# Improving Steerability Detection via an Aggregate Class Distribution Neural Network
## 3 Detecting the Steerability by AGGNN
### Datasets  
In order to train the quantum steering classification models, we need to collect the data for quantum states and select the features for the data. Inspired by [45], we generate two random \(4\times 4\) complex matrices, \(A\) and \(B\). Then we use the two matrices to generate a Hermitian matrix \(C\equiv(A+\mathrm{i}B)(A+\mathrm{i}B)^{\dagger}\), where \(\dagger\) means conjugate transpose. Finally, we obtain a density matrix \(\rho\equiv C/\mathrm{tr}(C)\). Thus, we can determine the steerability of each sample quantum state by utilizing SDP. We label the steerable state by \"1\" and the unsteerable state by \"0\". In this paper, we use the datasets in [45], collected by the following four kinds of feature vectors.  
**F1:** Every feature vector in F1 is a 15-dimensional vector, and the component is one of the set \(\{\rho_{ii}\), the real and imaginary part of \(\rho_{ij},i>j\},i,j\in\{1,2,3\}\).  
**F2:** Every feature vector in F2 is a 9-dimensional vector, and the component is one of the set \(\{\mathrm{tr}[(\sigma_{k}\otimes\sigma_{l})\rho]\},k,l\in\{1,2,3\}\).  
The Bloch representation of an arbitrary two-qubit density operator \(\rho\) is as follows  
\[\rho=\frac{1}{4}\left(I+\sum_{i=1}^{3}x_{i}\sigma_{i}\otimes I^{B}+\sum_{j=1}^ {3}y_{j}I^{A}\otimes\sigma_{j}+\sum_{k,l=1}^{3}t_{kl}\sigma_{k}\otimes\sigma_{ l}\right), \tag{17}\]  
where \(t_{kl}=\mathrm{tr}[(\sigma_{k}\otimes\sigma_{l})\rho]\) is a component of the correlation matrix. We extract the partial information by computing \(\{t_{kl}\}\) as features, since the correlation matrix can demonstrate a certain quantum correlation. We aim to obtain a high-performance machine learning model with partial information for only three fixed measurement directions \(x,y,z\).  
**F3:** Every feature vector in F3 is a 9-dimensional vector, and the component is one of the set \(\{\mathrm{tr}[(\sigma_{k}\otimes\sigma_{l})\rho^{\dagger}]\},k,l\in\{1,2,3\}\), where \(\rho\rightarrow\rho^{\prime}\equiv(I^{A}\otimes\sqrt{\rho^{B}})\rho(I^{A} \otimes\sqrt{\rho^{B}})\).  
To further explore a high-performance machine learning model with partial information, we convert state \(\rho\) into the canonical form \(\rho^{\prime}\) by local unitaries, preserving the steerability of \(\rho\). As proven in [50], the map is given by  
\[\rho\to p^{\prime}\equiv(I^{A}\otimes\sqrt{\rho^{B}})\rho(I^{A} \otimes\sqrt{\rho^{B}}), \tag{18}\]  
where \(\rho^{B}=\mathrm{tr}_{A}\rho\).  
Similarly, we extract the coefficients of the correlation terms of \(\rho^{\prime}\) and \(t^{\prime}_{kl}\) to combine the feature vector. Similar to the F2 case, we only need to measure an arbitrary two-qubit state in three fixed directions \(x,y,z\) to predict the steerability of it.  
**F4:** Every feature vector is composed of six features in F4, that is, F3 except for the terms of \(\{\sigma_{2}\otimes\sigma_{1},\sigma_{3}\otimes\sigma_{1},\sigma_{3}\otimes \sigma_{2}\}\).  
To explore a high-performance machine learning model of steering with less information, according to the symmetry, we drop the coefficients of the correlation terms, \(\{\sigma_{2}\otimes\sigma_{1},\sigma_{3}\otimes\sigma_{1},\sigma_{3}\otimes \sigma_{2}\}\).  
According to the number of measurements \(m=2,3,\ldots,8\), we can divide the whole dataset into 28 datasets after selecting the feature vectors. For each \(m\), we have at least 5000 examples labelled \"1\" and 5000 examples with labelled \"0\" in the corresponding dataset. We then randomly select 1000 positive examples and 1000 negative examples as the test set, and the rest as the training set. We employ the AGGNN to train a model for every dataset.