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.
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 GHZ states in multipartite systems. For system (C^2)^(\otimes N) where N is large, using the language of group theory, we show that a tiny proportion Θ[1/2^(N/2)] 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)^(\otimes N), 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)^(\otimes N), 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.
Abstract A set of orthogonal product states is deemed genuinely nonlocal if they remain locally indistinguishable under any bipartition. In this paper, we first construct a genuinely nonlocal product basis B I(5, 3) in C 5 ⨂ C 5 ⨂ C 5 using a set of nonlocal product states in C 3 ⨂ C 4 . Then, we obtain a genuinely nonlocal product basis B II(5, 3) by replacing certain states in B I(5, 3) with some superposition states. We achieve perfect discrimination of the constructed genuinely nonlocal product basis, separately employing two EPR states and one GHZ state. Our protocol is more efficient than quantum teleportation.
Genuine nonlocality arises when a set of multipartite orthogonal states is locally indistinguishable under any bipartition of the subsystems. The entanglement-assisted discrimination of such genuinely nonlocal orthogonal product sets has attracted significant attention in quantum information. Based on the criterion of local irreducibility, genuine nonlocality is classified into Type I (reducible) and Type II (irreducible). We present entanglement-assisted discrimination schemes for both types of genuinely nonlocal sets that use minimal resources. For low-dimensional cases, Type I sets require only a single EPR pair, whereas Type II sets necessitate only one GHZ state. We extend these protocols to higher-dimensional systems: the discrimination of Type I sets requires only one maximally entangled state in a two-qutrit system, while that of Type II sets similarly demands a single maximally entangled state in a three-qutrit system. For
n-partite (
n>3) systems, Type I sets continue to require only one maximally entangled state, whereas Type II sets necessitate just one additional EPR pair compared to their Type I counterparts. These results provide a robust framework for the efficient discrimination of genuinely nonlocal sets using minimal quantum resources.