Inertia Sets of Three-qubit Entanglement Witnesses

Mutiqubit entanglement witnesses are essential in the verification of quantum entanglement of distributed systems. To characterize such witnesses, we investigate the inertia set of three partial transposed matrices of three-qubit entangled states, which are not positive semidefinite. Theoretically, there are totally 165 elements in the set, and we show some of them by constructing exactly their expressions. Our method relies on both theoretical analysis and program for numerical computing. We also apply our results to describe some physical systems from flavor neutrino systems.