Geodesic
Constructing $\mathrm{SU(2)\times U(1)}$ orbit space for qutrit mixed states
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 111-127 
V. Gerdt; A. Khvedelidze; Y. Palii. Constructing $\mathrm{SU(2)\times U(1)}$ orbit space for qutrit mixed states. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 111-127. http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a7/
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     author = {V. Gerdt and A. Khvedelidze and Y. Palii},
     title = {Constructing $\mathrm{SU(2)\times U(1)}$ orbit space for qutrit mixed states},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {111--127},
     year = {2015},
     volume = {432},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a7/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The orbit space $\mathfrak P(\mathbb R^8)/\mathrm G$ of the group $$ \mathrm{G:=SU(2)\times U(1)\subset U(3)} $$ acting adjointly on the state space $\mathfrak P(\mathbb R^8)$ of a $3$-level quantum system is discussed. The semi-algebraic structure of $\mathfrak P(\mathbb R^8)/\mathrm G$ is determined within the Procesi–Schwarz method. Using the integrity basis for the ring of $\mathrm G$-invariant polynomials $\mathbb R[\mathfrak P(\mathbb R^8)]^\mathrm G$, the set of constraints on the Casimir invariants of the group $\mathrm U(3)$ coming from the positivity requirement for Procesi–Schwarz gradient matrix, $\mathrm{Grad}(z)\geqslant0$, is analyzed in detail.

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