Multigraded Poincaré Series for Mixed States of Two Qubits and the Boundary of the Set of Separable States

We consider the mixed states of the two-qubit system under the action of the group G = SU(2) × SU(2) of local unitary transformations. The Hilbert space of this system is the tensor product H = C ⊗ C. Let P be the algebra of real polynomial functions on the affine space of all hermitian operators of trace 1 on H. The polynomials f ∈ P which are invariant under G form the subalgebra P ⊆ P. We compute the multigraded Poincaré series for P and verify that it agrees with Makhlin’s list of 18 invariants. By using the recent result of Augusiak et al. we show that the boundary of the set of separable states is not smooth. We also express the invariant polynomial function, defining this boundary, in terms of Makhlin’s invariants.