Monotones and invariants for multi-particle quantum states

We introduce new entanglement monotones which generalize, to the case of ma ny parties, those which give rise to the majorization-based partial orderin g of bipartite states' entanglement. We give some examples of restrictions they impose on deterministic and probabilistic conversion between multipart ite states via local actions and classical communication. These include res trictions which do not follow from any bipartite considerations. We derive supermultiplicativity relations between each state's monotones and the mono tones for collective processing when the parties share several states. We a lso investigate polynomial invariants under local unitary transformations, and show that a large class of these are invariant under collective unitary processing and also multiplicative, putting restrictions, for example, on the exact conversion of multiple copies of one state to multiple copies of another.