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

Abstract 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, an efficient algorithm that combines semidefinite programming with a gradient descent method is developed. 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, it is demonstrated that the superior performance of this approach compared to many of the best‐known methods available today. This 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.