A Gramian Description of the Degree 4 Generalized Elliptope

One of the most widely studied convex relaxations in combinatorial optimization is the relaxation of the cut polytope mathscr CN to the elliptope mathscr EN, which corresponds to the degree 2 sum-of-squares (SOS) relaxation of optimizing a quadratic form over the hypercube ± 1N. We study the extension of this classical idea to degree 4 SOS, which gives an intermediate relaxation we call the degree 4 generalized elliptope mathscr E4N. Our main result is a necessary and sufficient condition for the Gram matrix of a collection of vectors to belong to mathscr E4N. Consequences include a tight rank inequality between degree 2 and degree 4 pseudomoment matrices, and a guarantee that the only extreme points of mathscr EN also in mathscr E4N are the cut matrices; that is, mathscr EN and mathscr E4N share no "spurious" extreme point. For Gram matrices of equiangular tight frames, we give a simple criterion for membership in mathscrE4N. This yields new inequalities satisfied in mathscrE4N but not mathscrEN whose structure is related to the Schl"afli graph and which cannot be obtained as linear combinations of triangle inequalities. We also give a new proof of the restriction to degree 4 of a result of Laurent showing that mathscrE4N does not satisfy certain cut polytope inequalities capturing parity constraints. Though limited to this special case, our proof of the positive semidefiniteness of Laurent's pseudomoment matrix is short and elementary. Our techniques also suggest that membership in mathscrE4N is closely related to the partial transpose operation on block matrices, which has previously played an important role in the study of quantum entanglement. To illustrate, we present a correspondence between certain entangled bipartite quantum states and the matrices of mathscrE4NsetminusmathscrCN.