The 24-cell and 4 qubits

A few days ago Lubos reported on an intriguing new paper by Volker Braun describing how to construct a Calabi-Yau manifold with 6 real dimensions and minimal Hodge numbers using the 24-cell. Such manifolds can be applied to the compactification of superstring theory down to our familiar 4 dimensional spacetime. The predictions for physics based on this particular manifold would be unrealistic but its discovery is an important step towards understanding the fuller range of possibilities. It is also of considerable mathematical interest in its own right.

I’m not going to say anything more about that paper but I do want to say something about the 24-cell and its curious relation to 4 qubits as well as a surprising relationship between the invariants of 4 qubits and the platonic solids. I found out about these things after talking to Mike Duff and his coworkers on the qubit/black-hole correspondence (Borsten, Dahanayke, Duff,  Marrani, Rubens). I think this is a bit more specialized than the kind of stuff I usually report on here, but some of our regular commenters expressed an interest and I am always happy to oblige. For anyone who does not understand any of the mathematical terms here they are all explained in wikipedia.

The 24-cell is a very special regular polytope in 4-dimensional space. It has the special property of being self dual in the same sense as the tetrahedron is self-dual in 3 dimensions. It can also be tessellated to fill 4 dimensional space just as a cube can tessellate  to fill 3d space. in fact the 24-cell is the only regular polytope in more than 2 dimensions that has both of these properties. The only comparable shapes in this sense are the triangle, square and hexagon in two dimensions.

The vertices of the 24-cell can be plotted in 4D co-ordinates at the 24 points given by

(±1,0,0,0), (0,±1,0,0),(0,0,±1,0),(0,0,0,±1),(±½,±½,±½,±½)

It’s dual can be plotted at the points,

(±1,±1,0,0),(±1,0,±1,0),(±1,0,0,±1),(0,±1,±1,0),(0,±1,0,±1),(0,0,±1,±1)

As many readers of viXra Log are undoubtedly aware, there are many connected mysteries surrounding the number 24 in mathematics and the 24-cell is one of the more enigmatic. 24 is famous as the dimension of the Leech lattice which is connected to the significance of the number in the theory of finite simple groups, especially examples such as the Mathieu groups, the Conway groups and the Fisher groups. The existence of the Leech lattice can be explained in terms of the 24-bit Golay code which can in turn be constucted using special properties of quadratic residues in Z24. Alternatively the Leech lattice is a reduction of an alternating lattice in 25+1 dimensions using a null vector relying on the fact that  the sum of the first 24 square numbers is 70^2. This closely connects together one set of circumstances where the number 24 appears in mathematics.

Then there is a second interlinked set of places where the number 24 shows up in number theory and the theory of special functions. This includes the  Ramanujan discriminant function, a modular form that is the 24th power of the Dedekind eta function. This can be connected to the fact that the value of zeta(-1) is 1/12. It has implications in bosonic string theory where it is linked to the critical dimension where the two dimensional worldsheet vibrates in the remaining 24 dimensions.

These two sets of places where the number 24 appears, one in group theory and the other in number theory, do not seem to have a causal link. You cannot reason that one implies the other. Yet you can combine the two by compactifying bosonic string theory over the Leech lattice. This was a realisation that led to the famous proof of the monstrous moonshine conjectures and a Fields medal for Richard Borcherds. This much will be familiar to anyone who follows related discussions on the internet and especially if they have read John Baez’s lecture on the number 24. As far as I know there is nothing else that clarifies the mystery of this connection. For example there is nothing that directly links the Golay Code to the Ramanujan Discriminant function except Moonshine.


What about the 24 cell, where does that fit in? According to Baez it relates to the number theory side via elliptic curves, but in a way that involves a group of order 24. Perhaps this is a clue to a missing direct link between between number theory and group theory. The connection described by Baez points to the fact that the moduli space of elliptic curves is given by modding out the group SL(2,3). The vertices of the 24-cell when plotted as quaternions (Hurwitz quaternions)  also form a group, and it is the same one, also known as the ditetrahedral group because it is the double cover of the rotation group of the tetrahedron. This seems very nice but actually there are only 15 groups of order 24 and only seven that are not direct products of smaller groups, so saying that two structures form the same group of order 24 is only a small factor better than saying that they have the same size. What we really need to find is a more direct way in which the 24-cell relates to elliptic curves.

