M Theory Lesson 150
Quite a while back, kneemo recommended a paper by Duff and Ferrara on the two way entanglement of three qutrits. The paper actually begins by looking at qubits associated to $4D$ stringy black holes, and in particular the use of a hyperdeterminant to express the entropy.
This hyperdeterminant is invariant under the U triality, which is a kind of three (spatial) dimensional analogue of the two dimensional duality currently generating much interest. Thus it is no surprise that when they move on to seven qubits and tripartite entanglement (giving seven lines with three nodes on the Fano plane) we start to see the familiar circulants, this time associated with $E(7)$, namely the matrix
2 2 1 2 1 1 1
1 2 2 1 2 1 1
1 1 2 2 1 2 1
1 1 1 2 2 1 2
2 1 1 1 2 2 1
1 2 1 1 1 2 2
2 1 2 1 1 1 2
Observe that this circulant is basically the $7 \times 7$ circulant for the Hamming code, with $1$ added to each entry, and indeed this circulant is associated to the Fano plane and seven bits of information. Moreover, an $E(8)$ interpretation has the advantage of agreeing with the 3 Time interpretation of the spatial dimensions, at least in the context of M Theory.
By considering the entries of the matrix above to be qutrit elements, $A_{ij} \in \{ 0,1,2 \} = \mathbb{F}_{3}$, we see that the addition of $1$ to each entry again yields a circulant, which is twice the complement of the Hamming circulant. And finally, yet another addition of unit entries returns the matrix to the Hamming circulant. Thus a triality is made manifest by the root vector circulants.
Duff and Ferrara point out that the question of real forms for $E(7)$ is not really important in this context, since the coefficients defining the state are allowed to be complex. Hmm. This also sounds like something that came up recently.
This hyperdeterminant is invariant under the U triality, which is a kind of three (spatial) dimensional analogue of the two dimensional duality currently generating much interest. Thus it is no surprise that when they move on to seven qubits and tripartite entanglement (giving seven lines with three nodes on the Fano plane) we start to see the familiar circulants, this time associated with $E(7)$, namely the matrix
2 2 1 2 1 1 1
1 2 2 1 2 1 1
1 1 2 2 1 2 1
1 1 1 2 2 1 2
2 1 1 1 2 2 1
1 2 1 1 1 2 2
2 1 2 1 1 1 2
Observe that this circulant is basically the $7 \times 7$ circulant for the Hamming code, with $1$ added to each entry, and indeed this circulant is associated to the Fano plane and seven bits of information. Moreover, an $E(8)$ interpretation has the advantage of agreeing with the 3 Time interpretation of the spatial dimensions, at least in the context of M Theory.
By considering the entries of the matrix above to be qutrit elements, $A_{ij} \in \{ 0,1,2 \} = \mathbb{F}_{3}$, we see that the addition of $1$ to each entry again yields a circulant, which is twice the complement of the Hamming circulant. And finally, yet another addition of unit entries returns the matrix to the Hamming circulant. Thus a triality is made manifest by the root vector circulants.
Duff and Ferrara point out that the question of real forms for $E(7)$ is not really important in this context, since the coefficients defining the state are allowed to be complex. Hmm. This also sounds like something that came up recently.
posted by Kea | 3:52 PM
4 Comments:
Kea, thanks for the links. Yesterday I started adding an entry about hyperdeterminants into Wikipedia. I hope it will be useful.
Good to hear you are helping wikipedia, phil. I am quite interested not only in hyperdeterminants, but in all sorts of multidimensional operators to which they may be related. Your post on the j invariant was really cool.
When I wrote that and pointed out that the 2x2x2x2 hyperdeterminant is of degree 24 I was not very serious when I suggested that it might be connected to the dimension of the Leech lattice and other mysteries of the number 24, but then I learnt that the discriminant of the quartic used to construct the hyperdeterminant is linked to the 24th power of the eta function which makes the connection look more promising.
I looked for signs of the Golay code in the structure of the hyperdeterminant but it is not there. However, the connection you have highlighted between the Hamming code from the Fano plane and the hyperdeterminant in the quartic invariant of E_7 makes it look like there could be more to find. I fear it is buried too deeply for me.
There are also strong connections between hyperdeterminants and generalised hypergeometric functions. Hypergeometric functions are connected to all kinds of interesting and relevant things. Just follow the links in Wikipedia from here!
Dear Phil,
If you are interested in the structure of the hyperdeterminant of type 2x2x2x2
as the one connected to four qubit entanglement see my paper in J.Phys.A.39
(2006) 9533-9545 which is based on the work of Luque and Thibon.
Moreover, the connection between the Hamming code, block designs, the tripartite entanglement of seven qubits, and Cartan's quartic invariant is further clarified in my Frascati lecture notes to be published soon in the SAM2007 proceedings.
This is an extended version of my recent papers on the connection between error correction, stringy black holes, the Fano plane etc. (Phys. Rev. D76, 106011 (2007), Phys. Rev. D75, 024024 (2007),
Phys. Rev. D74, 024030 (2006)).
These can also be found on the hepth and quant-ph archives.
Best regards
Peter Levay
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