On the variety of X-states

We introduce the notion of an X-state on $n$-qubits. After taking the Zariski closure of the set of X-states in the space of all mixed states, we obtain a complex algebraic variety $\text{scr}X$ that is equipped with the action of the Lie group of local symmetries $G$. We show that the field of $G$-invariant rational functions on $\text{scr}X$ is purely transcendental over the complex numbers of degree $2^{2n-1}-n-1$.