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arXiv:2608.07145

From triangles to prisms: towards a geometric extension of concurrence fill for three-qubit mixed states

Divyanshu K. Verma, Girish Kulkarni

quant-phSubmitted Aug 7, 2026• Announced Aug 10, 2026• 0 citations
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Abstract

The quantification of multipartite entanglement remains a long-standing open challenge in quantum information theory with major implications for quantum foundations and quantum technologies. In the context of three-qubit pure states, the concurrence triangle underlies geometric interpretations for a variety of multipartite entanglement measures. In particular, the concurrence fill, which is proportional to the square root of the area of the concurrence triangle, has been shown to be a valid measure of genuine tripartite entanglement. As a result, there is now significant interest in extending the concurrence triangle and concurrence fill to three-qubit mixed states. In this work, we show that any rank-2 three-qubit mixed state can be visualized in terms of concurrence prisms associated with their pure state decompositions. Thus, the concurrence fill of the mixed state obtained through convex-roof extension is proportional to the total prism fill area, defined as the square root of the base area times the height, associated with the prisms minimized over all pure state decompositions. We then consider three-qubit mixed states that are incoherent mixtures of Greenberger-Horne-Zeilinger (GHZ) and W states and analytically perform their convex-roof extension to obtain a closed-form expression for the concurrence fill of these rank-2 mixtures. Finally, we apply the geometric picture of concurrence prisms to these mixtures and their pure state decomposition corresponding to minimum concurrence fill as a tool for visualization.

Comments
9 pages, 4 figures

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The quantification of multipartite entanglement remains a long-standing open challenge in quantum information theory with major implications for quantum foundations and quantum technologies. In the context of three-qubit pure states, the concurrence triangle underlies geometric interpretations for a variety of multipartite entanglement measures. In particular, the concurrence fill, which is proportional to the square root of the area of the concurrence triangle, has been shown to be a valid measure of genuine tripartite entanglement. As a result, there is now significant interest in extending the concurrence triangle and concurrence fill to three-qubit mixed states. In this work, we show that any rank-2 three-qubit mixed state can be visualized in terms of concurrence prisms associated with their pure state decompositions. Thus, the concurrence fill of the mixed state obtained through convex-roof extension is proportional to the total prism fill area, defined as the square root of the base area times the height, associated with the prisms minimized over all pure state decompositions. We then consider three-qubit mixed states that are incoherent mixtures of Greenberger-Horne-Zeilinger (GHZ) and W states and analytically perform their convex-roof extension to obtain a closed-form expression for the concurrence fill of these rank-2 mixtures. Finally, we apply the geometric picture of concurrence prisms to these mixtures and their pure state decomposition corresponding to minimum concurrence fill as a tool for visualization.
Quantifying entanglement in composite systems is a fundamental challenge, yet exact results are available in only a few special cases. This is because hard optimization problems are routinely involved, such as finding the convex decomposition of a mixed state with the minimal average pure-state entanglement, the so-called convex roof. We show that under certain conditions such a problem becomes trivial. Precisely, we prove by a geometric argument that polynomial entanglement measures of degree 2 are independent of the choice of pure-state decomposition of a mixed state, when the latter has only one pure unentangled state in its range. This allows for the analytical evaluation of convex roof extended entanglement measures in classes of rank-2 states obeying such a condition. We give explicit examples for the square root of the three-tangle in three-qubit states, and we show that several representative classes of four-qubit pure states have marginals that enjoy this property.
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.
Multipartite entanglement is an indispensable resource in quantum communication and computation; however, it is a challenging task to faithfully quantify this global property of multipartite quantum systems. In this work we study the concurrence fill, which admits a geometric interpretation to measure genuine tripartite entanglement for the three-qubit system [Xie and Eberly, Phys. Rev. Lett. 127, 040403 (2021)]. First, we use the well-known three-tangle and bipartite concurrence to reformulate this quantifier for all pure states. We then construct an explicit example to conclusively show that the concurrence fill can be increased under local operations and classical communication (LOCC) on average, implying it is not an entanglement monotone. Moreover, we give a simple proof of the LOCC monotonicity of the three-tangle and find that the bipartite concurrence and the squared concurrence can have distinct performances under the same LOCC. Finally, we propose a reliable monotone to quantify genuine tripartite entanglement, which can also be easily generalized to the multipartite system. Our results shed light on the study of genuine entanglement and also reveal the complex structure of multipartite systems.
This study presents an analytical investigation of entanglement quantification in three-qubit pure states through the Acín canonical representation, which serves as a generalization of the two-qubit Schmidt decomposition. Driven by the intricacies of multipartite entanglement and the shortcomings of current measures, we employ a global concurrence measure derived from the generalized concurrence of various bipartitions within the system. The characteristics of this measure are explored both analytically and numerically across a range of SLOCC entanglement classes, such as product, biseparable, GHZ-type, and W-type states. When compared to the three-tangle and tripartite negativity, our findings indicate that this measure yields complementary insights into multipartite correlations. The outcomes underscore the significance of this measure in elucidating the structure and distribution of entanglement within three-qubit quantum systems.
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.
An algorithm is proposed that serves to handle full-rank density matrices when coming from a lower-rank method to compute the convex roof. This is in order to calculate an upper bound for any polynomial SL-invariant multipartite entanglement measure EEE. This study exemplifies how this algorithm works based on a method for calculating convex roofs of rank-2 density matrices. It iteratively considers the decompositions of the density matrix into two states each, exploiting the knowledge for the rank-2 case. The algorithm is therefore quasiexact as far as the rank-2 case is concerned, and it also hints where it should include more states in the decomposition of the density matrix. Focusing on the measure of three-way entanglement of qubits (called three-tangle), I show the results the algorithm gives for two states, one of which is the Greenberger-Horne-Zeilinger--Werner (GHZ-WWW) state, for which the exact convex roof is known. It overestimates the three-tangle in the state, thereby giving insight into the optimal decomposition the GHZ-WWW state has. As a proof of principle, I have run the algorithm for the three-tangle on the transverse quantum Ising model. I give qualitative and quantitative arguments why the convex roof should be close to the upper bound found here.