Quantum steering is a unique known quantum correlation resource that lies between quantum entanglement and Bell nonlocality in discrete systems. Its most distinctive feature is its inherent asymmetry1,2,3,4,5,6, making it a crucial resource for various quantum information processes. This asymmetry is particularly valuable in tasks such as one-sided device-independent quantum key distribution7,8,9, quantum secret sharing10, secure quantum teleportation11,12,13, and subchannel discrimination14,15, where it not only enhances key generation rates but also improves the efficiency and security of the protocols. Thereby, it is an inevitable task in the process of quantum information to quickly detect whether a quantum state is steerable and to quantify the degree of steering it exhibits.

In quantum information science, continuousvariable (CV) systems demonstrate unique advantages due to their deterministic generation of quantum states, efficient quantum operations, and precise quantum measurements16,17,18,19. This is especially true in the context of Gaussian states and Gaussian operations16,17, which are not only theoretically tractable but also experimentally accessible. Against this background, Gaussian steering was introduced20 and has since attracted significant research attention. Theoretically, Gaussian steering has been confirmed to be a valuable quantum resource21, and this point was more rigorously proven22. As a quantum resource, to make the Gaussian steering applicable, easily accessible quantifications of Gaussian steering have become a goal to strive for. Along this direction, Kogias and Adesso23 proposed a steering quantification for two-mode Gaussian states based on the EPR paradox; Kogias et al.24 introduced a quantification for bipartite Gaussian steering based on the symplectic eigenvalues of the matrix; and more recently, Yan et al.22 provided two quantifications of Gaussian steering using the covariance matrices of Gaussian states. Theoretically, these quantifications may be used in detection of Gaussian steering. However, if the number of modes is large, it will take much longer time to detect Gaussian steering in a bipartite multimode Gaussian state while maintaining high accuracy.

To overcome the computational limitations of traditional quantification methods, the rapid development of machine learning in recent years offers a novel approach. By extracting features and labels from finite data, machine learning models can efficiently perform classification and prediction. In quantum information theory of finite dimensional systems, machine learning has already been successfully applied to the classification and quantification of quantum correlations in finite-dimensional systems, including quantum entanglement25,26,27,28, EPR steering29,30,31,32,33, and Bell nonlocality34,35,36, as well as in quantum feature detection37,38,39, solving many-body problems40,41,42, and identifying phase transitions43,44. Usually, in these literatures, Support Vector Machine (SVM) and Back Propagation neural network (BPNN) have been utilized.

SVM45 and BPNN46,47 are two typical techniques in the field of machine learning. SVM is renowned for its remarkable capability in handling high-dimensional data and its efficient performance when data boundaries are clear. However, this method entails a relatively high time cost for training classifiers, and the difficulty of training increases significantly as the complexity of the training data rises. In contrast, the BPNN is generally faster to train and excels in recognizing complex patterns and generalizing from training data, achieving high accuracy, particularly with balanced datasets. However, this still leaves a challenge in effectively addressing imbalanced datasets. To tackle this issue, Shu et al. have recently introduced a model called the Meta-Weight-Net Neural Network (MWN), which specifically targets the challenges posed by imbalanced data and enhances classification performance in such scenarios48,49,50. MWN leverages meta-learning strategy to adjust sample weights, effectively tackling the challenges posed by imbalanced datasets and significantly enhancing detection robustness. Furthermore, due to the complexity of the data and the effects of noise, models derived from single methods often exhibit unstable predictive performance, with overfitting and underfitting being common issues. To address this challenge, ensemble learning method is typically employed to enhance predictive performance51,52. The primary idea behind this method is to combine the diverse features and strengths of various models, thereby balancing the weaknesses of individual models, reducing prediction errors, and improving generalization capabilities.

Although machine learning has shown great success in finite-dimensional systems, its application to CV systems, particularly in the detection of Gaussian steering, remains largely unexplored. The primary challenge lies in the infinite-dimensional nature of CV systems. Fortunately, the quantification of Gaussian steering proposed in22 offers a solution by transforming the Gaussian steering detection problem into a classification problem involving the covariance matrices. This transformation effectively reduces the infinite-dimensional problem to a finite-dimensional one, making it possible to detect Gaussian steering by analyzing the properties of the covariance matrices of Gaussian states.

In this study, we innovatively apply machine learning techniques to detect Gaussian steering, specifically employing SVM, BPNN, and MWN methods. We then integrate the models obtained from these three approaches to create an ensemble model for further detection. Using the quantification proposed22, our dataset construction reveals a pronounced imbalance: unsteerable Gaussian states vastly outnumber steerable ones in CV systems. This effect is particularly evident when the untrusted party possesses significantly more modes than the trusted party, in stark contrast to observations in finite-dimensional discrete-variable (DV) systems. As MWN is specifically designed to effectively handle and optimize the processing of imbalanced datasets, to tackle this imbalance between steerable and unsteerable states, we utilize MWN as one of our methods. This is one point that distinguishes the present paper from others.

In order to compare the advantages and disadvantages of various methods, we construct practically three types of datasets: balanced dataset, naturally distributed dataset that preserves the original characteristics of the data, and augmented imbalanced dataset. We then conduct extensive training and testing of four machine learning models across these three datasets. Although the experimental results validate the effectiveness and efficiency of these methods in detecting Gaussian steering, revealing the significant potential and promising prospects of machine learning techniques in addressing complex quantum correlations in CV quantum systems, our results also give suggestions which method is better corresponding to different application scenarios. This research not only enriches the toolkit of quantum information science but also provides new ideas and directions for the future development of quantum technologies.

The paper is organized as follows. Section 2 details the preparation process for both balanced and imbalanced datasets, and applies data augmentation strategy to improve model training for the imbalanced dataset. We then outline the specific processes for training classifiers using SVM, BPNN, MWN, and ensemble learning methods based on different datasets (including balanced, imbalanced, and augmented imbalanced datasets). In Section 3, we conduct numerical experiments to provide a detailed evaluation by comparing the detection accuracy and speed of the trained classifiers on specific \((n+m)\)-mode Gaussian states. Then we shortly summarize our work in Section 4. Preliminaries concerning Gaussian steering and machine learning models involved and some more details on numerical experiments are provided in the Supplementary material.

Machine learning training on Gaussian steering

To overcome the computational limitations of traditional Gaussian steering quantification methods, we turn to machine learning methods, which have demonstrated outstanding performance in handling complex pattern recognition and prediction tasks. That is, with its strong generalization ability to adapt to various scenarios, we utilize machine learning serving as a powerful tool for detecting the steering contained in Gaussian states.

