Hierarchy of genuine multipartite quantum correlations

Abstract

Classifying quantum states which exhibit different statistical correlations is among the most important problems in quantum information science and quantum many-body physics. In bipartite case, there is a clear hierarchy of states with different correlations: total correlation (T) \(\supsetneq \) discord (D) \(\supsetneq \) entanglement (E) \(\supsetneq \) steering (S) \(\supsetneq \) Bell nonlocality (NL). However, very little is known about genuine multipartite correlations (GM\(\mathcal {C}\)) for both conceptual and technical difficulties. In this work, we show that, for any N-partite qudit states, there also exists such a hierarchy: genuine multipartite total correlations (GMT) \(\supseteq \) genuine multipartite discord (GMD) \(\supseteq \) genuine multipartite entanglement (GME) \(\supseteq \) genuine multipartite steering (GMS) \(\supseteq \) genuine multipartite nonlocality (GMNL). Furthermore, by constructing explicit states, we show that GMT, GME and GMS are inequivalent with each other and thus GMT \(\supsetneq \) GME \(\supsetneq \) GMS.