  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 $\mathfrak{P}_+$.
