# Improving Steerability Detection via an Aggregate Class Distribution Neural Network
## 2 Preliminaries  
### Quantum Steering  
In the modern view of quantum steering, it is a concept that is incompatible with the local-hidden-state (LHS) model [36]. For an unknown quantum state \(\rho\) shared by [PERSON] and [PERSON], assume that [PERSON] performs the quantum measurements \(\{\mathcal{M}_{x}^{A}=\{M_{a|x}^{A}\}\}\) on her subsystems, and the corresponding measurement outcomes are denoted by \(a\). Based on the quantum theory, after choosing the quantum measurement \(M_{a|x}^{A}\) and obtaining the measurement outcome \(a\), the non-normalized quantum-state assemblage of the subsystem \(\rho_{B}\) is \(\{p(a|x),\rho_{a|x}\}\), where  
\[\rho_{a|x}=\text{tr}_{A}[(M_{a|x}^{A}\otimes I^{B})\rho], \tag{1}\]  
and the probability distribution  
\[p(a|x)=\text{tr}[(M_{a|x}^{A}\otimes I^{B})\rho]. \tag{2}\]  
Quantum steering describes a scenario: [PERSON] can fully control their measurements and access the condition state \(\rho_{a|x}\) without characterization of [PERSON]'s measurements. Namely, [PERSON] performs tomography to reconstruct the set of conditional assemblages \(\{\rho_{a|x}\}\) and the results are not dependent on any particular information about how [PERSON]'s measurements work. In other words, [PERSON] can steer [PERSON]'s local state by performing local measurements and classical communication on the particle she owns. Utilizing the LHS model, quantum steering can be defined as the impossibility of remotely generating ensembles produced by an LHS model. That is, suppose that a source sends a classical message \(\lambda\) to [PERSON] and a corresponding state \(\sigma_{\lambda}\) to [PERSON]. If the measurement applied by [PERSON] is \(x\), the classical variable \(\lambda\) determines the probability of her obtaining the measurement outcome \(a\), which is represented by \(p(a|x,\lambda)\). Although the probability distribution of the classical message \(\lambda\)is \(p(\lambda)\), since [PERSON] cannot access the classical variable \(\lambda\), the final corresponding assemblage that [PERSON] observes is composed of the elements  
\[\rho_{a|x}=\int\mathrm{d}\lambda p(\lambda)p(a|x,\lambda)\sigma_{\lambda}. \tag{3}\]  
We say that [PERSON] cannot steer [PERSON]'s state if the LHS model can generate a conditional assemblage \(\rho_{a|x}\) that corresponds to the state \(\rho\). Otherwise, we say that [PERSON] can steer [PERSON]'s state. However, efficiently identifying the steerability of a quantum state is usually very difficult. [PERSON] and [PERSON] gave the numerical calculation for the quantum-steering criterion in terms of SDP [35]. Now let us introduce this computing process. For an arbitrary bipartite state \(\rho\), if [PERSON] performs \(m\) measurements \(x=\{0,\ldots,m-1\}\) with \(k\) outcomes \(a=\{0,\ldots,k-1\}\) each, and \(\lambda^{{}^{\prime}}\) is a map from the measurement set \(\{0,\ldots,m-1\}\) to the outcome set \(\{0,\ldots,k-1\}\), the deterministic probability distribution is defined as \(D(a|x,\lambda^{{}^{\prime}})=\delta_{a,\lambda^{{}^{\prime}}(x)}\). Then, Equation (3) can be rewritten as  
\[\rho_{a|x}=\sum_{\lambda^{{}^{\prime}}=1}^{d}D(a|x,\lambda^{{}^{\prime}})\rho _{\lambda^{{}^{\prime}}}, \tag{4}\]  
where \(p(a|x,\lambda)=\sum_{\lambda^{{}^{\prime}}=1}^{d}p(\lambda^{{}^{\prime}}| \lambda)D(a|x,\lambda^{{}^{\prime}})\), and \(\rho_{\lambda^{{}^{\prime}}}\equiv\int d\lambda p(\lambda)p(\lambda^{{}^{ \prime}}|\lambda)\sigma_{\lambda}\). We write the SDP that can identify the steerability of a state [35] as follows  
\[ \begin{array}{l}\mathrm{given}\{\rho_{a|x}\},\{D(a|x,\lambda^{{}^{\prime}}) \}_{\lambda^{{}^{\prime}}}\\ \min_{\{F_{\{x\}}\}}\mathrm{tr}\sum_{ax}F_{a|x}\rho_{a|x}\\ \mathrm{s.t.}\sum_{ax}F_{a|x}D(a|x,\lambda^{{}^{\prime}})\geqslant 0\quad\forall \lambda^{{}^{\prime}}\\ \mathrm{tr}\sum_{ax\lambda^{{}^{\prime}}}F_{a|x}D(a|x,\lambda^{{}^{\prime}})=1,\end{array}  \tag{5}\]  
where \(\{F_{a|x}\}\) are Hermitian matrices.