Multigraded Poincare series for mixed states of two qubits and the boundary of the set of separable states

摘要

Let M be the set of mixed states and S the set of separable states of thetwo-qubit system, and G = SU(2) x SU(2) the group of local unitarytransformations (ignoring the overall phase factor). We compute the multigradedPoincare series for the algebra of G-invariant polynomial functions on theaffine space of all Hermitian operators of trace 1. We check that this seriesis consistent with the list of invariants computed by Makhlin. By using therecent result of Augusiak et al., we show that the boundary of S decomposesnaturally into two pieces. We prove that the part of this boundary which iscontained in the relative interior of M is a smooth manifold.