Hybrid of Gradient Descent And Semidefinite Programming for Certifying Multipartite Entanglement Structure

Multipartite entanglement is a crucial resource for a wide range of quantum information processing tasks, including quantum metrology, quantum computing, and quantum communication. The verification of multipartite entanglement, along with an understanding of its intrinsic structure, is of fundamental importance, both for the foundations of quantum mechanics and for the practical applications of quantum information technologies. Nonetheless, detecting entanglement structures remains a significant challenge, particularly for general states and large-scale quantum systems. To address this issue, we develop an efficient algorithm that combines semidefinite programming with a gradient descent method. This algorithm is designed to explore the entanglement structure by examining the inner polytope of the convex set that encompasses all states sharing the same entanglement properties. Through detailed examples, we demonstrate the superior performance of our approach compared to many of the best-known methods available today. Our method not only improves entanglement detection but also provides deeper insights into the complex structures of many-body quantum systems, which is essential for advancing quantum technologies