Preparation of training samples

To train the Gaussian steering classifiers for an \((n+m)\)-mode CV system, it is necessary to collect covariance matrices \(\Gamma _\rho\) of \((n+m)\)-mode Gaussian states \(\rho\) as training samples and to extract relevant features from these matrices. We can generate covariance matrix \(\Gamma _\rho\) randomly in the following way: First, a random real matrix F of order \(2(n+m)\) is created. Then, we form a Hermitian matrix \(F^\dag F\). Since the covariance matrix of a Gaussian state must be invertible, a small perturbation term proportional to the identity operator is introduced to ensure invertibility. To increase the randomness of the generated data, we apply additional technical processing, resulting in the actual covariance matrix being \(k(F^\dag F + 0.0001I)\), which satisfies the condition \(k(F^\dag F + 0.0001I)+i(\Omega _{A}\oplus \Omega _{B})\ge 0\), where \(k\in \mathbb R^+\), \(\Omega _{A}=\Omega _{n}\in \mathcal {M}_{2n}({\mathbb R})\) and \(\Omega _{B}=\Omega _{m}\in \mathcal {M}_{2m}({\mathbb R})\) are the symplectic matrix units (see Supplementary material A for more details concerning Gaussian states).

The training samples are generated as follows. We create l random covariance matrices by the way mentioned above and derive labels based on the property of \({\mathcal {J}}(\rho ),\) where \({\mathcal {J}}(\rho )={\Vert \Gamma _\rho +0_A\oplus i\Omega _B\Vert _1}-{\textrm{Tr}(\Gamma _\rho )},\) \(\Gamma _\rho\) is the covariance matrix of \(\rho\) and \(\Vert \cdot \Vert _1\) stands for the trace-norm. As \({\mathcal {J}}(\rho )\) depends only on the covariance matrix of the Gaussian state, we may denote \(\mathcal {J}(\Gamma _\rho )= {\mathcal {J}}(\rho )\). For any \((n+m)\)-mode Gaussian state \(\rho\), \(\mathcal {J}(\Gamma _{\rho }) = 0\) if and only if \(\rho\) is unsteerable from A to B through Alice’s Gaussian measurements; conversely, \(\mathcal {J}(\Gamma _{\rho }) > 0\) indicates steering. This allows us to label the samples accordingly. If the covariance matrix \(\Gamma _{\rho }\) corresponds to a steerable Gaussian state \(\rho\), the label \(y = 1\); otherwise, \(y = 0\). Consequently, we collect a batch of labeled data \(\mathcal {D}= \{(\Gamma _{i}, y_{i})\}_{i=1}^{l}\), consisting of covariance matrices for both steerable and unsteerable Gaussian states. This labeled dataset serves as the foundation for training the classifiers.

Notably, during the process of constructing the dataset \(\mathcal {D}= \{(\Gamma _{i}, y_{i})\}_{i=1}^{l}\), we observe that the number of unsteerable Gaussian states far exceeded that of steerable states. Achieving a balance between the numbers of steerable and unsteerable states required extremely high computational costs. This made collecting samples labeled as +1 (indicating steerable states) especially challenging, particularly as the number of samples increased. For instance, in the case of randomly generating (5+1)-mode Gaussian states, despite generating over 100 million covariance matrices, only 100,000 covariance matrices of Gaussian steerable states were obtained, and the entire process took approximately 500 hours on a computer Intel(R) Xeon(R) Gold 5318Y CPU @ 2.10GHz. This phenomenon demonstrates that the cost of generating steerable states increases significantly with the number of modes in bipartite Gaussian states. Additionally, it is important to note that the covariance matrices corresponding to Gaussian states of different modes are completely random and independent.

We recognize that the generation of balanced datasets incurs significant computational cost, which can be prohibitive for large-scale tasks. To address the computational challenge associated with generating balanced datasets, particularly for high-mode systems like the (5+1)-mode Gaussian system, alternative strategies should be considered. In the present paper, to better reflect real conditions and develop models that are more suited to the practical imbalance in the distribution of Gaussian steerable and unsteerable states, we take three construction processes of the datasets: balanced datasets, naturally imbalanced datasets and augmented imbalanced datasets via data augmentation strategies.

Balanced dataset

For given large positive integer l, to construct a balanced dataset \(\mathcal {D}_{B}= \{(\Gamma _{i}, y_{i})\}_{i=1}^{l}\) of \((n+m)\)-mode Gaussian states for training the quantum steering classifier, it is essential to generate training data with equal representation. So our objective is to create a balanced dataset by generating an equal number of steerable and unsteerable Gaussian state covariance matrices, labeled \(+1\) and 0 respectively, according to the value of \(\mathcal {J}\). Given that the number of covariance matrices corresponding to steerable states is typically limited, we continue to randomly generate matrices until a sufficient number of steerable samples (i.e., \(\frac{l}{2}\)) is collected. During this process, many, say t unsteerable covariance matrices are produced. Note that, \(t\gg \frac{l}{2}\). For instance, in the situation of \((5+1)\)-mode, with \(l=200,000\), the value of t achieves over 100, 000, 000. To ensure randomness, \(\frac{l}{2}\) unsteerable matrices are randomly selected in such t produced unsteerable covariance matrices. This results in a dataset with half of the labels as +1 (indicating steerable states) and the other half as 0 (indicating unsteerable states). Such a balanced distribution of data enhances the ability to learn and differentiate between steerable and unsteerable states.

Naturally imbalanced dataset

In this scenario, the covariance matrices of Gaussian states are randomly generated, and, similar to the balanced approach, labels are assigned based on the value of \(\mathcal {J}\). However, unlike the balanced situation, there is no attempt to equalize the number of steerable and unsteerable states. In other words, labels are assigned simply upon generating the covariance matrices. As a result, a significant imbalance emerges in the dataset \(\mathcal {D}_{I}= \{(\Gamma _{i}, y_{i})\}_{i=1}^{l}\), with the number of unsteerable Gaussian states \(l_1\) vastly outnumbering the number of steerable states \(l_2\), such that \(l_1 + l_2 = l\). This natural distribution results in a pronounced imbalance within the dataset. To quantify this imbalance, the imbalance factor \(\xi = \frac{l_{1}}{l_{2}}\) is introduced, defined as the ratio of the number of training samples in the majority class (unsteerable Gaussian states) to the number of samples in the minority class (steerable Gaussian states). For example, in the case of \((5+1)\)-mode, when we prepare 100 million samples, the imbalance factor \(\xi\) achieves up to \(\xi \approx \frac{100,000,000-100,000}{100,000}\approx 1,000\). In this way, we generate an imbalanced dataset \(\mathcal {D}_{I}= \{(\Gamma _{i}, y_{i})\}_{i=1}^{l}\) directly.

