Towards parameterizing the entanglement body of a qubit pair

TL;DR

This paper introduces a new coordinate-based method to efficiently evaluate non-local properties of two-qubit states by characterizing entanglement space and separable states using polynomial inequalities and geometric structures.

Contribution

It presents a novel parametrization of the 2-qubit entanglement space that simplifies the analysis of entanglement and separability properties using algebraic and geometric tools.

Findings

Efficient evaluation of non-local qubit pair characteristics.

Characterization of separable states via polynomial inequalities.

Geometric representation of entanglement space.

Abstract

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.