On the geometry of a class of N-qubit entanglement monotones

A family of N-qubit entanglement monotones invariant under stochastic local operations and classical communication (SLOCC) is defined. This class of entanglement monotones includes the well-known examples of the concurrence, the 3-tangle and some of the four-, five- and N-qubit SLOCC invariants introduced recently. The construction of these invariants is based on bipartite partitions of the Hilbert space in the form C2N ≅ CL x Cl with L = 2N-n ≥ l = 2n. Such partitions can be given a nice geometrical interpretation in terms of Grassmannians Gr(L, l) of l-planes in CL that can be realized as the zero locus of quadratic polynomials in the complex projective space of suitable dimension via the Pluecker embedding. The invariants are neatly expressed in terms of the Pluecker coordinates of the Grassmannians