Augmented imbalanced dataset via data augmentation strategies

By analyzing above two scenarios, we aim to handle different data distributions more flexibly, thereby enhancing the performance of machine learning models in Gaussian steering detection. However, the significant imbalance factor in naturally distributed datasets, as discussed by Shu et al.48, results in suboptimal generalization capabilities for models derived using the MWN method. Inspired by data augmentation strategies in image processing, such as flipping and shifting, we explore whether similar approaches can be employed to increase the number of covariance matrices corresponding to steerable Gaussian states. This could help reduce the imbalance factor and improve the model’s generalization ability. Therefore, we apply data augmentation strategy to create a third dataset.

The properties of \(\mathcal {J}\) discussed in Yan et al.22 indicate that \(\mathcal {J}\) is invariant under orthogonal transformations of the covariance matrices of Gaussian states, that is, the property (2) mentioned in Supplementary material, which serves as the foundation for our augmentation strategy.

For any \((n+m)\)-mode Gaussian state \(\rho\) with covariance matrix \(\Gamma _{\rho }\), based on the aforementioned property of the quantification \(\mathcal {J}\), we have

$$\begin{aligned} \begin{array}{ll} \mathcal {J}((K_{A}\oplus K_{B})\Gamma _{\rho }(K_{A}\oplus K_{B})^\dag ) = \mathcal {J}(\Gamma _{\rho }), \end{array} \end{aligned}$$
(1)

where \(K_{A} \in \mathcal {M}_{2n}(\mathbb {R})\) is an orthogonal matrix and \(K_{B} \in \mathcal {M}_{2m}(\mathbb {R})\) is an orthogonal symplectic matrix. Let \(\mathcal {X}\) represent a set of covariance matrices of Gaussian states, and define the set \(\mathcal {M}(\mathcal {X})\) as the collection of covariance matrices obtained by performing random orthogonal transformations described as in Eq.(1) on steerable covariance matrices within \(\mathcal {X}\). This ensures that the number of covariance matrices of steerable Gaussian states matches that of unsteerable Gaussian states in \(\mathcal {X}\). Notably, our augmentation methods are non-destructive. Through this approach, we obtain the augmented dataset \(\mathcal {D}_{A}= \{(\Gamma _{i}, y_{i})\}_{i=1}^{l}\) which decreases the ratio \(\xi = \frac{l_{1}}{l_{2}}\) of the number of unsteerable Gaussian states to the number of steerable Gaussian states.

Training process

To explore high-performance quantum steering classifiers for Gaussian states in CV systems with limited information, and to enhance the detection efficiency of arbitrary bipartite Gaussian state steering, we select three datasets \(\mathcal {D}_{B}\), \(\mathcal {D}_{I}\), and \(\mathcal {D}_{A}\). The feature vectors for these datasets consist of \((n+m)[1+2(n+m)]\) elements, derived from the upper triangular part of the covariance matrices, which are Hermitian and symmetric.We then train Gaussian steering classifiers for each of these three datasets. Note that throughout the training and testing process, we consider an accuracy of over 80% to be effective.

The case of balanced dataset

For the \((n+m)\)-mode dataset \(\mathcal {D}_{B}= \{(\Gamma _{i}, y_{i})\}_{i=1}^{l}\) , we utilize SVM, BPNN, and MWN methods for training and testing. After obtaining the respective models, we proceed with ensemble learning. Each method employs four-fold cross-validation and grid search optimization to identify the corresponding hyperparameters. The specific training process is outlined as follows:

In the case of SVM, the radial basis function (RBF) kernel function, \(K\left( \textbf{x}_i, \textbf{x}_j\right) = \exp \left( -\gamma \Vert \textbf{x}_i - \textbf{x}_j\Vert ^2\right) ,\) is utilized as specified in Supplementary material. To achieve optimal performance, hyperparameters undergo fine-tuning through grid search during the model training process, aiming to minimize classification error effectively.

For the BPNN approach, the training process is guided by the Adam optimizer. Each node in the four fully connected hidden layers applies a unique linear transformation to the input vector from the previous layer, followed by a nonlinear Rectified Linear Unit (ReLU) activation function, as illustrated in Supplementary material. The loss function is calculated using binary cross-entropy, comparing the true Gaussian steering labels \(y_{\Gamma _{\rho }}^{\text {true}}\) with the predicted labels \(y_{\Gamma _{\rho }}^{\text {pred}}\) generated by the neural network. Training continues iteratively until the loss function converges. As detailed in Table 1, the network architecture consists of two convolutional layers followed by five fully connected layers. The convolutional layers handle the input processing, while each fully connected layer employs a ReLU activation. The final output layer contains two nodes, indicating the probability of a sample belonging to either class in this binary classification task. Outputs greater than or equal to 0.5 are classified as steerable, whereas those below 0.5 are classified as unsteerable.

Table 1 The Structure of the BPNN.
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Regarding the MWN neural network, the stochastic gradient descent (SGD) optimization algorithm is employed to train the loss function. Each training cycle samples an unbiased mini-batch of labeled data with a balanced distribution for meta-learning, which aims to derive the optimal parameters for the MWN. The hyperparameter learning process utilizes the neural network structure depicted in Supplementary material. This architecture features an multilayer perceptron (MLP) network with a hidden layer containing 100 nodes, as shown in Table 2. Each hidden node applies a nonlinear ReLU activation function, while the output layer employs a Sigmoid function to ensure output values fall within the [0, 1] range. This network acts as a universal approximator for nearly any continuous function, allowing it to adapt various weighting functions. The loss function used remains consistent with that applied in the BPNN, and the training process iteratively progresses toward reducing the target loss until convergence is achieved. The MLP network thus comprises one input layer, multiple hidden layers, and an output layer, with each hidden layer utilizing the ReLU activation function.

Table 2 The Structure of MLP Network.
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The integration of the three trained machine learning models aims to enhance overall classification performance through an ensemble approach. The process is executed as follows: each base learner is first trained on the training dataset and subsequently generates predictions on the test dataset. These predictions serve as new features for the logistic regression model, as outlined in Supplementary material. This logistic regression model ultimately produces the final predictions of the ensemble. By effectively combining the predictions from various models, this approach significantly improves classification performance.

