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Advanced Quantum Technologies

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

Ya Jia, Fenzhuo Guo, Mengyan Li, Haifeng Dong, Fei Gao

Published Jun 1, 2026• 0 citations
Publisher

Abstract

ABSTRACT 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.

DOI
10.1002/qute.70348

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ABSTRACT 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.
Not all entangled states can exhibit quantum steering, and determining whether a given entangled state is steerable is a crucial problem in quantum information theory. The main challenge lies in verifying the existence of a local hidden-state (LHS) model capable of reproducing all post-measurement assemblages generated by arbitrary measurements. To address this, we propose a machine learning-based framework that employs batch sampling of measurements and gradient-based optimization to construct an optimal LHS model. We validate our method by analyzing the steerability of two-qubit Werner and two-qutrit isotropic states. For Werner states, our approach saturates the 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. We further investigate the steerability of this class of states under arbitrary POVMs, and our results suggest that POVMs can offer an advantage over PVMs in revealing the steerability of such states.
Measurement correlations in quantum systems can exhibit nonlocal behavior, a fundamental aspect of quantum mechanics with applications such as device-independent quantum information processing. However, it is in general not known which states are local and which ones are not. In particular, it remains an outstanding challenge to explicitly construct local hidden-variable (LHV) models for arbitrary multipartite entangled states. To address this, we use gradient-descent algorithms from machine learning to find LHV models that reproduce the statistics of arbitrary measurements for quantum many-body states. In contrast to previous approaches, our method employs a general ansatz, enabling it to discover LHV models for all local states. Therefore, it for example provides actual estimates for the critical noise levels at which two-qubit Werner states and three-qubit Greenberger-Horne-Zeilinger and W states become local. Furthermore, we find evidence suggesting that two-spin subsystems in the ground states of translationally invariant Hamiltonians are genuinely local, while bigger subsystems are in general not. Our method now offers a quantitative tool for determining the regimes of nonlocality in any given physical context, such as nonequilibrium, decoherence, or disorder.
We present a general criterion for deciding the existence of local-hidden-state (LHS) models and consequently local-hidden-variable (LHV) models for arbitrary states that can be implemented by semidefinite programming. This method can be applied to any given state and general (POVM) measurements and can also be adapted to generate random states with LHS models. As applications we provide an estimate of the fraction of two-qubit entangled states that have LHS models, and present new explicit examples of such states, including Bell-diagonal states, and noisy GHZ and W states. We also generate a list of 750000 new entangled two-qubit states with LHS model for projective measurements, 1500 for POVM measurements, and 150 genuine multipartite entangled three-qubit local states.
We use an error backpropagation (BP) neural network to determine whether an arbitrary two-qubit quantum state is steerable and optimize the steerability bounds of the generalized Werner state. Results show that, regardless of how we select the features for the quantum states, we can use the BP neural network to construct several models to obtain high-performance quantum steering classifiers compared with the support vector machine. Moreover, we predict the steerability bounds of the generalized Werner states using the classifiers that are newly constructed by the BP neural network; that is, the predicted steerability bounds are closer to the theoretical bounds. In particular, high-performance quantum steering classifiers with partial information about the quantum states that we need to measure in only three fixed measurement directions are obtained.
According to the geometric characterization of measurement assemblages and local hidden state (LHS) models, we propose a steering criterion which is both necessary and sufficient for two-qubit states under arbitrary measurement sets. A quantity is introduced to describe the required local resources to reconstruct a measurement assemblage for two-qubit states. We show that the quantity can be regarded as a quantification of steerability and be used to find out optimal LHS models. Finally we propose a method to generate unsteerable states, and construct some two-qubit states which are entangled but unsteerable under all projective measurements.
Quantum steering is an important nonclassical resource for quantum information processing. However, even though lots of steering criteria exist, it is still very difficult to efficiently determine whether an arbitrary two-qubit state shared by Alice and Bob is steerable or not, because the optimal measurement directions of Alice are unknown. In this work, we provide an efficient quantum steering detection scheme for an arbitrary two-qubit state with the help of machine learning, where Alice and Bob only need to measure in a few fixed measurement directions. In order to prove the validity of this method, we first realize a high-performance quantum steering classifier with the whole information. Furthermore, a high-performance quantum steering classifier with partial information is realized, where Alice and Bob only need to measure in three fixed measurement directions. Our method outperforms the existing methods in generic cases in terms of both speed and accuracy, opening up the avenues to explore quantum steering via the machine learning approach.