  Local operations assisted by classical communication (LOCC) constitute the
free operations in entanglement theory. Hence, the determination of LOCC
transformations is crucial for the understanding of entanglement. We
characterize here almost all LOCC transformations among pure multipartite
multilevel states. Combined with the analogous results for qubit states shown
by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a
characterization of almost all local transformations among multipartite pure
states. We show that nontrivial LOCC transformations among generic, fully
entangled, pure states are almost never possible. Thus, almost all multipartite
states are isolated. They can neither be deterministically obtained from
local-unitary-inequivalent (LU-inequivalent) states via local operations, nor
can they be deterministically transformed to pure, fully entangled
LU-inequivalent states. In order to derive this result, we prove a more general
statement, namely, that, generically, a state possesses no nontrivial local
symmetry. We discuss further consequences of this result for the
characterization of optimal, probabilistic single copy and probabilistic
multi-copy LOCC transformations and the characterization of LU-equivalence
classes of multipartite pure states.
