Geodesic
On the ring of local unitary invariants for mixed $X$-states of two qubits
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 107-123

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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 $1,1,1,2,2$. 
@article{ZNSL_2016_448_a6,
     author = {V. Gerdt and A. Khvedelidze and Yu. Palii},
     title = {On the ring of local unitary invariants for mixed $X$-states of two qubits},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {107--123},
     publisher = {mathdoc},
     volume = {448},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a6/}
}
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V. Gerdt; A. Khvedelidze; Yu. Palii. On the ring of local unitary invariants for mixed $X$-states of two qubits. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 107-123. http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a6/