We study the quantification of genuine multipartite entanglement (GME) for general multipartite states. A set of inequalities satisfied by the entanglement of N-partite pure states is derived by exploiting the restrictions on entanglement distributions, showing that the bipartite entanglement between each part and its remaining ones cannot exceed the sum of the other partners with their remaining ones. Then a series of triangles, named concurrence triangles, are established corresponding to these inequalities. Proper genuine multipartite entanglement measures are thus constructed by using the geometric mean area of these concurrence triangles, which are non-increasing under local operation and classical communication. The GME measures classify which parts are separable or entangled with the rest ones for non genuine entangled pure states. The GME measures for mixed states are given via the convex roof construction, and a witness to detect the GME of multipartite mixed states is presented by an approach based on state purifications. Detailed examples are given to illustrate the effectiveness of our GME measures.
We study the quantification of genuine multipartite entanglement (GME) for general multipartite states. A set of inequalities satisfied by the entanglement of N-partite pure states is derived by exploiting the restrictions on entanglement distributions, showing that the bipartite entanglement between each part and its remaining ones cannot exceed the sum of the other partners with their remaining ones. Then a series of triangles, named concurrence triangles, are established corresponding to these inequalities. Proper genuine multipartite entanglement measures are thus constructed by using the geometric mean area of these concurrence triangles, which are non-increasing under local operation and classical communication. The GME measures classify which parts are separable or entangled with the rest ones for non genuine entangled pure states. The GME measures for mixed states are given via the convex roof construction, and a witness to detect the GME of multipartite mixed states is presented by an approach based on state purifications. Detailed examples are given to illustrate the effectiveness of our GME measures.
Genuine Multipartite Entanglement Measure Based on α-concurrence
Quantifying genuine entanglement is a crucial task in quantum information theory. Based on the geometric mean of bipartite α-concurrences among all bipartitions, we present a class of well-defined genuine multipartite entanglement (GME) measures GαC with one parameter α for arbitrary multipartite states. We show that the GαC is of continuity for any multipartite pure states. By utilizing the related symmetry, analytical results of GαC are derived for any n-qubit GHZ states and W states, which show that the GHZ states are more genuinely entangled than the W states. With explicit examples, we demonstrate that the GαC can distinguish different GME states that other GME measures fail to. These results justify the potential applications of GαC in characterizing genuine multipartite entanglements.
Genuine multipartite entanglement (GME) represents the strongest form of quantum correlation, where all parties of a system across each bipartition are inseparably linked. This resource is crucial for advanced quantum protocols, including multi-party cryptography and distributed quantum computing, yet its quantification remains a significant theoretical challenge. This review surveys a rapidly evolving geometric paradigm that translates this challenge into a problem of polygon inequalities. By representing marginal bipartite entanglements as side lengths, the distribution of entanglement in a multipartite state must satisfy geometric closure conditions akin to forming a polygon. These polygon-based constraints provide a unifying framework for detecting and quantifying GME, offering an intuitive bridge between bipartite entanglement measures and global multipartite structure. We chart the development of this approach from its monogamy-based origins to modern formulations using concurrence triangles, wedge products, and entropy-based polygons. Recent advances have established rigorous conditions for these geometric quantities to serve as faithful, operationally meaningful entanglement measures, while also extending the framework to mixed states and higher-dimensional systems. Despite progress, key challenges persist in ensuring universal monotonicity and experimental robustness. By synthesizing these geometric insights, this review highlights how polygon inequalities are reshaping the quantification of complex quantum correlations and providing novel diagnostic tools for the next generation of quantum technologies.
