Inductive classification of multipartite entanglement under stochastic local operations and classical communication

We propose an inductive procedure to classify $N$-partite entanglement under stochastic local operations and classical communication provided such a classification is known for $N-1$ qubits. The method is based upon the analysis of the coefficient matrix of the state in an arbitrary product basis. We illustrate this approach in detail with the well-known bipartite and tripartite systems, obtaining as a by-product a systematic criterion to establish the entanglement class of a given pure state without resourcing to any entanglement measure. The general case is proved by induction, allowing us to find an upper bound for the number of $N$-partite entanglement classes in terms of the number of entanglement classes for $N-1$ qubits.