This is where the 4-qubit system comes in. The wavefunction of 4 qubits is represented by a 2x2x2x2 hypermatrix of 16 complex numbers. Local transformation on these qubits take the form of SL(2,C) transformations applied to each qubit independently so the overall symmetry group of the system is SL(2,C) 4 . To understand the entanglement possibilities for 4 qubits the first step is to find the polynomial invariants under this group. This is a non-trivial computation but  it can be shown that there are 4 independent invariants of degree 2, 4, 4 and 6 in the 16 components of the hypermatrix (see e.g. http://arxiv.org/abs/quant-ph/0212069 for a construction.) However, there is a special invariant that is a combination of these known as the hyperdeterminant which is of degree 24. The hyperdeterminant is a discriminant for the hypermatrix that is zero iff the quadriliear form constructed from the hypermatrix has singular points where all derivatives vanish. You don’t have to understand the details, just notice that this is another structure where the number 24 has special significance.

It turns out that the 4 qubit hypermatrix is related in a fundamental way to elliptic curves with the hyperdeterminant being related to the Ramanujan discriminant modular form mentioned above. I have described this relationship in detail at http://arxiv.org/abs/1010.4219 so I won’t repeat it here. This makes a direct link between the number 24 that appears as the degree of the hyperdeterminant and its appearance in the theory of modular forms linked to bosonic string theory. After discussing this with Mike Duff I was also able to link the 4 qubit system directly to bosonic strings and I used this in my FQXi essay.

The classification of 4-qubit entanglement is a tricky business. The SL(2,C) 4 transformation group has 12 independent paprameters so it should be possible to use these transformations to reduce any state with its 16 components to representative states parameterised by just 16-12=4 variables.  A clean solution was provided by Verstraete et al in http://arxiv.org/abs/quant-ph/0109033 . They found nine perameterised classes of states where the largest class known as Gabcd has 4 parameters and includes all states whose hyperdeterminant is non-zero. For present purposes I am only interested in this class. It takes a form that can be written in qubit terms as

Φ =  x (|0000> +|1111>) + y (|0011>+|1100>) + z (|0110> + |1001>) + t (|1010> + |0101>)

For this class of states, we can work out any of the invariants including the hyperdeterminant which is going to be a polynomial of degree 24 in the four variables xyz and t. This has the potential to be a complicated expression, after all the full hyperdeterminant in 16 variables is an expression with 2894276 terms. In practice for the reduced state the hyperdeterminant simplifies and when you work it out you will notice that the result factorised into 24 simple factors

Det(Φ) = x2y2z2t2(x+y+z+t)2(x+y+z-t)2(x+y-z+t)2(x+y-z-t)2(x-y+z+t)2(x-y+z-t)2(x-y-z+t)2(x-y-z-t)2

These factors correspond in an obvious way to the Hurwitz quaternions and therefore the vertices of the 24-cell. This provides a direct link between the number of vertices in the 24-cell and the degree of the hyperdeterminant for 4-qubits which in turn is linked to the exponents in modular forms and the critical dimension of bosonic string theory, just as we wanted.

Is there a better way to understand why the hyperdeterminant factorises so conveniently? Yes there is. Although all the 12 dimensions of the group SL(2,C) 4 were used to reduce the hypermatrix to the class Gabcd , there remains a discrete subgroup that maps states of Gabcd , (in the form above) onto themselves, so this discrete subgroup provides a group of linear transformations on the 4D space parameterised by xy, z and t. This subgroup turns out to be the Weyl group of D4 whose system of root vectors is the 24-cell. The polynomial invariants of this reflection group as functions of the four parameters xyz and t are also of degree 2, 4, 4 and 6 and correspond to the 4 qubit invariants. The Weyl group is the symmetry group of the root system so it just permutes the 24 factors in the hyperdeterminant making it an obvious invariant. Notice that D4 as a Lie-algebra is SO(8) or its split form SO(4,4) which is the group used to construct the 4-qubit/black-hole correspondence. This was what Borsten et al used to classify the entanglement of four qubits using a classification of nilpotent orbits that had already been worked out for black holes in M-theory. Their answer matches the Verstraete classification nicely.

