The moduli space of three-qutrit states

Abstract

We study the invariant theory of trilinear forms over a three-dimensional complex vector space, and apply it to investigate the behavior of pure entangled three-partite qutrit states and their normal forms under local filtering operations (SLOCC). We describe the orbit space of the SLOCC group SLs3,Cd33 both in its affine and projective versions in terms of a very symmetric normal form parametrized by three complex numbers. The parameters of the possible normal forms of a given state are roots of an algebraic equation, which is proved to be solvable by radicals. The structure of the sets of equivalent normal forms is related to the geometry of certain regular complex polytopes.