,10) and SO(3,11) as the local gauge-like transformations with

    G_4/H_4 = E_{7(-25)}/(E_{6(-78)}xU(1))

    G_3/H_3 = E_{8(-24)}/(E_{7(-25)}xSL(2,C))

    with E__{6(-78)} the maximal compact subgroup or as E_{7(-25)}/SO(2,10) with a further decomposition. The \bf 56 of the E_{7(-25)} in the 4-qubit correspond to 28 = 1 graviton plus 27 gauge potentials and 28 = 1-NUT parameter gravi-like boson plus 27 magnetic dual gauge potentials. The intertwining of the E_{7(-25)} as the global and local groups for the 4 and 3 qubit case seems interesting.

    An additional feature that might be of interest is that the decompositions you cite have even orthogonal Lie groups, and this would fit within my idea of there being a Bott-like periodicity with the exceptional cases. The open question is still whether the nonvanishing commutation of the spinors can be accounted for in some general K-S theorem. It might be that the nonvansishing of these commutators is a blessing in disguise, for it could mean there is a massive reduction in the number of fundamental degrees of freedom.. This restriction might recover something like the K-S theorem. I am on travel this week and into next, so I am a bit constrained with time right now.

    In doing all of this, I tend to like to also focus on the physical aspects of these things. I am not the greatest master of group theory, especially large Lie groups and sporadic groups. I think there is some sort of phase change that occurs at the horizon of black holes for accelerated frames. There is a phase change in analogue with symmetry protected states and the phase change from conducting phase to a Mott insulator phase.

    Cheers LC

    The qubit interpretation is only valid in the complex case (e.g., E_7(C)). Otherwise, one has entangled real qubits i.e., rebits. The complex case has much more degrees of freedom than the entangled qubit picture, however. In other words, there is a deeper quantum information interpretation here, more similar to dyonic topological quantum computation. These systems are far more general than any existing quantum computational framework. One must define new systems to make proper sense of them.

      Dear Michael,

      Grothendieck's motives " may serve as an unreasonably effective tool leading to a unified theory of physics and mathematics". This is a very ambitious but credible hypothesis that can at least lead to new conceptual bridges between maths and physics. You have the great merit to introduce the topic in a short non technical essay with the relevant references. I would love to understand quantum field theory in terms of the associahedron and related structures. "Locality and unitary are emergent features": does it mean that non-locality which is specific to quantum theory is not part of the picture?

      A few topics you introduced are also in my essay of this year that you may find useful to read. It makes a big use of Grothendieck's dessins d'enfants. Trying to see the relationship to motives, I found "Gauge Theories and Dessins d'Enfants: Beyond the Torus" by S. Bose et al. May be there are other references you may mention to me.

      I wish you the best for your enjoyable essay.

      Michel

        Hello Michel

        I'm glad you find these topics of interest.

        Please read:

        Positive Amplitudes In The Amplituhedron

        Nima Arkani-Hamed, Andrew Hodges, Jaroslav Trnka

        arXiv:1412.8478 [hep-th]

        Emergent spacetime from modular motives

        Rolf Schimmrigk

        arXiv:0812.4450 [hep-th]

        From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group

        Robert de Mello Koch, Sanjaye Ramgoolam

        arXiv:1002.1634 [hep-th]

        Is quantum information meaningful in the quaternionic case, such as E_7(H), or OxH? I am by default thinking of the exceptional groups as complex or quaterionic.

        The "mod" on the K-S theorem might have something to do with Hermitean domains. As I look into this it appears that this might be a more general way of looking at the Cartan theory of decompositions of Lie groups. If this could be made to work in the exceptional domain that would be interesting.

        LC

        I have not looked into E_7(H) but I do have a quantum information interpretation for [math]\mathbb{H}\otimes\mathbb{O}[/math] and [math]\mathbb{O}\otimes\mathbb{O}[/math].

        I suppose I would like to have some reference for that. I have certain ideas along these lines. The one limitation I have is that I am not that highly knowledgeable on these large groups, though I know some of this.

        Cheers LC

        There is nothing formally written yet. However, I will say the interpretation involves Jordan quantum mechanics over the quater-octonions and octo-octonions. One can then define FTS-like structures over these Jordan *-algebras to recover symmetries akin to E_7(H) and E_7(O).

        It is possible to compute (anti)hermitean eigenvalues with the FTS.

        I presume by *-Jordan QM you mean a C* version of the FTS as some generalization of a quantum group. The quantum group properties are then generalized into this form.

        LC

        I mean the off-diagonal elements of the FTS, which are Jordan C*-Algebra matrices up to the bioctonions. Beyond that, they are *-algebras of Jordan type.

