  Characterizing the transformation and classification of multipartite
entangled states is a basic problem in quantum information. We study the
problem under two most common environments, local operations and classical
communications (LOCC), stochastic LOCC and two more general environments,
multi-copy LOCC (MCLOCC) and multi-copy SLOCC (MCSLOCC). We show that two
transformable multipartite states under LOCC or SLOCC are also transformable
under MCLOCC and MCSLOCC. What's more, these two environments are equivalent in
the sense that two transformable states under MCLOCC are also transformable
under MCSLOCC, and vice versa. Based on these environments we classify the
multipartite pure states into a few inequivalent sets and orbits, between which
we build the partial order to decide their transformation. In particular, we
investigate the structure of SLOCC-equivalent states in terms of tensor rank,
which is known as the generalized Schmidt rank. Given the tensor rank, we show
that GHZ states can be used to generate all states with a smaller or equivalent
tensor rank under SLOCC, and all reduced separable states with a cardinality
smaller or equivalent than the tensor rank under LOCC. Using these concepts, we
extended the concept of "maximally entangled state" in the multi-partite
system.
