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Hoping to simplify the classification of pure entangled states of multi (m)− p a r t i t e quantum systems, we study exactly and asymptotically (in n) reversible transformations among n th tensor powers of such states (ie, n copies of the state shared among the same m parties) under local quantum operations and classical communication (LOCC). For exact transformations, we show that two states whose marginal one-party entropies agree are either locally unitarily equivalent or else LOCC incomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger states are LOCC incomparable to three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among the three parties. Asymptotic transformations yield a simpler classification than exact transformations; for example, they allow all pure bipartite states to be characterized by a single parameter—their partial entropy—which may be …