Woit has been reasonably bashing the Imperial press office for hyping work of Duff et al about entanglement for multiple qudits. Now Philip Gibbs blogs about his own 2001 paper linking multidimensional determinants to properties of elliptic curves. These tori are secretly what the green book is about!

Hyperdeterminants are special cases of the general discriminants. Recall that typical integer coordinates for a set $A$ lie on a lattice in the continuum. The dimension depends on the number of variables, and by limiting the digits to qudits we limit the allowed degree of terms in our initial polynomials. So for the triangular associahedron, we had qutrit objects $0$ and $1$ and $2$ and monomials of degree two, like $Y^2$ for the point $(0,2,0)$ in the plane. We can do this for any type of qudit and any number of particles $n$. The dimension of the secondary polytope will go up with $n - d$ and its shape encodes the resultant (entanglement measure). It quickly becomes too lengthy to write down, but the algorithm is completely understandable.

7 years ago

Nice investigation by Gibbs. Indeed a 3x3x3 hyperdeterminant is very useful in the study entangled qutrit systems.

ReplyDeleteHmm, degree 36 according to the green book. Messy!

ReplyDelete