A General Physics-Informed Machine Learning Method for Determining the Steerability of Arbitrary Entangled States

Determining whether an arbitrary entangled state is steerable is a challenging task, as it requires assessing the existence of a local hidden-state (LHS) model across all relevant measurement settings. In this work, we develop a Physics-Informed Machine Learning method that reformulates the construction of the LHS model as a differentiable parameter optimization problem. By leveraging the vmap vectorized transformation in JAX for efficient batch-wise sampling, we efficiently explore the continuous measurement space. Furthermore, physical constraints are embedded into the parameterization, and automatic differentiation is used for gradient-based optimization to obtain the optimal LHS model. We validate our method on two-qubit Werner and two-qutrit isotropic states. For Werner states, our approach marks the first numerical success in precisely reproducing analytical visibility bounds under three Pauli measurements, arbitrary projective measurements (PVMs), and arbitrary positive operator-valued measurements (POVMs). For isotropic states, we achieve the known analytical bounds under arbitrary PVMs and explore the steerability under arbitrary POVMs, suggesting that POVMs can offer an advantage over PVMs in revealing the steerability of such states.