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arXiv:2502.11365 · Physical Review A

Machine learning for detecting steering in qutrit-pair states

Pu Wang, Zhongyan Li, Hui-Xian Meng

Published Sep 8, 2025• 1 citation
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Abstract

Only a few states in high-dimensional systems can be identified as (un)steerable using existing theoretical or experimental methods. We utilize semidefinite programming (SDP) to construct a dataset for steerability detection in qutrit-qutrit systems. For the full-information feature ${F}_{1}$, artificial neural networks achieve high classification accuracy and generalization, and perform better than the support vector machine. As feature engineering plays a pivotal role, we introduce a steering ellipsoidlike feature, ${F}_{2}$, which significantly enhances the performance of each of our models. To address quantum steering detection in isotropic states, partially entangled states, and random states, we explore, respectively, the most tailored feature and classifier. Given that the SDP method provides only a sufficient condition for steerability detection, we establish the first rigorously constructed, accurately labeled dataset based on theoretical foundations. This dataset enables models to exhibit outstanding accuracy and generalization capabilities, independent of the choice of features. As applications, we investigate the steerability boundaries of isotropic states and partially entangled states, discover new steerable states, and determine their parameter ranges. This work not only advances the application of machine learning for probing quantum steerability in high-dimensional systems but also deepens the theoretical understanding of quantum steerability itself.