To evaluate model performance on unknown data, we implemented a rigorous cross validation process to maintain the independence of the training and test sets. The average accuracy from the four fold cross validation, also referred to as the cross validation accuracy, served as the benchmark for model performance. This accuracy information is used to iterate the training process and generate the final classifier. Furthermore, the generalization ability is validated using an independent test set.

The case of naturally imbalanced dataset

When addressing the issue of imbalanced datasets \(\mathcal {D}_{I}= \{(\Gamma _{i}, y_{i})\}_{i=1}^{l}\), we first leverage the inherent advantages of MWN in handling imbalanced data by dynamically adjusting sample weights to focus on optimizing the recognition of minority class samples. During the training of models using SVM and BPNN, since the number of steerable Gaussian state samples \(l_2\) is significantly fewer than that of unsteerable samples \(l_1\) (i.e., \(l_2 \ll l_1\)), specific strategy is required to enhance model performance.

To address this, we adjust the class weight parameters in SVM to increase the impact of minority class samples during model training. The class weight parameter is defined as \(w_i = \frac{l}{l_i},\) where \(w_i\) is the weight for class i, l is the total number of samples, \(l_i\) is the number of samples in class i. This definition increases the weight of the class with fewer samples, ensuring the model pays more attention to minority class instances. Additionally, we introduce the Focal loss function for BPNN, which is expressed as \(\mathcal {L}(p_t) = -\alpha _t (1 - p_t)^{\gamma } \log (p_t),\) where \(p_t\) represents the predicted probability for the true class (i.e., the probability of the sample being correctly classified as steerable if \(t = 1\) or unsteerable if \(t = 0\)), \(\alpha _t\) denotes the weight that balances the steerable and unsteerable classes, and \(\gamma\) is the focusing parameter that highlights difficult-to-classify samples. These weighting mechanisms aim to mitigate the negative impact of data imbalance on model performance. The network structures for BPNN and MWN remain consistent with those used for balanced datasets, as detailed in Tables 1 and 2. Once training is complete, we use the predictions from these models on the test set for ensemble learning, integrating the three models trained on the imbalanced dataset. This integration process is consistent with that used for balanced datasets, allowing us to leverage the strengths of different models to enhance overall classification performance and improve robustness against imbalanced data. Ultimately, the results from ensemble learning further optimize the model predictive accuracy and stability, providing a more effective solution for the classification task.

The case of data augmentation strategy

Due to the significant imbalance factor in the naturally distributed dataset, the generalization ability of the model obtained using the MWN method may not be good enough48. Thus we use a data augmentation strategy to increase the number of steerable Gaussian state samples, aiming to enhance the generalization capability of the trained machine learning model. Specifically, we use orthogonal transformations to generate additional steerable state samples, guiding the dataset toward the hypothesized optimal imbalance state without significantly increasing computational costs. Through this approach, we create a new dataset with a specific imbalance factor \(\mathcal {D}_{A}= \{(\Gamma _{i}, y_{i})\}_{i=1}^{l}\). The details of this Data Augmentation Strategy are described in Supplementary material.

On the new enhanced imbalanced dataset \(\mathcal {D}_{A}= \{(\Gamma _{i}, y_{i})\}_{i=1}^{l}\), we also train SVM, BPNN, and MWN. Similar to our approach for handling the previous imbalanced dataset, we continue to use class weight parameters in SVM and the Focal loss function in BPNN. Furthermore, we proceed with the ensemble learning process to further enhance the classification performance of the models, consistent with the approach used for balanced datasets. Experimental results demonstrate that the data augmentation strategy significantly improves model classification performance. This strategy not only enhances the fairness and scientific validity of the evaluation but also provides a valuable reference for addressing imbalanced data issues in practical applications.

In addressing the two imbalanced data scenarios mentioned above, we employ four-fold cross-validation for all four machine learning algorithms, similar to our approach with balanced dataset. During this process, the entire training set is randomly divided into four equal parts, with each part serving sequentially as the validation set while the remaining parts act as the training set. This method aims to identify the optimal classifier. To validate the generalization ability of the models obtained through the four training processes, we use a separate test set. It is important to note that the average accuracy of the four classifiers obtained from cross-validation is referred to as cross-validation accuracy, while the accuracy on the test set is called test accuracy.

Numerical results

In this section, we take a few examples to examine how well the training scheme proposed in Section 2, which uses machine learning methods to identify \((n+m)\)-Gaussian states, with a particular focus on the performance of the trained Gaussian steering classifiers. We select seven different cases of mode-types: \((1+1)\), \((1+2)\), \((2+1)\), \((2+2)\), \((1+5)\), \((5+1)\) and \((5+5)\), and generate corresponding datasets of steerable and unsteerable bipartite Gaussian states for each mode. Subsequently, we train the Gaussian steering classifiers using these four machine learning methods described in Section 2 and analyze their classification performance. All our numerical experiments are conducted on an Intel(R) Xeon(R) Gold 5318Y CPU @ 2.10GHz platform and the datasets are available at https://github.com/GuoJie1112/Gauss-Steering.

Each case of seven mode-types has three scenarios in its datasets, which we discuss separately. First, we evaluate the performance of the Gaussian steering classifier on balanced datasets and analyze its classification accuracy across modes. Second, we examine how the classifier addresses data imbalance in naturally generated imbalanced datasets, focusing on its ability to identify minority class samples. Finally, we explore whether data augmentation strategy can significantly improve classification accuracy on the generated datasets. This analysis will help us gain valuable insights into the performance of Gaussian steering classifiers across various data distributions and modes, facilitating future research and practical applications.

Detecting Gaussian steering on balanced datasets

To validate the performance of SVM, BPNN, MWN, and ensemble learning algorithms under different Gaussian state modes

$$\begin{array}{rl} (n+m)\in & \{(1+1), (1+2), (2+1), \\ & (2+2), (1+5), (5+1), (5+5)\},\end{array}$$

by the approach described in Section 2, we create a balanced dataset \(\mathcal {D}_{B}= \{(\Gamma _{i}, y_{i})\}_{i=1}^{l}\) for each mode in a total sample size of \(l=200,000\) with 100,000 samples labeled as \(+1\), indicating Gaussian steerable, and 100,000 samples labeled as 0, indicating Gaussian unsteerable.

