Small sets of genuinely nonlocal Greenberger-Horne-Zeilinger states in multipartite systems

A set of orthogonal multipartite quantum states are called (distinguishability-based) genuinely nonlocal if they are locally indistinguishable across any bipartition of the subsystems. In this work, we consider the problem of constructing small genuinely nonlocal sets consisting of generalized Greenberger-Horne-Zeilinger (GHZ) states in multipartite systems. For system ({C^2) }^ where N is large, using the language of group theory, we show that a tiny proportion (1/2^{N}) of the states among the N-qubit GHZ basis suffice to exhibit genuine nonlocality. Similar arguments also hold for the canonical generalized GHZ bases in systems ({C^d) }^, wherever d is even and N is large. What is more, moving to the condition that any fixed N is given, we show that d+1 genuinely nonlocal generalized GHZ states exist in ({C^d) }^, provided the local dimension d is sufficiently large. As an additional merit, within and beyond an asymptotic sense, the latter result also indicates some evident limitations of the ``trivial othogonality-preserving local measurements'' (TOPLM) technique that has been utilized frequently for detecting genuine nonlocality.