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arXiv:2506.16935 · International Journal of Theoretical Physics

Classification of Three-Qubit Genuine Entangled States Using Concurrence Fill

Shruti Aggarwal

Published Oct 16, 2025• 1 citation
arXivHTMLPDF
DOI
10.1007/s10773-025-06152-w

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In this paper, we analyze in detail the dynamics of the negativity criterion, fidelity and concurrence fill for the Tavis-Cummings model in which three qubits are in a common single-mode ideal resonator and interact resonantly with the field mode via multiphoton processes. By solving the unsteady Schrödinger equation, we give analytic solutions for genuine entangled Werner- and Greenberger-Horn-Zeilinger states. It is shown that the multiplicity of photon transitions can significantly stabilize the genuine entangled states of both types. Key features in the dynamics pattern of the entanglement parameters are highlighted. We conclude that the negativity criterion is not suitable for describing the dynamics of entanglement of qubits in the case of three-qubit Greenberger-Horn-Zeilinger states. Main advantages of the concurrence fill over the negativity criterion are shown.
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.
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.