We can go one step further and extend the group of transformations to include permutations of the 4 qubits. This gives a larger discrete group acting on Gabcd ,which can be identified as the Weyl group of F4. The corresponding root system is now the 48 vertices of a 24-cell combined with its dual. The polynomial invariants of this group are of degree 2,6,8 and 12 and they correspond to the primitive invariants of the hypercube that is symmetric under the permutations of the qubits as well as the usual  SL(2,C) 4 transformations.

This leads to one last curious correspondence that I want to point out. The degrees of the primitive invariants of the hypercube (2,4,4,6) are not trivial to work out, but can you see what they are related to? Think of a tetrahedron with 4 vertices, 6 edges and 4 faces. The remaining number 2 corresponds to the inside and outside of the tetrahedron which can be regarded as the two three dimensional parts which cover space in combination with the vertices, edges and faces. Remember that the 24-cell as a group is the double cover of the rotation group of the tetrahedron so there is a connection. For the invariants that are symmetric under permutations, the larger root system of 48 vectors from the two 24-cells combined also forms a group when the root vectors are regarded as unit quaternions. The full set of unit quaternions form the group SU(2) which is the double cover of SO(3) so any finite subgroup must be the double cover of some rotation group in 3D. In this case it is the rotation group of the cube or octahedron. This corresponds to the fact that the symmetric invariants for four qubits are of degrees (2,6,8,12) because the cube has 6 faces, 8 vertices and 12 edges (or you can use the dual octahedron).

So despite the fact that the invariants of the 4 qubit system are non-trivial to construct, their polynomial degrees correspond to the geometric elements of three of the platonic solids. What about the other two regular solids, the dodecahedron and icosahedron? There is another reflection group H4 whose root system corresponds to these solids and it therefore has invariants of degree (2,12,20,30) . Since this group acts on the same 4D sapce you can use it to construct four invariants of the 4 qubit system with these degrees, but there is no lie algebra corresponding to H4 and its significance is not so obvious. However, these three cases are part of a system of mysterious “trinities” as noted by the mathematician Vladimir Arnold. This means that there must be a lot more going on that we don’t really understand yet.


27 Responses to The 24-cell and 4 qubits

  1. Alejandro Rivero says:

    Also, are the faces of the 24-cell useful for some construction? It is interesting that they are 96, and so it gives entry to some imaginative, Lisi-like, constructs, putting the 96 for quarks and leptons and the 24 for gauges.

    • Philip Gibbs says:

      I think you are asking if D4 or F4 could be used as an interesting gauge group. I think on their own they are not big enough, but the triality makes them interesting to look at. I think Lisi uses F4 within E8 for something but I have not really looked at his model in enough detail. Tony Smith includes them in his model (there is not much he misses out)

  2. Alejandro Rivero says:

    Ah, it is not well known, but the replacement of the positive sum by the alternating sum allows to find a family of solutions to the 1^2-2^2+3^2-4^2… Diophantine equation, the smallest of it happens for 8 instead of 24.

    • Philip Gibbs says:

      That’s interesting, I didn’t know it. It’s tempting to think that it would relate to E8 and superstring theories in an analogous way to the Leech lattice and bosonic string theory but I don’t see how that would work.

      • Alejandro Rivero says:

        Yeah, if you don’t know it, it is not well known :-D. And, actually, the alternating equation is easier to solve, so perhaps it is not mentioned because of it. I just got the idea and checked, I would be happy if someone can provide some reference.

        An alternating series also appears with superstrings when you try to calculate the normal ordering constant in the NS sector. Then instead of the Zeta function you get Dirichlet Eta function, because you must to add bosons and fermions with different sign; the fermions only contribute to the odd numbers but there is an extra factor two going around, and the total sum becomes the infinite alternating sum. And of course in s=-1 Eta and Zeta differ in a factor 3, so 1/12 becomes 1/4 and you could argue that it fixes the dimension of the superstring, but I have never seen the argument done in this way.

