Recall the entry $n = 2$, $d = 3$ of the path table, with its noncommutative path weights:

As shown by Duff et al, this diagram encodes the hyperdeterminant invariant for three qubits via the Freudenthal triple system for the Jordan algebra $C \oplus C \oplus C$. The FTS is built from two Jordan alegbra elements, corresponding to the triplets $XXY$ and $XYY$, and two complex numbers, corresponding to the unique paths $XXX$ and $YYY$, otherwise known as $a_{000}$ and $a_{111}$. The FTS neatly manifests the triplet symmetry $SL(2)^{3}$ of entanglement classification. We see this in the grading of (permutation reduced) hyperdeterminant terms. There are three three dimensional representations in the nine terms, given by

$a_{100}^{2} a_{011}^{2}$, $-2 a_{000} a_{111} a_{100} a_{011}$, $-2 a_{100} a_{010} a_{011} a_{101}$

plus permutations on the bits. The three remaining terms of Cayley's hyperdeterminant $\Delta$, fixed under permutations, are

$a_{000}^{2} a_{111}^{2}$, $4 a_{000} a_{011} a_{101} a_{110}$, $4 a_{111} a_{100} a_{010} a_{001}$

Note that all $12$ terms have a total bit weight of $6$, reflecting the invariance of the expression under bit flipping and qubit swapping. The invariant $\Delta$ may be expressed as the determinant of a symmetric $2 \times 2$ matrix with terms of degree $2$ in the $a_{ijk}$. The off diagonal entries of this matrix have coefficient $1$ and the diagonal has coefficient $2$, just like the qutrit matrix from the general theory, which was a reduction of the associahedron (secondary) polytope.

7 years ago

The bit weight is 6, but you can go further. The bits can be grouped into three groups of four bits, one for each of the SL(2) symmetries. Each of these four groups has bit weight 2.

ReplyDeleteThis reduces the dimension of the Newton Polytope for the hyperdeterminant from 8 dimensional to 5 dimensional.

Heh, that's cool Phil.

ReplyDeleteBut I don't think so much in terms of the classical symmetries, because QG is about a lot more than some classical entanglement classification. As kneemo points out, here we are seeing directly how the hyperdeterminant is 'commutative' maths, whereas the physics does not restrict itself to this domain.