# Improving Steerability Detection via an Aggregate Class Distribution Neural Network
## 3 Detecting the Steerability by AGGNN
### Training and Testing
Compared to cross-validation test sets and random test sets, having high accuracy on the Werner state validation sets is more likely to be a good model. In Figure 3a-d, the Werner state (\(\theta=\frac{\pi}{4}\)) validation set's accuracies of the classification models trained by the SVM [45], BPNN [48] and AGGNN are depicted as solid blue, solid green and solid red columns, respectively. The generalized Werner state is unsteerable from [PERSON] to [PERSON] if \(\theta\) and \(\beta\) satisfy the following inequality,  
Figure 4: The number of well-trained models by the BPNN, AGGNN on F1 features. The first (diagonal-filled green) and second (diagonal-filled blue) columns depict the number of well-trained models by the BPNN and AGGNN, respectively. The label m is the number of measurements. \[\cos^{2}2\theta\geqslant\frac{2\beta-1}{(2-\beta)\beta^{3}}. \tag{19}\]  
We can clearly see that the theoretical bound value that [PERSON] can steer [PERSON]'s state is determined by Equation (19). In this experiment we generate four Werner state test sets with \(\theta=\{\frac{\pi}{4},\frac{\pi}{6},\frac{\pi}{8},\frac{\pi}{12}\}\), respectively. For each Werner state test set, we generated 5000 steerable states and 5000 unsteerable states based on the uniform distribution of \(\beta\), utilizing them to construct a dataset for every feature (F1-F4), where each instance contains a feature F\(i\) (\(i\) = 1,2,3,4), and a label (\"0\" for steerable and \"1\" for unsteerable). Finally, to explored whether our model could achieve better robustness to inaccurate quantum states and a better learning ability. For F1 features we believe the model can be well trained if it reaches more than 99% accuracies on the four validation sets generated by the Werner state (\(\theta=\{\frac{\pi}{4},\frac{\pi}{6},\frac{\pi}{8},\frac{\pi}{12}\}\)). We conducted five training sets, obtained 20 models, and then counted the model's accuracy on the four validation sets to obtain the number of well-trained models. In Figure 4, the number of well-trained models by the BPNN are depicted by the first (solid-filled green) column, while those trained by the AGGNN are depicted by the second (solid-filled blue) column. The experiments show that the AGGNN model can be easily trained to produce a good model. Because the data generated by the SDP has errors and an uneven data distribution, there is no well-trained model when \(m=6\). The AGGNN will train better models if the datasets are regenerated or the amount of data increases. Interestingly, for the F1 feature with whole information of a quantum state, the classification accuracy of the AGGNN models on the Werner state validation sets is higher than the SVM and BPNN models. The classification accuracy of the AGGNN for most cross-validation and random test sets is higher than the BPNN, and the AGGNN is more accessible to train than the BPNN. However, the classification accuracy of the AGGNN for cross-validation test sets is lower than the SVM, but not by much. This indicates that the AGGNN has good generalization abilities and can effectively prevent overfitting. For F2-F4 features with partial information, the classification accuracy of the AGGNN in Werner state validation sets is higher than the SVM and BPNN. However, the classification accuracy of the AGGNN for the cross-validation and random test sets decreases because there are many misjudments and inaccurate quantum states in the training data. The AGGNN can reduce the learning ability of error data and enlarge the difference between different class sample features extracted by the model to improve the model's robustness and prediction ability, so the classification accuracy of Werner state validation sets can further increase. In contrast, the classification accuracy of cross-validation and random test sets decreases.