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 $${{\mathcal{E}}_{{2 \times 2}}}$$ represented as the direct product of an ordered 3-dimensional simplex and the double coset $${\text{SU(2)}} \times {\text{SU(2)}}{\kern 1pt} \backslash {\kern 1pt} {\text{SU(4)}}{\kern 1pt} /{\kern 1pt} {{{\text{T}}}^{{\text{3}}}}{\kern 1pt} .$$ Within this framework, the subset $$\mathcal{S}{{\mathcal{E}}_{{2 \times 2}}} \subset {{\mathcal{E}}_{{2 \times 2}}}$$ 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.