  A set of orthogonal multipartite quantum states is said to be
distinguishability-based genuinely nonlocal (also genuinely nonlocal, for
abbreviation) if the states are locally indistinguishable across any
bipartition of the subsystems. This form of multipartite nonlocality, although
more naturally arising than the recently popular "strong nonlocality" in the
context of local distinguishability, receives much less attention. In this
work, we study the distinguishability-based genuine nonlocality of a typical
type of genuine multipartite entangled states -- the d-dimensional GHZ states,
featuring systems with local dimension not limited to 2. In the three-partite
case, we find the existence of small genuinely nonlocal sets consisting of
these states: we show that the cardinality can at least scale down to linear in
the local dimension d, with the linear factor l = 1. Specifically, the method
we use is semidefinite program and the GHZ states to construct these sets are
special ones which we call "GHZ-lattices". This result might arguably suggest a
significant gap between the strength of strong nonlocality and the
distinguishability-based genuine nonlocality. Moreover, we put forward the
notion of (s,n)-threshold distinguishability and utilizing a similar method, we
successfully construct (2,3)-threshold sets consisting of GHZ states in
three-partite systems.
