Multipartite correlation measures and framework for multipartite quantum resources theory

In recent years, it has been recognized that properties of multipartite physical systems, such as genuine multipartite entanglement, can be considered as important resources for quantum information and other areas of physics. However, the current framework of multipartite quantum resource theory is flawed. In this paper, we propose a more reasonable framework for multipartite quantum resource theory with a particular focus on axiomatic definition for true measures of multipartite quantum correlations (MQC) that regulates how to measure the correlation in part systems (the unification condition) and how to describe the requirement from many-body resource theory that the correlation hold by part system does not exceed that of the entire system (the hierarchy condition). We find that, due to the inherent characteristics of MQCs, the true measures of different MQCs exhibit distinct hierarchy conditions. Based on this framework, we verify that multipartite entanglement, $k$-entanglement, $k$-partite entanglement, multipartite non-PPT, multipartite coherence, multipartite imaginarity, multipartite multi-mode Gaussian non-product correlation, multipartite multi-mode Gaussian imaginarity, and multipartite single-mode Gaussian coherence are all symmetric multipartite quantum resources. We also show that, multipartite steering is an asymmetric multipartite quantum resource. Finally, the monogamy relations for true measures of symmetric MQCs are discussed.