# Improving Steerability Detection via an Aggregate Class Distribution Neural Network
## 3 Detecting the Steerability by AGGNN
### Predicting the Steerability Bounds  
In this part, the steerability bounds of the generalized Werner state are predicted by the quantum steering classification models trained by AGGNN and compared with the bounds computed by SDP [35], trained by SVM [45], and trained by BPNN [48].  
Figure 5a-d depict the classification models trained with the F1-F4 features, respectively. Figure 5a shows the four subgraphs with \(\theta=\{\frac{\pi}{4},\frac{\pi}{6},\frac{\pi}{8},\frac{\pi}{12}\}\), respectively, as illustrated above each chart. In all charts, the cyan-square ties depict the bounds predicted by the classification models trained by the SVM for \(m=2,\ldots,8\). The blue-star lines depict the bounds predicted by the classification models trained by the BPNN for \(m=2,\ldots,8\). The black-plus lines depict the bounds predicted by the classification models trained by the AGGNN for \(m=2,\ldots,8\). The yellow-circle lines depict the bounds computed by SDP with \(m=2,\ldots,8\). The red-cross lines depict the steerability bounds from Alice to Bob which are determined by Equation (19). We can clearly see that the predicted steerability bounds by the SVM [45], BPNN [48] and AGGNN are more similar to the theoretical bounds than those calculated by SDP [35], especially concerning the F1 feature. The steerability bounds predicted by the AGGNN are more similar to the theoretical bounds than those predicted by the SVM and BPNN.  
Figure 5a-d illustrate that the steerability bounds predicted by the AGGNN are more similar to the theoretical bounds than those predicted by the SVM [45] and BPNN [48]. This is because the AGGNN makes samples with similar features have a similar class prediction distribution according to the distance measure of the sample features; therefore, the model can reduce the learning of mislabelled or inaccurate quantum states, making it easier for the model to converge to solutions similar to the theoretical boundaries in the training process. Furthermore, when the similarity between the predicted steerability and theoretical bounds increases, the AGGNN will improve the classification accuracy of the Werner state validation sets. For features F2-F4 with partial information of a quantum state, in some cases, the boundary predicted by the SDP is more similar to the theoretical boundaries than the machine learning method, possibly due to data distribution problems caused by inaccurate data.  
The above results indicate that the AGGNN is an efficient model for steerability detection. For a quantum state with all information, our model performs better than the BPNN and SVM. For partial information, the predicted steerability boundary of our model is more similar to the theoretical boundary than the BPNN and SVM. Furthermore, the AGGNN can predict the steerability bounds of other states' ensembles, such as Tstate [52]. Moreover, our model has a roughly similar time consumption to the BPNN and SVM. Taking \(m=8\) as an example, the classification model trained by these three machine learning methods spends about \(10^{-2}\) s predicting a quantum state, while the SDP [35]  
Figure 5: (**a–d**) The predictions of the steerability bounds for the generalized Werner states by the trained classification models and SDP with the features F1–F4, respectively. The cyan-square lines depict the bounds predicted by the SVM-trained classification model, the blue-star lines depict the bounds predicted by the BPNN-trained classification model, the black-plus lines are the bounds predicted by the AGGNN-trained classification model, the yellow-circle lines depict the bounds computed by SDP, and the red-cross lines depict the steerability bound from Alice to Bob.  
program spends about \(10^{2}\) s. This demonstrates the time superiority of the machine learning methods.