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arXiv:2501.10747 · Advanced Quantum Technologies

Quantifying Quantum Steering with Limited Resources: A Semi‐supervised Machine Learning Approach

Yansa Lu, Zhihua Chen, Zhihao Ma, Shao‐Ming Fei

Published Jan 18, 2025• 3 citations
arXivHTMLPDF

Abstract

Abstract Quantum steering, an intermediate quantum correlation lying between entanglement and nonlocality, has emerged as a critical quantum resource for a variety of quantum information processing tasks such as quantum key distribution and true randomness generation. The ability to detect and quantify quantum steering is crucial for these applications. Semi‐definite programming (SDP) has proven to be a valuable tool to quantify quantum steering. However, the challenge lies in the fact that the optimal measurement strategy is not priori known, making it time‐consuming to compute the steerable measure for any given quantum state. Furthermore, the utilization of SDP requires full information of the quantum state, necessitating quantum state tomography, which can be complex and resource‐consuming. In this work, the semi‐supervised self‐training model is used to estimate the steerable weight, a pivotal measure of quantum steering. The model can be trained using a limited amount of labeled data, thus reducing the time for labeling. The features are constructed by the probabilities derived by performing three sets of projective measurements under arbitrary local unitary transformations on the target states, circumventing the need for quantum tomography. The model demonstrates robust generalization capabilities and can achieve high levels of precision with limited resources.

DOI
10.1002/qute.202400521

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Abstract Quantum steering, an intermediate quantum correlation lying between entanglement and nonlocality, has emerged as a critical quantum resource for a variety of quantum information processing tasks such as quantum key distribution and true randomness generation. The ability to detect and quantify quantum steering is crucial for these applications. Semi‐definite programming (SDP) has proven to be a valuable tool to quantify quantum steering. However, the challenge lies in the fact that the optimal measurement strategy is not priori known, making it time‐consuming to compute the steerable measure for any given quantum state. Furthermore, the utilization of SDP requires full information of the quantum state, necessitating quantum state tomography, which can be complex and resource‐consuming. In this work, the semi‐supervised self‐training model is used to estimate the steerable weight, a pivotal measure of quantum steering. The model can be trained using a limited amount of labeled data, thus reducing the time for labeling. The features are constructed by the probabilities derived by performing three sets of projective measurements under arbitrary local unitary transformations on the target states, circumventing the need for quantum tomography. The model demonstrates robust generalization capabilities and can achieve high levels of precision with limited resources.
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.
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.
Abstract Quantum coherence, as one of the most fundamental non-classical features in quantum mechanics, plays a pivotal role in various quantum information processing tasks, including quantum computing and quantum metrology. The robustness of quantum coherence (RoC) offers an operational interpretation by quantifying the advantage provided by a quantum state in phase discrimination tasks. To achieve verification RoC with high precision via semidefinite programming (SDP), complete knowledge of quantum states is typically required. Relying solely on expectation values of observables may introduce significant errors in SDP-based estimations. To estimate RoC with high precision using limited data extracted from quantum states, firstly, we propose a semi-supervised K-nearest neighbor (KNN) algorithm (semi-KNN) and a semi-supervised method that combines the KNN and random forest (RF) models with a dynamical threshold (semi-KNN-RF). Then we implement the semi-KNN and semi-KNN-RF models to efficiently estimate quantum coherence by analyzing statistical data obtained from randomly generated local projective measurements performed on unknown quantum states. The semi-KNN-RF model performs better than the semi-KNN model. This innovative methodology allows for accurate coherence estimation, even for high-dimensional quantum 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.
Quantum entanglement is a key resource in quantum computing and quantum information processing tasks. However, its quantification remains a major challenge since it cannot be directly extracted from physical observables. To address this issue, we study a few machine-learning based models to estimate the amount of entanglement in two-qubit as well as three-qubit systems. We use measurement outcomes as the input features and entanglement measures as the training labels. Our models predict entanglement without requiring the full state information. This demonstrates the potential of machine learning as an effcient and powerful tool for characterizing quantum entanglement
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.