Multipartite correlation measures and framework for multipartite quantum resources theory

TL;DR

This work introduces a coherent framework for multipartite quantum resource theory by defining true MQC measures that obey faithfulness, monotonicity, a unification condition across all part reductions, and a hierarchy condition linking sub-repartitions to the whole system. It shows how multiple symmetric MQCs (e.g., multipartite entanglement, k-entanglement, non-PPT, and several Gaussian resources) admit true measures within this framework, while proving that multipartite steering is an asymmetric resource. The paper also develops a formal monogamy structure for symmetric MQCs, with global, complete, tight, and strong variants, and provides concrete examples of monogamy behavior in known MQCs. Overall, the framework clarifies how to quantify and compare correlations in part versus whole systems, with implications for resource allocation and quantum information processing in complex multipartite settings.

Abstract

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.