M Theory Lesson 355

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.