  In an effort to simplify the classification of pure entangled states of multi
(m) -partite 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). With regard to exact transformations, we
show that two states whose 1-party entropies agree are either locally-unitarily
(LU) equivalent or else LOCC-incomparable. In particular we show that two
tripartite Greenberger-Horne-Zeilinger (GHZ) states are LOCC-incomparable to
three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among
the three parties. Asymptotic transformations result in a simpler
classification than exact transformations. We show that m-partite pure states
having an m-way Schmidt decomposition are simply parameterizable, with the
partial entropy across any nontrivial partition representing the number of
standard ``Cat'' states (|0^m>+|1^m>) asymptotically interconvertible to the
state in question. For general m-partite states, partial entropies across
different partitions need not be equal, and since partial entropies are
conserved by asymptotically reversible LOCC operations, a multicomponent
entanglement measure is needed, with each scalar component representing a
different kind of entanglement, not asymptotically interconvertible to the
other kinds. In particular the m=4 Cat state is not isentropic to, and
therefore not asymptotically interconvertible to, any combination of bipartite
and tripartite states shared among the four parties. Thus, although the m=4 cat
state can be prepared from bipartite EPR states, the preparation process is
necessarily irreversible, and remains so even asymptotically.
