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 $\mathfrak{ P}_+$.
Categories: quant-ph