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Physica Scripta

Genuine multipartite entanglement measure based on G ω -concurrence

Wen-Hui Liu, Xinbo Liu, Zhi‐Xiang Jin

Published Oct 14, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/aeb451

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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.
Genuine Multipartite Entanglement Measure Based on ααα-concurrence
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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.
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.
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.