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 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 X is purely transcendental over the complex numbers of degree 2²ⁿ⁻¹-n-1.
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Candelori et al. (2024) studied this question.
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