# Improving Steerability Detection via an Aggregate Class Distribution Neural Network
## 1 Introduction  
[PERSON] presented the concept of quantum steering [1] based on the well-known EPR paradox [2] in 1935. Although the EPR argument led to long-lasting discussions, there was not much interest in quantum steering. It was not until 2007 when [PERSON], [PERSON], and [PERSON] presented a precise definition and systematic criteria [3] that people began to attach importance to steering. Steering is a quantum correlation between entanglement [4, 5] and non-locality [6], wherein one party can steer the state of another distant party by choosing appropriate local measurements. Currently, steering plays an essential role in diverse quantum information-processing tasks [3, 7, 8] such as one-sided device-independent quantum key distribution [9, 10, 11, 12, 13, 14], channel discrimination [15, 16], randomness certification [17, 18, 19], and tele-amplification [20]. Given the importance of steering in quantum information-processing tasks, it is necessary to detect the steerability of a given bipartite quantum state [3, 7, 8, 15, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. Although various criteria and inequalities for steering have been presented [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32], it is sometimes difficult to determine whether an arbitrary unknown quantum state is steerable, even in the most simple case of a two-qubit state. Theoretically, these criteria can be estimated through semidefinite programming (SDP) [35], but it is very time-consuming and accompanied by misjudgment because of having to consider a wide range of measurement directions in the real computing process [36]. Machine learning can effectively enhance the accuracy and speed of prediction by learning from known data and experience, which has already been adopted for quantum information-processing tasks such as entanglement classification [37], non-locality discrimination [38, 39], phase-transition identification [40, 41], quantum state tomography [42], and Markovianity [43]. In addition, several machine learning methods, such as the artificial neural network, support vector machine (SVM), decision tree, and semi-supervised machine learning, have been used to detect and quantify the steerability of a given state [44, 45, 46, 47, 48]. Among these methods, the SVM [45] and backpropagation neural network (BPNN) [48] are both supervised machine learning methods used to detect steerability. Compared to the SVM [45], BPNN [48] has better learning ability, simpler parameter tuning, and better performance in processing large amounts of data and continuous data features. Therefore, [PERSON], [PERSON], et al. used this method to construct several models to obtain high-performance quantum-steering classifiers. Nevertheless, the BPNN still has the problem of overfitting inaccurate data. The AggMatch algorithm improves the performance of semi-supervised learning by integrating the prediction results of similar samples [49]. Inspired by this method, we present a new aggregate class distribution neural network (AGGNN) structure to overcome the problem of overfitting inaccurate quantum states. The AGGNN creates samples of similar features that have a similar prediction distribution based on the similarity measure in order to improve the generalization ability of networks. Thus, for both the whole information and partial information of a quantum state, AGGNN can construct higher-performance quantum-steering classification models compared to the BPNN and SVM models. In addition, we use the AGGNN classification models to predict the steerability bounds of the generalized Werner states. The steerability bounds of the generalized Werner states predicted by the AGGNN are more similar to the theoretical bounds compared to the SVM [45] and BPNN [48] models. This paper is organized as follows. In Section 2, we introduce the quantum-steering problem and the AGGNN classification model.