The genuinely multipartite nonlocality of graph states is model-dependent

Bell’s theorem proves that some quantum state correlations can only be explained by bipartite non-classical resources. The notion of genuinely multipartite nonlocality (GMNL) was later introduced to conceptualize the fact that nonclassical resources involving more than two parties in a nontrivial way may be needed to account for some quantum correlations. In this letter, we first recall the contradictions inherent to the historical definition of GMNL. Second, we turn to one of its redefinitions, called Local-Operations-and-Shared-Randomness GMNL (LOSR-GMNL), proving that all caterpillar graph states (including cluster states) have this second property. Finally, we conceptualize a third, alternative definition, which we call Local-Operations-and-Neighbour-Communication GMNL (LONC-GMNL), that is adapted to situations in which short-range communication between some parties might occur. We show that cluster states do not have this third property, while GHZ states do. Beyond its technical content, our letter illustrates that rigorous conceptual work is needed before applying the concepts of genuinely multipartite nonlocality, genuine multipartite entanglement or entanglement depth to benchmark the nonclassicality of some experimentally-produced quantum system. We note that most experimental works still use witnesses based on the historical definitions of these notions, which fail to reject models based on bipartite resources.

💡 Research Summary

This paper provides a rigorous analysis of the concept of Genuinely Multipartite Nonlocality (GMNL), highlighting that its proper definition and application are highly model-dependent.

The authors begin by recalling the historical definition of GMNL proposed by Svetlichny within the Local Operations and Classical Communication (LOCC) framework. They point out its inherent contradictions: it allows signaling distributions in its explanatory model, and more critically, systems composed solely of multiple bipartite entangled states (like EPR pairs) can be classified as GMNL. This undermines its utility for device-independent benchmarking of a source’s multipartite character.

To address these issues, the paper turns to a redefinition of GMNL within the Local-Operations-and-Shared-Randomness (LOSR) framework. The LOSR-GMNL model excludes any communication between parties and only allows for shared randomness and combinations of multipartite (specifically, tripartite) nonsignalling resources. The main technical contribution is proving that all caterpillar graph states—a family that includes linear cluster states—possess LOSR-GMNL in a noise-robust manner. Using the inflation technique, the authors derive a Bell-like inequality for the 4-qubit cluster state and show a quantum violation, demonstrating that its correlations cannot be explained without a genuine four-way nonclassical cause. This result significantly advances prior work that was confined to the weaker, device-dependent setting of Genuine Multipartite Entanglement (GME).

Motivated by practical experimental considerations—such as in condensed matter systems where short-range communication between neighboring sites might be unavoidable—the authors then introduce a novel framework: Local-Operations-and-Neighbour-Communication (LONC). This model permits a limited number (t) of synchronous communication rounds along a given network graph, leading to a quantitative notion called LONC-GMNL_t. A key conceptual finding is that under this new definition, cluster states and GHZ states occupy opposite ends of the spectrum. While cluster states are LOSR-GMNL, they can be simulated with only two rounds of neighbour communication on a path graph (they are LONC-GMNL2). In contrast, GHZ states exhibit the maximum level of nonlocality under the LONC definition (LONC-GMNL). This stark difference illustrates that the perceived “genuineness” of a state’s multipartite nonlocality critically depends on the specific causal explanatory model one adopts to define it.

Beyond the technical proofs, the paper’s overarching message is a cautionary one. It emphasizes that rigorous conceptual work is essential before applying terms like GMNL, GME, or entanglement depth to benchmark experimentally produced quantum systems. Most current experimental witnesses are still based on the historical, flawed definitions, which fail to rule out models based merely on combinations of bipartite resources. Therefore, the choice of the underlying causal model (LOCC, LOSR, LONC, etc.) must align carefully with the physical and operational context of the experiment to draw meaningful conclusions about a system’s multipartite nonclassical nature.