Once the dataset was constructed, we proceeded with the machine learning training process. During the training process, we reserved the last 20,000 positive and 20,000 negative samples as a test set \(\mathcal {D}_{C}\), while the remaining 80,000 positive and 80,000 negative samples were used for training. To select the best hyperparameters, we employed four-fold cross-validation and grid search methods. Specifically, the SVM, BPNN, and MWN used cross-validation to tune the model hyperparameters, dividing the dataset into 80% for training and 20% for testing. In the training phase, four-fold cross-validation randomly divided the training set into four equal parts, each time selecting one part as the validation set while using the others for training. The average accuracy across the four validations is referred to as the cross-validation accuracy and is used for iterative model optimization to produce the final classifiers. After obtaining the four final classifiers, we performed ensemble learning to combine the diverse features and strengths of different models, thereby mitigating the weaknesses of individual models. The specific training results are presented in Fig. 1.

Fig. 1
figure 1

Accuracy of different classifiers for various \((n+m)\)-modes Gaussian states in balanced datasets.

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Figure 1 illustrates eight different performance metrics, including the cross-validation and test accuracies for SVM, BPNN, MWN, and ensemble learning. The accuracy for each test set is evaluated based on 20,000 randomly generated steerable and unsteerable Gaussian state samples. In the figure, the final cross-validation accuracy is indicated by blue shading, while the classification accuracy on the random test set formed from the retained 20,000 samples is indicated by green shading. The results show that all accuracies exceed 98%, clearly indicating that the models were well trained. The ensemble learning model performed exceptionally well, achieving an overall accuracy above 99%. Aside from the ensemble learning model, the BPNN also demonstrated stable results across all possible Gaussian state modes, with an accuracy consistently above 99%, indicating its strong generalization capability. However, due to the lengthy training time of the SVM, which exceeded one week, it is recommended to directly use the BPNN for further experiments in practical applications.

However, as point out in Section 2, a big shortcoming of preparing balanced datasets is the high computational cost. For instance, in the \((5+1)\)-mode case, generating a balanced dataset of 100,000 samples labeled as \(+1\) requires the random generation of up to 100 million covariance matrices, which consumes substantial computational resources and requires considerable amounts of time.

Detecting Gaussian steering on naturally imbalanced datasets

In this subsection we examine the performance of classifiers trained on naturally generated imbalanced datasets across different Gaussian state modes using four machine learning algorithms SVM, BPNN, MWN, and ensemble learning algorithms. Specifically, we create corresponding datasets \(\mathcal {D}_I= \{(\Gamma _{i}, y_{i})\}_{i=1}^{l}\) as described by Section 2 for each of modes \((n+m)\in \{(1+1)\), \((1+2)\), \((2+1)\), \((2+2)\), \((1+5)\), \((5+1), (5+5)\}\). We in fact generate randomly a middle size datasets containing \(l=300,000\) covariance matrices of Gaussian states, labeled each sample with \(+1\) or 0 according to whether the Gaussian state associated is steerable or unsteerable. It is important to note that the distribution of positive and negative labels varies across different modes. Detailed information on label generation for each mode can be found in Table 3.

Table 3 Imbalance factors of \((n+m)\)-mode Gaussian state covariance matrices.
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It is observed that, in naturally generated datasets, class imbalance is particularly severe: the number of unsteerable states (negative samples) significantly exceeds that of steerable states (positive samples). When the modes of the untrusted party are greater than that of the trusted party, this imbalance is more prominent.

As \(\mathcal {D}_I\) contains no sufficient many positive samples (steerable ones), we selected randomly 20,000 positive samples (steerable ones) and 20,000 negative samples (unsteerable ones) from the balanced dataset \(\mathcal {D}_B\) created in Subsection A as a common test set \(\mathcal {D}_C\), on which each model is tested and its classification accuracies is predicted. For a more intuitive comparison of the precision values of the four models, we present Fig. 2. In this figure, the purple line with “\(\circ\)” markers represents the precision achieved by the SVM model, the green line with “\(\square\)” markers indicates the precision obtained by the BPNN model, and the orange line with “\(\star\)” markers shows the precision of the MWN model. The blue line with triangular markers represents the precision obtained by the ensemble learning model.

Fig. 2
figure 2

Accuracy of different classifiers for various \((n+m)\)-modes in naturally imbalanced datasets.

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From Fig. 2, it is evident that all four models exhibit high training accuracy, exceeding 98%. This directly contributes to the ensemble learning model achieving an accuracy of over 99%. However, due to the influence of data imbalance, their testing accuracy shows a stark contrast. None of the models reached a classification accuracy of over 73% on the test set. The SVM model had the lowest accuracy, ranging from 60% to 63% across different \((n+m)\)-mode Gaussian states. Notably, its accuracy declined further with increasing imbalance. The performance of the BPNN model is significantly affected by changes in the imbalance factor of the dataset, with testing accuracy ranging from 60% to 67%. The MWN model achieved an accuracy ranging from 68.6% to 70%, demonstrating relatively better generalization capability. As a result of the influence from all three models, the testing accuracy of ensemble learning model ranged from 69% to 71%. In all cases, the test accuracy failed to reach the 80% target. Our results indicate that the low test accuracy is attributed to the high imbalance in the datasets. The greater the imbalance factor \(\xi =\frac{l_1}{l_2}\) is, the lower the test accuracy becomes. This indicates that structural bias in the data distribution introduces intrinsic difficulties for the learning task.

This imbalance is not merely a statistical artifact but arises from the underlying physical properties of Gaussian quantum steering in CV systems. Our analysis reveals, for the first time, that when the number of modes on the untrusted party exceeds that of the trusted party, the likelihood of generating steerable states drops significantly. This behavior reflects the nature of Gaussian steering asymmetry. For instance, in the \((5+1)\)-mode configuration, when we generate randomly 300,000 Gaussian states, steerable states are extremely rare among naturally generated samples, and the class imbalance factor reaches 365.75 (see Table 3). Under such extreme imbalance, classifiers tend to overfit the majority class (unsteerable states), leading to reduced overall accuracy and especially poor recall on steerable states. This physically induced skew in the sample distribution distorts the learned decision boundaries, weakening generalization in the positive class region and resulting in significant performance fluctuations across different mode configurations (see Fig. 2b). To address this structurally driven imbalance, we propose a data augmentation strategy to enrich the set of steerable state samples in Subsection 2 A. By applying symplectic transformations within the steerable region, we generate a diverse and physically consistent set of additional samples. This approach not only reduces the imbalance factor but also preserves the Gaussian structure and steering properties of the original states. Consequently, the classifier becomes more sensitive to the asymmetry inherent in Gaussian steering and learns more robust and physically meaningful decision boundaries. In the following subsections, we will demonstrate how this augmentation strategy improves classification performance and robustness across various CV quantum systems with different mode configurations.

