Optimizing quantum state transformation under locality constraint

In this paper, we present a general numerical framework for both deterministic and probabilistic quantum state transformations, under locality constraints. For a given arbitrary bipartite initial state and a desired bipartite target state, we construct an optimized local quantum channel that transforms the initial state into the target state with high fidelity. To achieve this goal, local quantum channels are parametrized on a complex Stiefel manifold and optimized using gradient-based methods. We demonstrate that this approach significantly enhances entanglement distillation for weakly entangled states via two complementary strategies: optimized local state transformation and probabilistic local transformation. These results establish our method as a powerful and versatile tool for a broad class of quantum information processing tasks.