        • [deleted]

        • Post #35

        The bioctonion plane has E_6 symmetry, which is is a decent GUT type of theory. E_6 embeds SO(10). For gravitation one might have to go to quateroctonions HâŠ--O with E_7 symmetry or ocyo-octonions with E_8 symmetry and dimension = 128. The 128 dimensions, the last in the secquence of O, RâŠ--O, CâŠ--O HâŠ--O with dim(P^2(KâŠ--O) ) = 16, 32, 64, 128. In this way one can get the G_3/H_3 and G_4/H_4 coset constructions.

        The pairing up of division algebras K' ,K permits the definition of a Lie algebra with derivations (derivatives) such der(K')âŠ--J3(K)⊕(K'_tf)âŠ--J3(Y), where tf means tracefree. Is it then possible with this sort of gadget to construct E_{7(-25} and E_{8(24)}? If so then maybe the qubit SLOCC coset construction can go through.

        I think that Hermitian symmetry forms might be a way of appealing to the Cartan decompositions in the Kostant-Sekiguchi theorem. I am not sure how this Tits construction of the Freudenthal triple system can facilitate that.

        LC

        • [deleted]

        • Post #36

        It is possible. In noncommutative and nonassociative geometry, one uses derivations of the algebra to define vector fields and differential forms. Your decomposition for division algebra pairs, follows from the definition of a derivation for the observable algebra over that pair.

        For example, over the quater-octonions, we have an observable agebra derivation of dimension 5 x dim tf(J(3,O))+dim(Aut(H))=5x26+3=133. This gives the Lie algebra for E_7(-25).

        For the octo-octonions, we have 9 x dim tf(J(3,O))+dim(Aut(O))=9x26+14=248. This gives the Lie algebra for E_8(-24).

        I discovered these derivations long ago, as generalizations of the E_6(-26) derivations given by P. Ramond and S. Cato. I presume the E_6(-26) derivation was introduced to them by F. Gürsey. Such derivations make use of the associator over J(3,O). This is why one can rightfully refer to the geometry as nonassociative, as it is built from a nonassociative structure algebra.

        Hi again,

        I had an earlier post, now my essay has been posted. I know it is probably not to your liking. But I think it will give you more confidence in your own system.

        Essay

        Thanks and good luck.

          This is application of the Freudenthal-Tits der(K)вЉ--J3(O)вЉ•(K_tf)вЉ--J3(O)_tf for K = C would then I presume be dim(Der(C)вЉ--J3(O) + aut(C)) = 3в€™26 + 0 = 78 corresponding to E_{6(-26)}.

          If this is the case then it seems we should be able to work out the geometry of 3 and 4 qubits according to cobordism or Morse theory. My idea is that the Kostant-Sekiguchi theorem has a Morse index interpretation. The nilpotent orbits N on an algebra g = h + k, according to Cartan's decomposition with [h,h] вЉ‚ h, [h,k] вЉ‚ k, [k,k] вЉ‚ h

          N∩G/g = N∩K/k.

          For map Ој:P(H) --- > k on P(H) the projective Hilbert space. The differential dОј = = П‰(V, V') is a symplectic form. The variation of ||Ој||^2 is given by a Hessian that is topologically a Morse index. The maximal entanglement corresponds to the ind(Ој).

          In general orbit spaces are group or algebraic quotients. Given C = G valued connections and A = automorphism of G the moduli or orbit space is B = C/A. The moduli space is the collection of self or anti-self dual orbits M = {∇ \in B: self (anti-self) dual}. The moduli space for gauge theory or a quaternion bundle in 4-dimensions is SO(5)/SO(4), or for the hyperbolic case SO(4,1)/SO(3,1) = AdS_5. The Uhlenbeck-Donaldson result for the hyperbolic case is essentially a form of the Maldacena duality between gravity and gauge field.

          In computing the topologies of four manifolds the important quantity evaluated is the intersection form. This may be realized with cobordism of 3-spaces in the 4-dim space. This computes across, admittedly I am being a bit glib here, across the Morse indices.

          Physically the idea is that entanglement is itself uncertain in quantum gravity. The fluctuations of spacetime with certain topologies correspond to different entanglement geometries. Consequently the entanglement of states with a quantum black hole are themselves uncertain. This might be one route to looking at this problem called the firewall.

          Cheers LC

          Ah yes, firewalls and the black hole information paradox. At the level of E8, transformations of quantum information are quasiconformal. They preserve a 57-dimensional non-linear norm form. This is far from the usual linear unitary transformations one usually considers over qubits.