Detecting Gaussian steering on imbalanced datasets via augmentation strategy

Due to the inherent severe imbalance between the steerable Gaussian states and the unsteerable Gaussian states, the generalization ability of models trained from naturally generated datasets fall short of expectations, as shown in previous subsection. Consequently, in this subsection, we proceed to examine the impact of applying data augmentation strategy.

Given the significant imbalance between steerable and unsteerable Gaussian state datasets \(\mathcal {X}\) in CV systems, for example, the naturally imbalanced dataset \(\mathcal {D}_I = \{ (\Gamma _{i}, y_{i}) \}_{i=1}^{l}\) as discussed above. To increase the number of covariance matrices for steerable Gaussian states, we implemented data augmentation strategy to generate an augmented collection \(\mathcal {M}(\mathcal {X})\) of steerable Gaussian state covariance matrices by the method described in Section 2.

A key question during the data augmentation process is how to appropriately set the imbalance factor \(\xi\) of the augmented dataset to optimize the performance of the classification model. The choice of \(\xi\) directly determines the proportion of minority class samples (steerable states) in the augmented dataset \(\mathcal {D}_A = \{(\Gamma _i, y_i)\}_{i=1}^{l}\) from the naturally imbalanced dataset \(\mathcal {D}_I\), thereby influencing the model’s ability to learn the characteristics of steerable states. When \(\xi = 1\), the dataset \(\mathcal {D}_A\) is fully balanced, making the model effectively learn their features and improve classification performance. However, this fully balanced strategy may lead to overfitting while significantly increasing computational costs. This issue is particularly critical in high-mode Gaussian states classification tasks, where dataset construction costs grow exponentially with data dimensionality, rendering fully balanced strategies impractical under limited computational resources. Conversely, when \(\xi\) is set to a large value, approaching the level of the naturally imbalanced dataset \(\mathcal {D}_I\) (e.g., \(\xi = 169.65\) in \((1+1)\)-mode), computational costs remain relatively low. However, under this setting, the augmented dataset remains highly imbalanced, limiting the ability of model to learn the features of steerable states and ultimately impairing overall classification performance. Therefore, in data augmentation strategies, selecting an appropriate \(\xi\) is crucial for balancing the detectability of steerable states and the associated computational costs. The augmented dataset must contain a sufficient number of steerable states to facilitate effective learning while avoiding excessive computational burdens.

To systematically evaluate the impact of \(\xi\), we conduct experiments on the \((1+1)\)-mode quantum state classification task, focusing on how different values of \(\xi\) affect the ability of model to recognize steerable states. The evaluation not only considers accuracy as a global performance metric but also incorporates confusion matrix analysis to provide a more detailed examination of classification outcomes. The experiments are performed with candidate values \(\xi \in \{120, 100, 80, 50, 40, 30, 20\}\), compared against two baseline settings: \(\xi = 1\) (fully balanced) and \(\xi = 169.65\) (naturally imbalanced). The analysis comprehensively examines of the generated BPNN model and MWN model classification accuracy, recall of steerable states, and dataset construction costs. The detailed experimental results are presented in Fig. 3, and Table 4.

Fig. 3
figure 3

The test accuracy of different classifiers for various imbalance factors of augmentation strategy for \((1+1)\)-mode.

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Table 4 The time of augmenting datasets for different imbalance factors of \((1+1)\)-mode Gaussian states.
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Experimental results indicate that as the imbalance factor \(\xi\) decreases, the ability of the model to detect minority class samples, which correspond to quantum steerable states, improves significantly. When \(\xi \ge 80\), the confusion matrix reveals a high misclassification rate for steerable states, with some of them being incorrectly classified as unsteerable states (the majority class), thereby degrading overall classification performance. Reducing \(\xi\) to 50 leads to a notable increase in the recall of steerable states, demonstrating that data augmentation effectively enhances the ability of model to learn their distinguishing features. Furthermore, as \(\xi\) decreases, overall classification accuracy improves, suggesting that while the representation of steerable states is strengthened, the classification of unsteerable states remains stable. However, this improvement comes at the cost of a significant increase in computational overhead.

Based on the preceding experimental analysis, we initially adopt \(\xi = 50\) as the standard imbalance factor for data augmentation. This configuration is uniformly applied across all modes \((n+m) \in \{(1+1), (1+2), (2+1), (2+2), (1+5), (5+1), (5+5)\}\). Relevant experimental results and detailed analysis are provided in Supplementary material C. As illustrated by the accuracy metrics and confusion matrices Figures 7–9 in Supplementary material C, this setting offers a favorable balance between classification performance and computational efficiency. Specifically, it ensures an overall classification accuracy consistently close to 80%, representing a clear improvement over the baseline with naturally imbalanced data.

However, further inspection of the confusion matrices and accuracy indicators reveals that, although the overall performance has improved and meets the target 80%, the recall for steerable states, which are typically the minority class, remains suboptimal in highly imbalanced configurations such as the \((5+1)\)-mode, where the number of modes held by the trusted party is significantly smaller than that of untrusted one. For example, in \((5+1)\)-mode, the recall rate for steerable states reaches only 53.33% with BPNN, 67.11% with MWN, and 71.55% with the ensemble model, still falling short of the desired 80%. This suggests that a persistent bottleneck remains in recognizing the minority class under such imbalance conditions. To further improve the ability of model to detect steerable states, we further reduce the imbalance factor to \(\xi = 30\), thereby constructing an augmented dataset \(\mathcal {D}_A\) with improved minority class representation (see Table 5 for specific class distributions). The classification performance is evaluated across different mode configurations using four machine learning models: SVM, BPNN, MWN, and an ensemble model. All models are trained on such new augmented datasets \(\mathcal {D}_A\) with imbalance factor \(\xi =30\) and tested on a common test set \(\mathcal {D}_C\) to evaluate generalization performance.

Table 5 Information on the augmented datasets \(\mathcal {D}_{A}\) of Gaussian states.
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Fig. 4
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Classification accuracy of different models for various \((n+m)\)-mode with augmentation strategy (imbalance factor \(\xi =30\)).

