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 ${\mathfrak{P}}_{+}\mathrm{/}\mathrm{U}(d)$ 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 ${\mathfrak{P}}_{+}\mathrm{/}\mathrm{U}(d)$ 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 ${\mathfrak{P}}_{+}$.