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

Entangling properties of a mixed two-qubit system can be described by local homogeneous unitary invariant polynomials in the elements of the density matrix. The structure of the corresponding ring of invariant polynomials for a 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 $SU(2)\times SU(2)$-invariant polynomials modulo its syzygy ideal to the $SO(2)\times SO(2)$-invariant ring freely generated by five homogeneous polynomials of degrees $1,1,1,2,2$.