        The whole thing is amusing because number theoreticians mechanically substitute Riemann Zeta by Dirichlet Eta as a routine: it has the same zeros but not the pole in s=1, and then better analiticity that Zeta.

      • Philip Gibbs says:

        The alternating series gives the triangle numbers which increases in size quadratically so there will be an infinite number of values for which it is a square. They can be generated recursively. The straight sum increases as a cubic polynomial so you can use the theory of elliptic curves to show that it has a unique solution.

  3. Luboš Motl says:

    Dear Phil, it’s hard to read your text because you often seem to worship trivial numerology. The number 24=4! is way too simple for anyone sensible to be surprised that it appears at many places of maths and mathematical physics.

    It’s somewhat hard to filter out the text and find out whether it contains anything beyond the excitement that 4! appears at several places. Could you please prepare a filtered version of the text for the readers who realize that it is a totally irrational numerology to be impressed by the repeated appearance of the number 24 only?

    You seem to combine descriptions of some basic structures where the number 24 plays a role but is there anything beyond the number 24 that they share? If it’s just the number 24, then of course it is as silly to describe these texts in one article as it would be to talk about two-year, 24-month guarantees for electronics and the switching to a 24-hour format for time in the same article. ;-)

    Do you at least agree with this proposition? Because if you don’t, I am not gonna read any article of yours about numbers.

    • Philip Gibbs says:

      Dear Lubos, yes that is exactly the point. The appearance of the number 24 on its own would not be very interesting but in each case the underlying structures are linked in some way that matches the number 24 between the two structures. Most of this has been noted before and summarised by John Baez in the document linked and elsewhere. I have added that the 24-cell is linked to the hyperdeterminant because the hyperdterminant factories into 24 factors that correspond to the root vectors of D4 which form the 24-cell. The hyperdeterminant is also linked to the discriminant modular form in such a way that its degree must match the power of the eta function. In this way all the structures are linked so it is not pure numerology.

    • Alejandro Rivero says:

      The post is not very different of Baez’s “24.pdf”, which seems to pivot around Dedekind function and its connections to modular forms and to lattices. Perhaps it should also compare to mod 8 periodicities in lattices, for instance the theorems of Milgram about exp(2 pi i signature(L)/8) for any even lattice L, and similar scents. I asked about all of this to some authorities years ago, and I was redirected to some ICM lectures by Mike Hopkins, supposedly touching the topic of 24-periodicities, but I never found them.

    • Luboš Motl says:

      Dear Phil,

      Baez’s talk about “24” is a populist superficial piece of nonsense that doesn’t do a good job in any of the topics involved, because Baez really doesn’t understand them. But do I understand you well that your article is saying nothing beyond what he did?

      Thanks
      Lubos

    • Philip Gibbs says:

      Only a small part of this post overlaps with what Baez wrote. Things mentioned here that were not in the Baez article include qubits, hyperdeterminants, D4, F4, platonic solids, etc.

  4. kneemo says:

    Fascinating post, Phil.

    Did you by chance ever see Robert Wilson’s O^3 construction of the Leech lattice using octonions? He mentions relations to E8 and the Golay code as well.

    And yes, O^3 is octonionic 3-space, which has real dimension 24. :)

    • Philip Gibbs says:

      kneemo, thanks.
      I remember seeing something about such a connection but I have not studied it in detail. There are indeed a lot of things to learn.

  5. Yuri Danoyan says:

    Dear Phil
    Could you pass me reference to system of mysterious “trinities” as noted by the mathematician Vladimir Arnold.

  6. Yuri Danoyan says:

    Sorry i fond in Russian

    http://elementy.ru/lib/430178/430282

  7. Kea says:

    Excellent. So we note once again that the 24-cell has much more to do with qubits, operads (and octonions) than with the so called physics of string theory.

