Hierarchy of Genuine Multipartite Quantum Correlations

TL;DR

This paper establishes a hierarchy of genuine multipartite quantum correlations, demonstrating their inequivalence and extending the known bipartite correlation hierarchy to multipartite systems.

Contribution

It introduces a hierarchy of genuine multipartite correlations and shows their inequivalence through explicit state constructions.

Findings

Established a hierarchy: GMT ≥ GMD ≥ GME ≥ GMS ≥ GMNL.

Proved that GMT, GME, and GMS are inequivalent.

Constructed specific states to demonstrate the hierarchy and inequivalence.

Abstract

Classifying states which exhibiting 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 exist 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 precise states, we show that GMT, GME and GMS are inequivalent with each other, thus GMT $\supsetneq$ GME $\supsetneq$ GMS.