Exact Compatibility Geometry of Three-Qubit Entanglement

Abstract

We present a permutation-symmetric supporting relation between the pairwise concurrences and the three-tangle in three-qubit system, whose extrema are attained only by the symmetric $W$ and GHZ local-unitary classes. Furthermore, by introducing the concept of a four-dimensional compatibility body, we show that our proposed global relation and the previous CKW relation can be uniformly viewed as different supporting directions of the same achievable set. Subsequently, we provide a necessary-and-sufficient semi-algebraic characterization of the complete four-dimensional compatibility of a three-qubit pure state. Based on this, we can conversely describe the compatibility between entangled components: given the remaining entangled components, the missing two-body entangled component can only take values within a precise parameter-free interval. We demonstrate that the above construction also applies to arbitrary concurrence-generated entanglement measures.