Geodesic
Describing orbit space of global unitary actions for mixed qudit states
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Tome 421 (2014), pp. 68-80 
V. P. Gerdt; A. M. Khvedelidze; Yu. G. Palii. Describing orbit space of global unitary actions for mixed qudit states. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Tome 421 (2014), pp. 68-80. http://geodesic.mathdoc.fr/item/ZNSL_2014_421_a5/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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 $\mathfrak P_+$.

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