Describing orbit space of global unitary actions for mixed qudit states

The unitary $\mathrm U(d)$-equivalence relation between elements of the space $\mathfrak P_+$ of mixed states of $d$-dimensional quantum system defines the orbit space $\mathfrak P_+/\mathrm U(d)$ and provides its description in terms the ring $\mathbb R[\mathfrak P_+]^{\mathrm U(d)}$ of $\mathrm U(d)$-invariant polynomials. We prove that the semi-algebraic structure of $\mathfrak P_+/\mathrm U(d)$ is determined completely by two basic properties of density matrices, their semi-positivity and Hermicity. Particularly, it is shown that the Processi–Schwarz inequalities in elements of integrity basis for $\mathbb R[\mathfrak P_+]^{\mathrm U(d)}$ defining the orbit space, are identically satisfied for all elements of $\mathfrak P_+$.