Describing the Orbit Space of the Global Unitary Actions for Mixed Qudit States

The unitary U(d)-equivalence relation on the space β + of mixed states of a d-dimensional quantum system defines the orbit space β+/U(d) and provides its description in terms of the ring R[β+] U(d) of U(d)-invariant polynomials. We prove that the semi-algebraic structure of β+/U(d) is completely determined by two basic properties of density matrices, their semi-positivity and Hermiticity. In particular, it is shown that the Procesi–Schwarz inequalities in the elements of the integrity basis for R[β+]U(d) defining the orbit space are identically satisfied for all elements of β+. Bibliography: 9 titles.