Optimizing LOCC Protocols on Product Stiefel Manifold

Local operations and classical communication (LOCC) is a foundational framework in quantum information from both theoretical and experimental perspectives. However, designing and optimizing LOCC protocols is intractable due to their complex structure. Determining achievable bounds and designing practically implementable LOCC protocols remain crucial challenges when the number of communication rounds is finite. In this work, we develop a framework to optimize fixed-round LOCC via Riemannian optimization on the product Stiefel manifold, which not only yields near-optimal objective function values but also produces fully implementable protocols. We demonstrate the applicability of this framework through key tasks in quantum information processing, such as entanglement distillation and state merging. Our results provide new insights into the achievable bounds for entanglement distillation and block entanglement state merging. We obtain improved distillation and state merging protocols, some of which match the upper bounds derived via positive partial transpose relaxations. These results demonstrate that optimizing LOCC via manifold optimization can serve as a powerful tool to advance research on distributed quantum information processing.