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Annalen der Physik

A Genuine Multipartite Entanglement Measure: The Geometric Mean of Total Concurrence

Rui‐Zhou Liang, Xinbo Liu, Zhi‐Xiang Jin

Published Mar 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1002/andp.202500554

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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.
Genuine Multipartite Entanglement Measure Based on ααα-concurrence
Quantifying genuine entanglement is a crucial task in quantum information theory. Based on the geometric mean of bipartite ααα-concurrences among all bipartitions, we present a class of well-defined genuine multipartite entanglement (GME) measures GαααC with one parameter ααα for arbitrary multipartite states. We show that the GαααC is of continuity for any multipartite pure states. By utilizing the related symmetry, analytical results of GαααC are derived for any nnn-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 GαααC can distinguish different GME states that other GME measures fail to. These results justify the potential applications of GαααC in characterizing genuine multipartite entanglements.
In this paper, we propose the geometric mean of bipartite concurrences as a genuine multipartite entanglement measure. This measure achieves the maximum value for absolutely maximally entangled states and has desirable properties for quantifying potential quantum resources. The simplicity and symmetry in the definition facilitates its computation for various multipartite entangled states including the GHZ states and the WWW states. With explicit examples, we show that our measure results in distinct entanglement orderings from other measures and can detect differences in certain types of genuine multipartite entanglement while other measures cannot. These results justify the potential application of our measure for tasks involving genuine multipartite entanglement.
Abstract Quantum entanglement is a core resource for quantum communication and computation. Genuine multipartite entanglement (GME), as a critical quantum correlation, underpins measurement-based quantum computation and multipartite quantum simulation, but existing GME measures cannot well characterize uneven entanglement distributions and are insensitive to single-partition entanglement failure in practical bottleneck scenarios. To address this problem, we propose a novel GME measure termed harmonic total concurrence (HTC), defined as the harmonic mean of total concurrence over all nontrivial bipartitions, and place it within a one-parameter power-mean family of total concurrence measures. HTC satisfies the standard entanglement hierarchy and effectively distinguishes GHZ and W states. Its computable lower bounds for mixed states outperform geometric total concurrence in entanglement detection under typical noise, and it extends the effective time window for entanglement certification while enabling early warning of asymmetric entanglement decay. We further define an entanglement non-uniformity parameter and prove HTC induces a state ordering distinct from existing measures. This work provides a new theoretical tool for multipartite entanglement characterization, with good experimental accessibility on photonic, atomic and optomechanical platforms.
Abstract We introduce G ω -GME, a family of genuine multipartite entanglement measures defined via the geometric mean of a parameterized bipartite concurrence. We prove that G ω -GME satisfies all GME axioms, derive analytical expressions for GHZ and W states, and establish a strict entanglement hierarchy. Continuity under state perturbations is proven with explicit bounds. Compared with genuinely multipartite concurrence, GqC, and G α C, G ω -GME resolves finer entanglement structures and offers greater flexibility through its extended parameter range. These features make G ω -GME a powerful and practical tool for multipartite entanglement quantification and quantum network certification.
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.
Genuine multipartite entanglement (GME) is a valuable resource for quantum technologies, but quantifying it in many-body systems can be demanding since measures defined through optimizations over bipartitions are often assumed to require large joint measurements and many measurement settings. We introduce the concepts of GME evaluation length and GME evaluation number, which capture, respectively, the largest subsystem that must be jointly measured and the number of distinct subsystems that need to be measured to evaluate multipartite entanglement measures defined via bipartition optimizations. Taking the genuine multipartite concurrence (GMC) as a canonical example, we show that, for NNN-qubit Haar-random pure states, the GMC can be evaluated using only NNN single-qubit purity measurements. Thus, for typical many-body states, evaluating the GMC requires considering only the NNN single-site bipartitions, rather than the exponentially many distinct bipartitions. We further corroborate this behavior numerically for physically accessible states generated by local random quantum circuits. Our results show that the practical cost of quantifying GME can be dramatically lower than suggested by its formal definition, establishing measurement complexity as a complementary operational lens for multipartite entanglement.