Three-Tangle of a General Three-Qubit State in the Representation of Majorana Stars

Research paper by Chon-Fai Kam, Ren-Bao Liu

Indexed on: 02 Feb '19Published on: 02 Feb '19Published in: arXiv - Quantum Physics


Majorana stars, the $2j$ spin coherent states that are orthogonal to a spin-$j$ state, offer a geometric representation of the quantum state and many interesting quantum characteristics. 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 applied to general non-symmetric $n$-qubit states. We show that after a series of SL$(2,\mathbb{C})$ transformations, non-symmetric 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. We further discuss the extension to the Majorana-star representation of the entanglement of more qubits.