  It is known that there exist sets of pure orthogonal product states which
cannot be perfectly distinguished by local operations and classical
communication (LOCC). Such sets are nonlocal sets which exhibit nonlocality
without entanglement. These nonlocal sets can be completable or uncompletable.
In this work both completable and uncompletable small nonlocal sets of
multipartite orthogonal product states are constructed. Apart from nonlocality,
these sets have other interesting properties. In particular, the completable
sets lead to the construction of a class of complete orthogonal product bases
with the property that if such a basis is given then no state can be eliminated
from that basis by performing orthogonality-preserving measurements. On the
other hand, an uncompletable set of the present kind contains several Shifts
unextendible product bases (UPBs) that belong to qubit subspaces. Identifying
these subspace UPBs, it is possible to obtain a class of high-dimensional
multipartite bound entangled states. Finally, it is shown that a two-qubit
maximally entangled Bell state shared between any two parties is sufficient as
a resource to distinguish the states of any completable set (of the above kind)
perfectly by LOCC. This constitutes an example where the amount of
entanglement, sufficient to accomplish the aforesaid task, depends neither on
the dimension of the individual subsystems nor on the number of parties.
