On the ring of local invariants for a~pair of the entangled $q$-bits

V. Gerdt ; Yu. Palii ; A. Khvedelidze

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 104-123

Résumé

The entanglement characteristics of two $q$-bits are encoded in the invariants of the adjoint action of the group $\mathrm{SU}(2)\otimes\mathrm{SU}(2)$ on the space of the density matrices $\mathfrak P_+$, i.e., space of $4\times4$ non-negative Hermitian matrices. The corresponding ring $\mathbb C[\mathfrak P_+]^{\mathrm{SU}(2)\otimes\mathrm{SU}(2)}$ in elements of the density matrix is studied. The special integrity basis for $\mathbb C[\mathfrak P_+]^{\mathrm{SU}(2)\otimes\mathrm{SU}(2)}$ is described and constraints on its elements due to the semi-definiteness of the density matrix are given explicitly in the form of inequalities. This basis has the property that only a minimal number of primary invariants of degree 2, 3 and one lowest degree 4 secondary invariant that appear in the Hironaka decomposition of $\mathbb C[\mathfrak P_+]^{\mathrm{SU}(2)\otimes\mathrm{SU}(2)}$ are subject to the polynomial inequalities. Bibl. – 32 titles.