DOI
10.1103/h7qp-63dg

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Only a few states in high-dimensional systems can be identified as (un)steerable using existing theoretical or experimental methods. We utilize semidefinite programming (SDP) to construct a dataset for steerability detection in qutrit-qutrit systems. For the full-information feature F1{F}_{1}F1​, artificial neural networks achieve high classification accuracy and generalization, and perform better than the support vector machine. As feature engineering plays a pivotal role, we introduce a steering ellipsoidlike feature, F2{F}_{2}F2​, which significantly enhances the performance of each of our models. To address quantum steering detection in isotropic states, partially entangled states, and random states, we explore, respectively, the most tailored feature and classifier. Given that the SDP method provides only a sufficient condition for steerability detection, we establish the first rigorously constructed, accurately labeled dataset based on theoretical foundations. This dataset enables models to exhibit outstanding accuracy and generalization capabilities, independent of the choice of features. As applications, we investigate the steerability boundaries of isotropic states and partially entangled states, discover new steerable states, and determine their parameter ranges. This work not only advances the application of machine learning for probing quantum steerability in high-dimensional systems but also deepens the theoretical understanding of quantum steerability itself.
Inspired by the research on detecting the steerability of two-qubit states via machine learning, this study explores the detection of steerability in qubit-qutrit systems using various machine learning methods. It is found that, (1) for random states, Werner states, and UN states (a newly constructed class of entangled states where Alice cannot steer Bob), there always exist supervised or semi-supervised machine learning methods with a steering detectability accuracy rate of over 95%; (2) the steering bounds predicted by the supervised machine learning methods are mostly higher than the non-steerable bounds provided by theory and lower than the steering bounds determined by the semidefinite programming (SDP). This demonstrates that the machine learning methods are reliable for detecting the steerability of quantum states in qubit-qutrit systems and can detect more steering states compared with the SDP method. It lays the foundation for utilizing machine learning methods to detect quantum states' steerability in bipartite and high-dimensional systems.
Abstract Quantum steering is an important form of nonclassical correlation and plays a central role in one-sided device-independent quantum information processing. However, efficiently determining whether an arbitrary two-qubit state is steerable remains challenging, since semidefinite-programming-based certification generally depends on the choice of measurement settings and can become computationally demanding. In this work, we propose an interpretable quantum steering detection scheme based on Kolmogorov–Arnold Networks (KANs). The training labels are generated through semidefinite programming for randomly sampled two-qubit density matrices. Several physically motivated feature representations are constructed to investigate the influence of measurement and feature reduction. Owing to their learnable univariate edge functions and symbolic formula extraction capability, KANs provide not only accurate classification but also a more transparent representation of the learned steering decision boundary. Numerical experiments show that the proposed KAN-based classifiers can distinguish states with SDP-detected steering from states for which no witness is found under the sampled measurements. In particular, canonical-correlation features provide an effective representation for learning the steering boundary, and further feature compression can still retain competitive detection performance. These results indicate that KANs offer a promising, efficient, and interpretable machine-learning approach for finite-measurement steering detection of arbitrary two-qubit states.
Abstract Quantum steering is an important nonlocal resource and has a wide range of applications in quantum information processing. Although a lot of steering criteria have been proposed, it is still very difficult to efficiently detect quantum steering in experiment. Here we employ machine learning techniques to tackle the problem of quantum steering detection in two-qubit system. The quantum steering and un-steering inequalities are combined together, so as to construct quantum steering classifiers for the generalized Werner states via artificial neural networks. More steerable and unsteerable quantum states can be identified by the classifiers proposed here than by the quantum steering inequality or un-steering inequality, which provides a new way to detect steering with only partial information of the given quantum states. We consider two types of artificial neural networks, one is the single-layer perceptron and the other is the multi-layer perceptron. The result shows that the multi-layer perceptron outperforms the single-layer perceptron in terms of accuracy. Compared with the existing quantum steering criteria, our methods do not require the whole information of the quantum state, and the steering of it is detected by using state-independent measurements, so it is easy to realize in experiment.
Abstract Generalization is essential for reliable machine-learning detection of quantum steerability. In this work, we develop a CNN-based approach using the matrix versions of the full-information feature F 1 and the steering-determined feature F 2 as inputs for two-qubit steerability classification. Several physically distinct state families with reliable labels are employed to evaluate classifier generalization. Vector-input classifiers based on F 1 and F 2 are included as benchmarks under a unified classification protocol. Across the considered classifiers and test datasets, F 2 -based models exhibit consistently more stable generalization than their F 1 -based counterparts, indicating that steering-determined feature extraction plays a more dominant role than classifier selection in the observed performance improvement in the considered settings. Meanwhile, no single classifier is uniformly optimal across all physical state families. Since CNN– F 2 achieves the best overall performance on the T -diagonal benchmarks, it is further employed to visualize and estimate the projective-measurement steering hierarchy of axially symmetric states, a special subclass of T -diagonal states.
Einstein-Podolsky-Rosen (EPR) steering is a kind of powerful nonlocal quantum resource in quantum information processing such as quantum cryptography and quantum communication. Many criteria have been proposed in the past few years to detect the steerability both analytically and numerically. Supervised machine learning such as support vector machines and neural networks have also been trained to detect the EPR steerability. To implement supervised machine learning, one needs a lot of labeled quantum states by using the semidefinite programming, which is very time consuming. We present a semisupervised support vector machine method which only uses a small portion of labeled quantum states in detecting quantum steering. We show that our approach can significantly improve the accuracies by detailed examples.
Network quantum steering plays a pivotal role in quantum information science, enabling robust certification of quantum correlations in scenarios with asymmetric trust assumptions among network parties. The intricate nature of quantum networks, however, poses significant challenges for the detection and quantification of steering. In this work, we develop a neural network-based method for measuring network quantum steerability, which can be generalized to arbitrary quantum networks and naturally applied to standard steering scenarios. Our method provides an effective framework for steerability analysis, demonstrating remarkable accuracy and efficiency in standard bipartite and multipartite steering scenarios. Numerical simulations involving isotropic states and noisy Greenberger-Horne-Zeilinger states yield results that are consistent with established findings in these respective scenarios. Furthermore, we demonstrate its utility in the bilocal network steering scenario, where an untrusted central party shares two-qubit isotropic states of different visibilities, $\ensuremath{\nu}$ and $\ensuremath{\omega}$, with trusted end-point parties and performs a single Bell state measurement. Through explicit construction of a network local hidden state model derived from numerical results and incorporation of the entanglement properties of network assemblages, we analytically demonstrate that the network steering thresholds are determined by the curve $\ensuremath{\nu}\ensuremath{\omega}=1/3$ under the corresponding configuration.