Inequivalent Multipartite Coherence Classes and New Coherence Monotones

Quantum coherence has received significant attention in recent years, but its study is mostly conducted in single party settings. In this paper, we generalize important results in multipartite entanglement theory to their counterparts in quantum coherence theory. First, we give a necessary and sufficient condition for when two pure multipartite states are equivalent under local quantum incoherent operations and classical communication (LICC), i.e., two states can be deterministically transformed to each other under LICC operations. Next, we investigate and give the conditions in which such a transformation succeeds only stochastically. Different from entanglement case for two-qubit states, we find that the SLICC equivalence classes are infinite. Moreover, it's possible that there are some classes of states in multipartite entanglement can convert into each other, while, they cannot convert into each other in multipartite coherence. In order to show the difference among SLICC classes, we introduce two coherence monotones: accessible coherence and source coherence, following the logistics given in [\emph{Phys.~Rev.~Lett. 115,~150502 (2015)}]. These coherence monotones have a straightforward operational interpretation, namely, the accessible coherence characterizes the proficiency of a state to generate other states via quantum incoherent operations, while the source coherence characterizes the set of states that can be reached via quantum incoherent operations acting on the given state of interest.