Suppose a set of
m-partite, $m\ensuremath{\ge}3$, pure orthogonal fully separable states is given. We consider the task of distinguishing these states perfectly by local operations and classical communication (LOCC) in different
k partitions,
1<k<m. Based on this task, it is possible to classify the sets of product states into different classes. For tripartite systems, a classification of the sets with explicit examples is presented. Few important cases related to the aforesaid task are also studied when the number of parties, $m\ensuremath{\ge}4$. These cases never appear for a tripartite system. However, to distinguish any LOCC indistinguishable set, entanglement can be used as resource. An important objective of the present study is to learn about the efficient ways of resource sharing among the parties. We also find an interesting application of multipartite product states, which are LOCC indistinguishable in a particular
k partition. Starting from such product states, we constitute a protocol to distribute bound entanglement between two spatially separated parties by sending a separable qubit.
Local distinguishability of orthogonal quantum states can effectively reduce the consumption of quantum resources and lower economic costs in quantum protocols. Although numerous achievements have been made regarding local distinguishability of orthogonal quantum states, some fundamental issues have not been effectively addressed. For example, the local distinguishability of five orthogonal product states (OPSs) is still unknown up to now. In this paper, we give the properties of local distinguishability of five OPSs on bipartite and tripartite quantum systems. Firstly, to characterize the structure of a set of bipartite OPSs, we propose the concept of the vector of orthogonal relations for a set of bipartite OPSs. Secondly, we classify the structures of five bipartite OPSs into six categories by this concept and prove that five of these six categories can be perfectly distinguished by local operations and classical communication (LOCC). Thirdly we show that the local distinguishability of each case of the sixth category singly. On the other hand, we first divide the structures of five tripartite OPSs into eight categories by the vectors of orthogonal relations of five tripartite OPSs. Then we give the local distinguishability of each category. Our work enriches the research results of quantum nonlocality and will provide a clear understanding of the local distinguishability of five OPSs.