  8. Lawrence B. Crowell says:

    There is an interplay between F_4, D_4 ~ SO(8) and B_4 ~ SO(9). The 24-cell is composed of 24 octahedral cells. The F_4 as 24 octahedral cells can be partition into three sets of rectified tetrahedral cells. These are formed from trunctions. Another construction is with 16 rectified tetrahedral cells and 8 octahedral cells. The first construction is a form of the semi-direct product of SO(8) with S^3. The next is a composition of the 16 cell with 8 octahedral cells. The Schlafli numbers for these constructions are:

    F4, [3,4,3] — (1152)
    B4, [4,3,3] — (384)
    D4, [3^{1,1,1}] — (192)

    And the 1152 is the number of Hurwitz quaternions. With F_4 there is a quotient system with B_4 ~ SO(9), and the B_4 representation with 16 roots plus 8 long roots. The 52 dimensional space of F_4 contains the 36-dim of SO(9), and the remaining 16 dimensions is the 16-dimensional Cayley plane. I am less certain about what is meant by F_4/D_4. The quotient with the 28-dimensional SO(8) leaves a 24-dimensional subspace. The short exact sequence

    F_4/B_4:1 – -> so(9) – ->F_{52/16} – -> OP^2 – -> 1,

    Where F_{52/16} restricts from 52 dimension of 16 spinors plus the 36 of SO(9). OP^2 is the projective Cayley plane in 16 dimensions, where F_4 is the automorphism of the Jordan matrix algebra.

    This relationship between D_4 ~ SO(4,4) and B_4 ~ SO(5,4) seems to be suggest a structure for 5-qubit entanglements with holographic content. The BFSS M-theory is on the infinite momentum frame an SO(9) theory. This suggests that the 5-qubit system might be a holographic projection from a 4-qubit system.

  9. Kea says:

    Right, so now you should really read kneemo’s work on Jordan algebras.

    • Lawrence B. Crowell says:

      I have read a number of Rios papers. These structures do not exclude string theory. In fact most of this is just a way of working such theories in different or more general formats.

      • Kea says:

        On their own, it is true that these papers do not exclude string theory. Taken in the right context, however …

  10. Mitchell Porter says:

    You should love this paper which just came out. They want to use methods inspired by lattice gauge field theory to classify n-qubit entanglement states. You consider the qubits as arranged in a loop, and then perform an operation on nearest neighbors which symmetrizes their correlations. You work your way around the loop doing this – one one pair, then the next pair, and so on – until you return to where you started. At this point, the symmetrizing operation on the last pair may asymmetrize the first pair, and the degree of this mismatch is a holonomy! And, their gauge group is the Lorentz group! Surely you can turn this into an argument for the emergence of special relativity from event symmetry… :-)

    • Mitchell Porter says:

      And while we’re on the subject of getting something for nothing, please note this new paper, which generalizes a well-known technique for efficient simulation of quantum systems, the Multi-Scale Entanglement Renormalization Ansatz (MERA), to the continuum, and which concludes that the resulting new type of RG flow might explain the emergent extra dimension in AdS/CFT. I guess that means that the derivation of string theory from quantum information is complete now; only the details need to be worked out. :-)

      • Lawrence B. Crowell says:

        This sounds a bit like a Moose in compactified gauge theory with extra large dimensions. As energy E – -> 0 these transition into CY spaces, but then at higher energy they link up into these chains.

        Cheers LC

    • Philip Gibbs says:

      I saw the LGT theory paper. It could be interesting but it is dealing with SU(2) invariants which I am less familiar with in this context

  11. Lawrence B. Crowell says:

    The root system for F_4 corresponds to the D_4 lattice. The lattice obeys {(x_1, …, x_n) \in Z_n} such that the sum of x_i is even. This is analogous to a checkerboard lattice where you select one color for the set. For n = 4 the autmorphism group includes the Hadamard matrix, an important matrix in qubit manipulation. The D_n lattice has a θ-function series

    D_n: Θ_{D_n} = ½(θ(z)_3^n + θ(z)_4^n) = sum_{m=0}^∞r_n(2m)q^{2m}

    Or equal to

    Θ_{D_n} = Π_{m=1}^∞(1 – q^{2m})^n((1 + q^{2m-1})^2n + (1 – q^{2m-1})^{2n}).

    Cheers LC

  12. [...] the 24-cell is a unique mathematical structure that comes up in the context of systems of qubits as I discussed just [...]

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