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New Journal of Physics

Learning the hierarchy of steering measurement settings of qubit-pair states with kernel-based quantum models

Zheng-Lin Tsai, Hongming Wang, Hongbin Chen

Published Aug 22, 2025• 2 citations
Publisher

Abstract

Abstract Quantum steering has been proven to be a unique quantum correlation sandwiched between Bell nonlocality and quantum entanglement. Due to its fundamental importance, quantum steering has been studied extensively. To demonstrate steerability, one relies on a particular resource referred to as steerable assemble on one side of a two-party system. However, it is generically unclear how to reach such steerable resource from a bipartite quantum state. For this purpose, one must optimize over all possible measurement settings, which constitute a hierarchical structure. On the other hand, in light of the rapid development of quantum computing technology, quantum machine learning (QML) has emerged as a field with a promising potential in demonstrating quantum advantage. Here we leverage the power of kernel-based QML models to infer the hierarchy of steering measurement settings. By using a computational protocol, we can generate an appropriate dataset necessary for training the models. Additionally, based on the physics of quantum steering, we encode the states to be recognized into five different types of features. This helps us identify the most compact characterization of Alice-to-Bob steerability, which is Alice’s regularly aligned steering ellipsoid. We then apply the well-trained models to predict the hierarchy for three specific families of states. We also compare the QML model with two classical models, demonstrating its superior performance, as well as the practical quantum advantage.

DOI
10.1088/1367-2630/adfe89

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Quantum steering exhibits an inherent asymmetric property that differs from both quantum entanglement and Bell nonlocality, which leads to no-way, one-way, and two-way steering. Flexibly manipulating the direction of quantum steering without reducing the quantum steerability is very important for information processing. In this paper, implementing sequential unsharp measurements and the steering-radius criterion, we show how to manipulate that steering direction among multiple Alices and Bobs. We demonstrate that the change in steering direction depends on the number of measurement settings and the sharpness of measurements. For a given two-qubit asymmetric state, we obtain 13 kinds of steering regions with different steering directions in the case of one Alice and three Bobs and obtain 11 kinds in the case with two Alices and two Bobs. For the case with one Alice and three Bobs, we further find a volume bound to visualize quantum steering. Our manipulation scheme is experimentally easy to achieve and rather assigns the steerability to more observers without reducing it, which offers a direct reference for the practical application of quantum steering.

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Abstract Quantum steering has been proven to be a unique quantum correlation sandwiched between Bell nonlocality and quantum entanglement. Due to its fundamental importance, quantum steering has been studied extensively. To demonstrate steerability, one relies on a particular resource referred to as steerable assemble on one side of a two-party system. However, it is generically unclear how to reach such steerable resource from a bipartite quantum state. For this purpose, one must optimize over all possible measurement settings, which constitute a hierarchical structure. On the other hand, in light of the rapid development of quantum computing technology, quantum machine learning (QML) has emerged as a field with a promising potential in demonstrating quantum advantage. Here we leverage the power of kernel-based QML models to infer the hierarchy of steering measurement settings. By using a computational protocol, we can generate an appropriate dataset necessary for training the models. Additionally, based on the physics of quantum steering, we encode the states to be recognized into five different types of features. This helps us identify the most compact characterization of Alice-to-Bob steerability, which is Alice’s regularly aligned steering ellipsoid. We then apply the well-trained models to predict the hierarchy for three specific families of states. We also compare the QML model with two classical models, demonstrating its superior performance, as well as the practical quantum advantage.
Abstract Quantum steering has attracted increasing research attention because of its fundamental importance, as well as its applications in quantum information science. Here we leverage the power of the deep learning model to infer the steerability of quantum states with specific numbers of measurement settings, which form a hierarchical structure. A computational protocol consisting of iterative tests is constructed to overcome the optimization, meanwhile, generating the necessary training data. According to the responses of the well-trained models to the different physics-driven features encoding the states to be recognized, we can numerically conclude that the most compact characterization of the Alice-to-Bob steerability is Alice’s regularly aligned steering ellipsoid; whereas Bob’s ellipsoid is irrelevant. We have also provided an explanation to this result with the one-way stochastic local operations and classical communication. Additionally, our approach is versatile in revealing further insights into the hierarchical structure of quantum steering and detecting the hidden steerability.
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.
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.
Quantum entanglement is a key resource for foundational quantum information and communication tasks, such as teleportation, entanglement swapping and quantum key distribution [1]. Quantum steering is a quantum correlation that is distinct from other nonclassical correlations such as Bell nonlocality [2] and entanglement. Steering describes the effect of measuring one system, let's say Alice, and affecting the measurement results on the other system, let's say Bob. Moving through the hierarchy of nonlocal classes, from Bell nonlocality towards nonsepa-rability, requires an increasing number of parties and apparatus that need to be trusted but gives access to protocols which are more robust to noise for projective measurements [3,4]. In a quantum steering task [5] the correlations of Alice and Bob's measurement outcomes are tested against a classical probability bound which can be only violated if the state shared between the parties is quantum steerable.
Phenomena that slip beyond the grasp of our classical intuition reveal uniquely quantum effects that deepen our understanding of the physical world and enable advances in information processing, particularly in quantum communication and computation. One such phenomenon is quantum steering, whereby measurements performed by one party influence the conditional states of another when the two share an entangled quantum system. If the observed correlations cannot be explained by a local hidden state model, the state is said to be steerable. In this work, we investigate the typicality of this behavior: given a generic two-qubit state and mmm Haar-random projective measurements, what is the probability of observing steering? We derive analytical expressions for the steering probability PS\mathcal{P}_SPS​ of Werner states in two- and three-setting scenarios, the latter restricted to coplanar projective measurements on the Bloch sphere. For larger numbers of settings and various random states ensembles, we perform numerical analyses showing that PS\mathcal{P}_SPS​ increases systematically with the number of measurements and substantially exceeds the corresponding probabilities associated with Bell nonlocality. Our results demonstrate that random states with minimal environmental coupling exhibit a high probability of steering for finite mmm and approach genuine typicality, PS=100%\mathcal{P}_S=100\%PS​=100%, as the number of settings increases. We provide a detailed characterization of PS\mathcal{P}_SPS​ across different state ensembles and specific families, including Bell-diagonal and Werner states, identifying those with the greatest non-classical potential and highlighting their relevance for protocols in which steering serves as a key resource.
Quantum steering exhibits an inherent asymmetric property that differs from both quantum entanglement and Bell nonlocality, which leads to no-way, one-way, and two-way steering. Flexibly manipulating the direction of quantum steering without reducing the quantum steerability is very important for information processing. In this paper, implementing sequential unsharp measurements and the steering-radius criterion, we show how to manipulate that steering direction among multiple Alices and Bobs. We demonstrate that the change in steering direction depends on the number of measurement settings and the sharpness of measurements. For a given two-qubit asymmetric state, we obtain 13 kinds of steering regions with different steering directions in the case of one Alice and three Bobs and obtain 11 kinds in the case with two Alices and two Bobs. For the case with one Alice and three Bobs, we further find a volume bound to visualize quantum steering. Our manipulation scheme is experimentally easy to achieve and rather assigns the steerability to more observers without reducing it, which offers a direct reference for the practical application of quantum steering.