The orbit space of the group G := SU(2) × U(1) ⊂ U(3) acting adjointly on the state space of a three-level quantum system is discussed. The semialgebraic structure of is determined within the Procesi–Schwarz method. Using an integrity basis for the ring of G-invariant polynomials , the set of constraints on the Casimir invariants of the group U(3) coming from the positivity requirement Grad(z) ≥ 0 for the Procesi–Schwarz gradient matrix is analyzed in detail.

Journal of Mathematical SciencesAug 17, 20151 citation

arXiv:1608.05213 · Annals of Physics

Super-quantum states in SU(2) invariant 3×N level systems

Soumik Adhikary, Ipsit Kumar Panda, V. Ravishankar

Published Dec 13, 2016• 2 citations

arXiv HTML PDF

DOI: 10.1016/j.aop.2016.12.006

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Super-quantum correlation for SU(2) invariant state in

4\otimes 2

4 ⊗ 2 system

The orbit space

\mathfrak{P}(\mathbb{R}^8)/\mathrm{G}

, of the group

\mathrm{G}:=\mathrm{SU(2)\times U(1)}\subset\mathrm{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 G-invariant polynomials,

\mathbb{R}[\mathfrak{P}(\mathbb{R}^8)]^{\mathrm{G}}

, the set of constraints on the Casimir invariants of

\mathrm{U}(3)

group coming from the positivity requirement of Procesi-Schwarz gradient matrix,

\mathrm{Grad}(z)\geqslant 0

, is analyzed in details.