ABSTRACT Entanglement is a fundamental resource in quantum information science. In this work, we introduce the geometric total concurrence (GTC), a new measure of genuine multipartite entanglement (GME) for ‐partite systems based on the concept of total concurrence. The GTC satisfies a key criterion by assigning higher entanglement to GHZ states than to W states. We derive analytically computable lower bounds for the GTC using PPT and realignment criteria. Furthermore, we show that the GTC induces an entanglement ordering distinct from existing measures, providing a unique perspective on multipartite correlations. Through detailed examples, we demonstrate that the GTC offers improved sensitivity in detecting entanglement and sharper discrimination among different types of entangled states, establishing it as a practical tool for characterizing and quantifying GME.
ABSTRACT Characterizing the entanglement in multipartite quantum systems remains a central challenge in the theory of quantum entanglement. We establish a comprehensive framework for constructing ‐entanglement measures by leveraging bipartite entanglement measures and general ‐entanglement measures () with arithmetic and geometric averaging schemes. Numerical examples demonstrate that these measures outperform the existing ones, such as ‐ME concurrence, geometric mean (k‐GM) concurrence, and the generalized ‐entanglement measure by inducing distinct entanglement orderings and offering superior resolution. Furthermore, we reveal the operational relationship between the ‐entanglement and the local quantum coherence, verifying that the persistence of entanglement is intrinsically constrained by the system's capacity to maintain the local coherence. Finally, to overcome the inherent limitation of the conventional three‐tangle in characterizing the genuine multipartite entanglement of the GHZ‐W mixed states, we design a novel GME measure. Numerical results confirm that this measure effectively eliminates the detection blind spots where vanishes, providing a complete and practical mathematical tool for the quantitative analysis of genuine multipartite quantum correlations.
Abstract Quantifying genuine entanglement is a crucial task in quantum information theory. In this work, a new genuine multipartite entanglement measure based on informationally complete entanglement is proposed, termed the informationally complete genuine multipartite entanglement measure(ICGMEM). A set of entanglement inequalities for pure states in ‐partite systems is first established based on constraints on entanglement distribution. These inequalities reveal that the pairwise entanglement between any individual part of the system cannot exceed the collective entanglement of the remaining parts. Building on these inequalities, a series of geometric structures called informationally complete entanglement measure triangles is constructed and the ICGMEM is defined based on the geometric mean area of these triangles. These results demonstrate that the ICGMEM provides a more refined characterization of entanglement compared to existing well‐known measures. Furthermore, it is shown that the ICGMEM significantly enhances computational efficiency for analyzing genuinely entangled pure states, achieving up to a twofold improvement when the number of subsystems is sufficiently large. Through detailed examples, the advantages of the ICGMEM are illustrated, highlighting its effectiveness in distinguishing and quantifying entanglement in multipartite systems. This work can better advance the theoretical understanding of genuine multipartite entanglement.
Xie and Eberly introduced a genuine multipartite entanglement (GME) measure ‘concurrence fill’ (Xie and Eberly, 2021) for three-party systems. It is defined as the area of a triangle whose side lengths represent squared concurrence in each bi-partition. However, it has been recently shown that concurrence fill is not monotonic under LOCC, hence not a faithful measure of entanglement. Though it is not a faithful entanglement measure, it encapsulates an elegant geometric interpretation of bipartite squared concurrences. There have been a few attempts to generalize GME quantifier to four-party settings and beyond. However, some of them are not faithful, and others simply lack an elegant geometric interpretation. The recent proposal from Xie et al.. constructs a concurrence tetrahedron, whose volume gives the amount of GME for four-party systems; with generalization to more than four parties being the hypervolume of the simplex structure in that dimension. Here, we show by construction that to capture all aspects of multipartite entanglement, one does not need a more complex structure, and the four-party entanglement can be demonstrated using 2D geometry only. The subadditivity together with the Araki-Lieb inequality of linear entropy is used to construct a direct extension of the geometric GME quantifier to four-party systems resulting in quadrilateral geometry. Our quantifier can be geometrically interpreted as a combination of three quadrilaterals whose sides result from the concurrence in one-to-three bi-partition, and diagonal as concurrence in two-to-two bipartition.