Optimal entanglement witness of multipartite systems using support vector machine approach

Abstract: An entanglement witness (EW) is a Hermitian operator that can distinguish an entangled state from all separable states. We drive and implement a numerical method based on machine learning to create a multipartite EW. Using support vector machine (SVM) algorithm, we construct EW's based on local orthogonal observables in the form of a hyperplane that separates the separable region from the entangled state for two, three and four qubits in Bell-diagonal mixed states, which can be generalized to multipartite mixed states as GHZ states in systems where all modes have equal size. One of the important features of this method is that, when the algorithm succeeds, the EWs are optimal and are completely tangent to the separable region. Also, we generate non-decomposable EWs that can detect positive partial transpose entangled states (PPTES).