Three-tangle of a general three-qubit state in the representation of Majorana stars
Chon‐Fai Kam, Ren‐Bao Liu · Physical Review A · 2020
Majorana stars, the $2j$ spin coherent states that are orthogonal to a spin-$j$ state, offer a visualization of general quantum states and may disclose deep structures in quantum states and their evolutions. In particular, the genuine tripartite entanglement---the three-tangle of a symmetric three-qubit state, which can be mapped to a spin-3/2 state---is measured by the normalized product of the distance between the Majorana stars. However, the Majorana representation cannot be applied to general nonsymmetric $n$-qubit states. We show that, after a series of $\mathrm{SL}(2,\mathbb{C})$ transformations, nonsymmetric three-qubit states can be transformed to symmetric three-qubit states, while at the same time the three-tangle is unchanged. Thus the genuine tripartite entanglement of general three-qubit states has the geometric representation of the associated Majorana stars. The symmetrization and hence the Majorana star representation of certain genuine high-order entanglement for more qubits are possible for some special states. In general cases, however, the constraints on the symmetrization may prevent the Majorana star representation of the genuine entanglement.