  Understanding multipartite entanglement is vital, as it underpins a wide
range of phenomena across physics. The study of transformations of states via
Local Operations assisted by Classical Communication (LOCC) allows one to
quantitatively analyse entanglement, as it induces a partial order in the
Hilbert space. However, it has been shown that, for systems with fixed local
dimensions, this order is generically trivial, which prevents relating
multipartite states to each other with respect to any entanglement measure. In
order to obtain a non-trivial partial ordering, we study a physically motivated
extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC
transformations acting on a finite number of entangled pure states. We study
both multipartite and bipartite multi-state transformations. In the
multipartite case, we demonstrate that one can change the stochastic LOCC
(SLOCC) class of the individual initial states by only applying Local Unitaries
(LUs). We show that, by transferring entanglement from one state to the other,
one can perform state conversions not possible in the single copy case; provide
examples of multipartite entanglement catalysis; and demonstrate improved
probabilistic protocols. In the bipartite case, we identify numerous
non-trivial LU transformations and show that the source entanglement is not
additive. These results demonstrate that multi-state LOCC has a much richer
landscape than single-state LOCC.
