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 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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. 
@article{ZNSL_2015_432_a7,
     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/}
}
TY  - JOUR
AU  - V. Gerdt
AU  - A. Khvedelidze
AU  - Y. Palii
TI  - Constructing $\mathrm{SU(2)\times U(1)}$ orbit space for qutrit mixed states
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2015
SP  - 111
EP  - 127
VL  - 432
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a7/
LA  - en
ID  - ZNSL_2015_432_a7
ER  - 
%0 Journal Article
%A V. Gerdt
%A A. Khvedelidze
%A Y. Palii
%T Constructing $\mathrm{SU(2)\times U(1)}$ orbit space for qutrit mixed states
%J Zapiski Nauchnykh Seminarov POMI
%D 2015
%P 111-127
%V 432
%U http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a7/
%G en
%F ZNSL_2015_432_a7
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/

[1] C. Procesi, G. Schwarz, “The geometry of orbit spaces and gauge symmetry breaking in supersymmetric gauge theories”, Phys. Lett. B, 161 (1985), 117–121 | DOI | MR

[2] C. Procesi, G. Schwarz, “Inequalities defining orbit spaces”, Invent. Math., 81 (1985), 539–554 | DOI | MR | Zbl

[3] V. Gerdt, A. Khvedelidze, Yu. Palii, “Describing the orbit space of the global unitary actions for mixed qudit states”, J. Math. Sci., 200:6 (2014), 682–689 ; arXiv: 1311.4649[quant-ph] | DOI | Zbl

[4] E. Vinberg, V. Popov, “Theory of invariants”, Itogi Nauki i Techniki, Ser. Sovremennie problemi matematiki. Fundam. napravl., 55, 1989, 137–309 (in Rissian) | MR | Zbl

[5] M. Forger, “Invariant polynomials and Molien functions”, J. Math. Phys., 39:2 (1998), 1107–1141 | DOI | MR | Zbl

[6] C. Procesi, An approach to Lie Theory through Invariants and Representations, Springer, 2007 | MR

[7] D. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms, Third Ed., Springer, 2007 | MR | Zbl

[8] L. Michel, L. A. Radicati, “The geometry of the octet”, Ann. Inst. Henri Poincare Section A, 18 (1973), 185–214 | MR | Zbl

[9] V. Gerdt, A. Khvedelidze, Yu. Palii, “On the ring of local polynomial invariants for a pair of entangled qubits”, J. Math. Sci., 168:3 (2010), 368–378 ; arXiv: 1007.0968[quant-ph] | DOI | MR | Zbl