Linear semi-infinite programming approach for entanglement quantification

TL;DR

This paper introduces a linear semi-infinite programming approach to quantify entanglement in three-qubit states, providing a new algorithm with proven convergence and applications to distinguish types of three-qubit entanglement.

Contribution

It formulates the entanglement quantification as an LSIP problem, proving duality and boundedness, and develops a cutting-plane algorithm for practical computation.

Findings

The LSIP formulation has no duality gap even for non-continuous measures.

The central cutting-plane algorithm converges globally and provides lower bounds.

Application to three-qubit states distinguishes different entanglement types.

Abstract

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.