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Figure  4 presents the classification accuracy of different models across various \((n+m)\)-mode configurations, while Figure 13 in Supplementary material C further illustrates the class-wise performance through confusion matrices for the case \(\xi =30\). Compared to both the natural distribution and the \(\xi = 50\) augmentation strategy, the dataset generated with \(\xi = 30\) consistently improves the recall of the minority class (i.e., steerable states) across all four models, without compromising overall classification performance. For the ensemble model, the classification accuracy for steerable states exceeds 80% in all modes except \((5+1)\)-mode, where it still reaches 78.57%, closely approaching the target threshold. In the \((5+5)\)-mode, the performance peaks at 81.16%. Meanwhile, the recognition accuracy for the majority class remains largely unaffected, and the overall accuracy consistently stays above 85%, indicating strong balance and generalization capability.

The comparative results shown in Figs.  2, 4, and 7–13 systematically illustrate the specific pathway and effectiveness of the data augmentation strategy in improving model performance. Under the naturally imbalanced data condition, where positive (steerable) state samples are extremely scarce, models trained using machine learning typically exhibit high false negative (FN) rates, leading to significant performance degradation. For instance, the accuracy of MWN for steerable states in the \((1+5)\)-mode is only 47.64%. After applying the imbalance factor \(\xi = 50\), this accuracy improves significantly to 70.59%. Further increasing the augmentation strength to \(\xi = 30\) boosts the accuracy to 75.88%. The ensemble model, which combines the strengths of the three base models, not only achieves the lowest FN rate but also maintains overall accuracy above 85%, making it the better solution for steerable state identification. This also highlights the effectiveness of the augmentation strategy in addressing the class imbalance issue.

Although the augmentation strategy with imbalance factor \(\xi = 30\) significantly improves the detection of steerable states in high-mode scenarios, particularly when the number of modes held by the untrusted party greatly exceeds that of the trusted party, it also incurs increased computational and storage costs. As the augmentation strength increases (i.e., as the imbalance factor decreases), the resources required to generate a balanced dataset grow substantially. Nevertheless, such augmentation leads to higher overall classification accuracy as well as improved precision for each individual class. Therefore, in practical applications, the choice of imbalance factor should be made by carefully balancing the available computational resources against the desired level of detection accuracy.

Following extensive experimental validation and analysis, \(\xi = 30\) is adopted in this paper as the standard imbalance factor for data augmentation. This configuration not only stabilizes overall classification performance but also markedly enhances the model’s ability to identify steerable states in challenging regimes, such as when the number of modes held by the untrusted party far exceeds that of trusted ones. It thus achieves a favorable balance between accuracy and efficiency in classifying Gaussian steerable versus unsteerable states in CV systems. All subsequent analyses are performed under this setting.

Detection time for Gaussian steering

We further analyze the time required to detect Gaussian steering. With four classifiers established across three datasets, we can directly utilize these classifiers to recognize the steering in Gaussian states. Using the third augmented dataset as example, we discuss the detection time for evaluating the steering capability of any Gaussian quantum state, comparing it to the time needed to compute the Gaussian steering quantification \(\mathcal {J}\). All computation times are expressed in seconds, and the specific results are presented in Table 6.

Due to the very fast decay rate of quantum steering, a sufficiently fast detection and identification speed is required to meet the needs of quantum information processes. Therefore, the detection speed is also an important indicator for judging the quality of the detection method. Table 6 presents the average detection times across different modes, including SVM, BPNN, MWN, three models’ ensemble learning (Ensem3), as well as the ensemble learning of only BPNN and MWN models (Ensem2), and the traditional \(\mathcal {J}\) computation method. The data clearly indicate that although the traditional \(\mathcal {J}\) computation method is efficient in small-scale systems, it becomes less applicable as the size and dimensionality of the Gaussian system increase. Specifically, as the mode increases, the computation time for \(\mathcal {J}\) grows significantly, particularly in high-mode cases (e.g., the (5+5)-mode). In these high-mode cases, traditional methods not only face computational bottlenecks but also experience significant inefficiencies in time. For instance, in the (5+5)-mode, computing \(\mathcal {J}\) takes \(1.48 \times 10^{-2}\) seconds, whereas machine learning methods only require \(9.22 \times 10^{-5}\) seconds, speeding up the process by a factor of 600. This stark difference underscores the potential of machine learning methods in large-scale, high-mode, and high-dimensional systems.

Table 6 Average detection time for Gaussian steering.
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Against this backdrop, we focus on ensemble learning, specifically using the stacking ensemble method to construct the final model. This method integrates the predictions of three base models: SVM, BPNN, and MWN. At the same time, we also considered an ensemble containing only BPNN and MWN models. In each test, the ensemble model independently runs each base model and passes the resulting predictions to a meta-model (in this case, logistic regression) for the final decision. To further enhance inference efficiency, we introduce a parallel computing strategy. By leveraging parallel computation, the predictions of all base models are obtained simultaneously, which significantly reduces inference time and avoids the delays associated with serial execution. This parallelization enables the ensemble model to quickly aggregate the predictions from each base model and pass the final output to the meta-model, thus accelerating the overall inference process. However, in modes such as (1+1), (1+2), (2+1), (2+2), and (5+1), the ensemble learning model successfully combines the predictions from SVM, BPNN, and MWN for inference. In contrast, in higher modes like (1+5) and (5+5), as the mode increases, the computation time for the SVM model rises significantly, creating a bottleneck in the ensemble inference process. Specifically, in the (1+5)-mode, the average detection time for SVM is \(1.81 \times 10^{-4}\) seconds, resulting in an overall ensemble detection time of \(2.12 \times 10^{-4}\) seconds when using all three models. Similarly, in the (5+5)-mode, the detection time with all three models increases to \(1.48 \times 10^{-4}\) seconds. It is important to note that, as the mode of the system increases, this bottleneck effect becomes more pronounced, further hindering overall performance. To address this issue, we experimented with using only BPNN and MWN as the base models for the ensemble. The results indicate that this approach does not lead to a significant decrease in prediction accuracy (see Table II in Supplementary material), but detection time is significantly reduced, effectively improving computational efficiency. Based on these findings, we recommend using the BPNN and MWN ensemble, rather than one that includes SVM, in high-mode scenarios. This adjustment can significantly improve inference speed without sacrificing much accuracy. The values of Ensem2 in Table 6 reflect the inference times when using only BPNN and MWN in the ensemble. Building on these experimental results, we further suggest that in high-mode scenarios, SVM should be avoided in favor of base models with higher computational efficiency, such as BPNN and MWN. This approach not only accelerates inference speed but also optimizes overall computational efficiency, resulting in faster and more resource-efficient performance.

