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arXiv:2609.03864

Five-Dimensional Compatibility Geometry of Pure Three-Qubit Entanglement

Wei Song, Xiao-Lan Zong, Ming Yang

quant-phSubmitted Sep 3, 2026• Announced Sep 4, 2026• 0 citations• Revised Sep 6, 2026 (v2)
v1 · Sep 3, 2026v2 · Sep 6, 2026v3 · Sep 10, 2026v4 · Sep 13, 2026
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Abstract

We investigates the joint compatibility problem of the three pairwise concurrences, the three-tangle, and the Kempe invariant in arbitrary pure three-qubit states, and gives necessary and sufficient conditions for these five quantities to be simultaneously realizable by the same pure state. This result extends the existing four-dimensional tangle compatibility relation to the five-coordinate case, while also incorporating the known Kempe-invariant compatibility structure in permutation-symmetric states.

Comments
7 pages,The manuscript has been fully revised, and the new version will be coming out soon

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For pure symmetric three-qubit states there are only three algebraically independent entanglement measures; one choice is the pairwise concurrence C\mathcal{C}C, the three-tangle $\ensuremath{\tau}$, and the Kempe invariant $\ensuremath{\kappa}$. Using a canonical form for symmetric nnn-qubit states derived from their Majorana representation, we derive the explicit achievable region of triples $(\mathcal{C},\ensuremath{\tau},\ensuremath{\kappa})$.
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Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.