A Non-Convex Optimization Strategy for Computing Convex-Roof Entanglement

TL;DR

This paper introduces a novel non-convex optimization approach using a modified genetic algorithm and quasi-Newton refinement to compute convex roof entanglement measures in quantum states, demonstrating reliable results on various test cases.

Contribution

It presents the first derivative-free, non-convex computational method for convex roof entanglement measures, combining genetic algorithms with QR factorization and quasi-Newton methods.

Findings

Method reliably reproduces entanglement curves for large systems.

Approach effectively computes entanglement in pure dephasing evolutions.

Successfully studies temperature dependence of Gibbs state entanglement.

Abstract

We develop a numerical methodology for the computation of entanglement measures for mixed quantum states. Using the well-known Schr\"odinger-HJW theorem, the computation of convex roof entanglement measures is reframed as a search for unitary matrices; a nonconvex optimization problem. To address this non-convexity, we modify a genetic algorithm, known in the literature as differential evolution, constraining the search space to unitary matrices by using a QR factorization. We then refine results using a quasi-Newton method. We benchmark our method on simple test problems and, as an application, compute entanglement between a system and its environment over time for pure dephasing evolutions. We also study the temperature dependence of Gibbs state entanglement for a class of block-diagonal Hamiltonians to provide a complementary test scenario with a set of entangled states that are qualitatively different. We find that the method works well enough to reliably reproduce entanglement curves, even for comparatively large systems. To our knowledge, the modified genetic algorithm represents the first derivative-free and non-convex computational method that broadly applies to the computation of convex roof entanglement measures.