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The unitary $ \mathrm{U}(d)$-equivalence relation between elements of thespace $\mathfrak{P}_+\,$ of mixed states of $d$-dimensional quantum systemdefines the orbit space $ \mathfrak{P}_+/ \mathrm{U}(d)\,$ and provides itsdescription in terms the ring $\mathbb{R}[\mathfrak{P}_+]^{\mathrm{U}(d)}\,$ of$\mathrm{U}(d)$-invariant polynomials. We prove that the semi-algebraicstructure of $ \mathfrak{P}_+/ \mathrm{U}(d)\, $ is determined completely bytwo basic properties of density matrices, their semi-positivity and Hermicity.Particularly, it is shown that the Processi-Schwarz inequalities in elements ofintegrity basis for $\mathbb{R}[\mathfrak{P}_+]^{\mathrm{U}(d)}\,$ defining theorbit space, are identically satisfied for all elements of $\mathfrak{P}_+$.

Describing orbit space of global unitary actions for mixed qudit states