Distinguishability-based genuine nonlocality with genuine multipartite entanglement

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 Greenberger-Horne-Zeilinger (GHZ) states, featuring systems with local dimension not limited to two. 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 programming 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.