Linear semi-infinite programming approach for entanglement quantification

We explore the dual problem of the convex roof construction by identifying it as a linear semi-infinite programming (LSIP) problem. Using the LSIP theory, we show the absence of a duality gap between primal and dual problems, even if the entanglement quantifier is not continuous, and prove that the set of optimal solutions is non-empty and bounded. In addition, we implement a central cutting-plane algorithm for LSIP to quantify entanglement between three qubits. The algorithm has global convergence property and gives lower bounds on the entanglement measure for non-optimal feasible points. As an application, we use the algorithm for calculating the convex roof of the three-tangle and $\pi$-tangle measures for families of states with low and high ranks. As the $\pi$-tangle measure quantifies the entanglement of W states, we apply the values of the two quantifiers to distinguish between the two different types of genuine three-qubit entanglement.