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