This stark difference in computation time fully highlights the advantages of machine learning methods when dealing with large-scale and high-dimensional systems. Machine learning models such as BPNN, MWN, and their ensemble learning not only greatly outperform traditional methods in terms of speed but also can significantly reduce the computational overhead without sacrificing the prediction accuracy. These methods perform particularly well when handling large-scale datasets and complex Gaussian states, and they are especially suitable for practical applications with high requirements for time and resource efficiency. Machine learning methods can significantly reduce the computational complexity while ensuring high precision. This advantage is difficult for traditional methods to match in larger and more complex systems, providing a more practical and feasible solution for the detection of Gaussian steering in CV systems with a large number of modules. Our experiment also suggests that, to improve the accuracy and speed of integrated machine learning predictions, the basic learning models should select those that are heterogeneous and have fast prediction speed.

Conclusion and discussion

This research delves into machine learning methodologies for detecting Gaussian steering in any \((n+m)\)-mode bipartite continuous-variable (CV) systems, utilizing algorithms such as Support Vector Machines (SVM), Backpropagation Neural Networks (BPNN) and Meta-Weight-Net Neural Network (MWN). To enhance detection performance, an ensemble learning approach that integrates SVM, BPNN and MWN is also implemented. A computable Gaussian steering quantification introduced22, denoted as \(\mathcal {J}\), serves as a pivotal tool for labeling the samples.

During the dataset construction process, we observed that the majority of randomly generated Gaussian states are unsteerable, which naturally leads to a significant class imbalance. A particularly noteworthy finding is that this imbalance becomes more pronounced when the number of modes held by the untrusted party significantly exceeds that of the trusted party. For example, in the \((5+1)\)-mode system, the imbalance factor reaches as high as 365.75 when naturally generate 3000,000 samples, indicating a severe skew in the distribution of steerable versus unsteerable states. In response to this phenomenon, three schemes of dataset preparation were proposed: balance dataset, naturally generated dataset and augmented dataset. We trained these three kinds of datasets using the above mentioned four machine learning methods and then analyzed and compared the behaviours of the classifiers, aiming to find a practical approach to detect Gaussian steering, maintaining a relative high test accuracy. A remarkable feature of this work is that we introduce MWN method into our consideration, which is developed recently for effectively tackling the imbalances datasets.

As numerical experiments, we conduct machine learning on \((n+m)\)-mode CV systems with \((n+m)\in \{(1+1),(1+2),(2+1),(2+2),(1+5), (5+1),(5+5)\}\). Our results exhibit that training Gaussian steering detectors by SVM, BPNN, MWN and ensemble learning methods on balanced datasets results highest test accuracy \((\ge 98\%)\). However, preparing a balanced dataset incurs substantial computational and time costs, reaching hundreds of times that of preparing a naturally generated dataset. Although preparing a naturally generated dataset is the least costly, due to the extreme imbalance, the classifiers exhibit poor generalization ability on naturally generated dataset (with test accuracy between \(60\%\) and \(71\%\)). To address this issue, we propose a notion of imbalance factor \(\xi (\ge 1)\) to describe the degree of imbalance of the datasets, which is the ratio of the number of unsteerable samples and the number of steerable samples in the dataset. This enables us to conduct a data augmentation strategy by applying a result on \(\mathcal {J}\) to decrease the imbalance factor from the naturally generated dataset and improve the test accuracy of the classifiers. To get a suitable factor of augmented datasets, we first examine the selection of imbalance factors, using \((1+1)\)-mode Gaussian states as an example. We find that, the smaller \(\xi\) is, the higher the detection accuracy is. If a test accuracy exceeding 80% is considered, numerical experiments indicate that setting the data imbalance factor to \(\xi = 50\) offers a favorable trade-off between detection accuracy and computational efficiency. However, further analysis using confusion matrices reveals that the classification accuracy for the minority class can be substantially improved by adopting a more aggressive augmentation strategy. Therefore, we ultimately consider \(\xi = 30\) as the good imbalance factor, as it significantly enhances the recall of the minority class while maintaining stable overall performance across all models.

Analyzing and comparing the numerical results for seven mode scenarios, we conclude that: (1) Generating balanced datasets facilitate model training but demand significant computational resources; (2) The naturally generated imbalanced datasets reflect real-world distributions but may reduce model accuracy; (3) Applying data augmentation strategy to decrease the ratio \(\xi\) of unsteerable Gaussian states and steerable can improving classifier performance on naturally generated imbalanced datasets. Hence the data augmentation strategy helps maintain model efficiency across varying data distributions, particularly enhancing accuracy and robustness when dealing with imbalanced data; (4) The ensemble learning consistently performs the best, regardless of the dataset structure. This success is attributed to its ability to combine the diverse strengths of multiple models, effectively balancing the weaknesses of individual models, thus reducing prediction errors and enhancing generalization; (5) BPNN excels on balanced datasets comparing with SVM and MWN, demonstrating strong generalization, rapid training speed, and high classification accuracy; (6) MWN network achieves the best performance on imbalanced datasets comparing with SVM and BPNN, particularly when combined with data augmentation, yielding detection speeds at least 100 times faster than the traditional Gaussian steering quantification \(\mathcal {J}\), with detection times reduced to \(10^{-5}\) seconds per instance. Based on the experimental results, we recommend avoiding SVM in high-mode scenarios and instead using more computationally efficient base models like BPNN and MWN for ensemble learning. Generally, to improve the accuracy and speed of essemble machine learning predictions, the basic learning models should select those that are heterogeneous and have fast prediction speed. This approach not only significantly accelerates inference speed but also optimizes overall computational efficiency, ensuring robust performance in complex, large-scale settings.

This research demonstrates the effectiveness of machine learning in improving Gaussian steering detection accuracy and reveals its potential to uncover new physical phenomena in quantum systems. A key achievement is the successful application of data augmentation techniques, which significantly enhanced classification performance. After augmentation, the ensemble learning model achieved over 85% accuracy, with the highest recall rate for steerable states reaching 81.16%. The study also suggests that future research should dynamically adjust the imbalance factor for different modes to optimize model performance and improve the balance between accuracy and computational cost. Furthermore, this work highlights the broader potential of machine learning to explore new quantum phenomena. Future investigations also include adapting these methods for detecting other Gaussian correlations, such as entanglement and Gaussian coherence, optimizing their use in large-scale multi-mode systems, developing techniques for quantum correlation detection in discrete-variable (DV) systems using high-dimensional feature learning, and integrating quantum machine learning technologies, such as quantum neural networks, support vector machines and Meta-Weight-Net Neural Network, to further enhance precision and efficiency in quantum information processing. By continuously refining these strategies, we aim to push the boundaries of quantum machine learning for broader and more effective applications in quantum information science.