Towards parameterizing the entanglement body of a qubit pair

A method allowing to increase a computational efficiency of evaluation of non-local characteristics of a pair of qubits is described. The method is based on the construction of coordinates on a generic section of 2-qubit's entanglement space \(\mathcal{E}_{2\times2}\) represented as the direct product of an ordered 3-dimensional simplex and the double coset \(\mathrm{SU(2)}\times\mathrm{SU(2)}{\backslash} {\mathrm{SU(4)}}/ \mathrm{T^3}\,.\) Within this framework, the subset \(\mathcal{SE}_{2\times2} \subset\mathcal{E}_{2\times2}\) corresponding to the rank-4 separable 2-qubit states is described as a semialgebraic variety given by a system of 3rd and 4th order polynomial inequalities in eigenvalues of the density matrix, whereas the polynomials coefficients are trigonometric functions defined over a direct product of two regular octahedra.