Special core tensors of multi-qubit states and the concurrency of three lines

Classification of multipartite states aims to obtain a set operationally useful and finite entanglement classes under the action either local unitary (LU) or stochastic operation classical communication (SLOCC). In this work, we propose computationally simple approach find these by using higher order singular value decomposition (HOSVD) concurrency three lines. Since HOSVD simultaneously diagonalizes one-body reduced density matrices (RDM) states, core tensor is pure-state representation such diagonalized RDM. We identified special tensors four qubits, which are also genuinely entangled default. The further categorized into families based on their first $n$-mode values, $\sigma_1^{(i)2}$. current proposal limited multi-qubit system, but it scales well with large systems produces number states.