Genuine Multipartite Nonlocality with Causal-Diagram Postselection

The generation and verification of genuine multipartite nonlocality is of central interest for both fundamental research and quantum technological applications, such as quantum privacy. To demonstrate genuine multipartite nonlocality in measurement data, the statistics is commonly postselected by neglecting undesired data. Until now, valid postselection strategies have been restricted to local postselection. A general postselection that is decided after communication between parties can mimic nonlocality, even though the complete data are local. Here, we establish conditions under which genuine multipartite nonlocality is demonstrable even if observations are postselected collectively. Intriguingly, certain postselection strategies that require communication among several parties still offer a demonstration of genuine multipartite nonlocality shared between all parties. The results are derived using the causal structure of the experiment and the no-signalling condition imposed by relativity. Finally, we apply our results to show that genuine three-partite nonlocality can be created with independent particle sources.