# A Convex Hull-Based Machine Learning Algorithm for Multipartite Entanglement Classification
## 1 Introduction  
Machine learning was born from pattern recognition, which possesses the ability to make decisions without explicit programming after learning from large amounts of data. Up to now, machine learning has been employed to quantum areas. Thus far, a number of promising applications have been proposed, such as quantum metric learning [1], the gate decomposition problem [2], quantum states discrimination [3], quantum discrete feature encoding [4], quantum nodes based on variational unsampling protocols [5] and quantifying steerability [6]. Entanglement was first described by [PERSON], [PERSON] and [PERSON] [7]. Later, quantum entanglement became a useful resource, enabling tasks such as quantum cryptography [8], quantum teleportation [9] and driving fields on the spectrum [10]. There are also many methods which have been proposed to distinguish and quantify entanglement, including Tsallis-q entanglement [11], device-independent entanglement witnesses [12] and the geometric measure of entanglement [13]. When it comes to the number of parties involved in entanglement, there are two typical classes: bipartite entanglement and multipartite entanglement. When there are more than two parties involved, the situation gets complicated. For example, when there are three qubits in the Hilbert space \(\mathcal{H}_{A}\), \(\mathcal{H}_{B}\) and \(\mathcal{H}_{C}\), a state is called a fully separable state if it can be written as  
\[\left|\Phi^{\prime s}\right\rangle_{A\left|\beta\right|C}=\left|\alpha\right\rangle _{A}\otimes\left|\beta\right\rangle_{B}\otimes\left|\gamma\right\rangle_{C} \tag{1}\]  
where \(\left|\alpha\right\rangle_{A}\in\mathcal{H}_{A}\left|\beta\right\rangle_{B} \in\mathcal{H}_{B},\left|\gamma\right\rangle_{C}\in\mathcal{H}_{C}\). Biseparable states can be written as a product state in the bipartite system. A biseparable state can be created if two of the three qubits are grouped together into one party. There are three possibilities for grouping two qubits together, and hence there are three classes of biseparable states. There are three possibilities: \(\left|\Phi^{\text{bs}}\right\rangle_{A\left|BC\right\rangle}=\left|\alpha \right\rangle_{A}\otimes\left|\delta\right\rangle_{BC^{\prime}}\left|\Phi^{ \text{bs}}\right\rangle_{B\left|AC\right\rangle}=\left|\alpha\right\rangle_{B} \otimes\left|\delta\right\rangle_{AC}\) and \(\left|\Phi^{\text{bs}}\right\rangle_{C\left|AB\right\rangle}=\left|\alpha \right\rangle_{C}\otimes\left|\delta\right\rangle_{AB}\), where \(\left|\delta\right\rangle\) denotes a two-party state that might be entangled. Finally, a state is called genuine entangled if it is neither fully separable nor biseparable. There are two main families of multipartite entanglement: one is the GHZ states [14,15], and the other is the so-called W states [16]. A schematic picture of the structure of mixed states for three qubits is shown in Figure 1[17]. Given two three-qubit states \(|\phi\rangle\) and \(|\psi\rangle\), one can ask whether it is possible to transform a single copy of \(|\phi\rangle\) into \(|\psi\rangle\) with local operations and classical communication (LOCC) without requiring that this is done with certainty. This operation is called stochastic local operations and classical communication (SLOCC). Compared with the well-known local operations and classical communication (LOCC), SLOCC has a non-unit probability. For systems of \(N\) qudits described by Hilbert spaces of the form \(\mathcal{H}:=\mathbb{C}^{d_{1}}\otimes\ldots\otimes\mathbb{C}^{d_{N}}\), SLOCC operations are mathematically described by the group \(G:=SL(d_{1},\mathbb{C})\times\ldots\times SL(d_{N},\mathbb{C})\), and the action is given by the tensor product. We call two states equivalent if there is a non-vanishing probability of success when trying to convert one to another through SLOCC. The distinction between a GHZ state and a W state is that one cannot be transferred to another through SLOCC [18].