On the ring of local unitary invariants for mixed X-states of two qubits

Entangling properties of a mixed 2-qubit system can be described by the local homogeneous unitary invariant polynomials in elements of the density matrix. The structure of the corresponding invariant polynomial ring for the special subclass of states, the so-called mixed X−states, is established. It is shown that for the X−states there is an injective ring homomorphism of the quotient ring of SU(2)×SU(2) invariant polynomials modulo its syzygy ideal and the SO(2) × SO(2)−invariant ring freely generated by five homogeneous polynomials of degrees 1,1,1,2,2.