Multipartite states under elementary local operations

Multipartite pure states are equivalent under stochastic local operations and classical communication (SLOCC) whenever they can be mapped into one another by invertible local operations. It is shown that this is equivalent to the interconvertibility through finite sequences of elementary local operations. A multipartite version of the Gauss-Jordan elimination strategy is then obtained, enabling the analytical SLOCC classification of previously unknown examples. It is argued that the problem of SLOCC classification is equivalent to the problem of classifying the multipartite fully reduced forms that the coefficient matrix of a state can assume after being subjected to this Gauss-Jordan elimination procedure. The method is applied to examples of so-called locally maximally entangleable and hypergraph states, showing different SLOCC equivalences. Moreover, possible physical implications for states with a Dicke-like structure are sketched.