Determining the existence of the common entanglement witnesses for some entangled states
Yu-Chun Wu, Guang‐Can Guo · Physical Review A · 2007
Entanglement witnesses (EWs) are nonpositive Hermitian operators which can detect the presence of entanglement. In this paper, we prove that for a given pair of entangled states ${\ensuremath{\rho}}_{1}$ and ${\ensuremath{\rho}}_{2}$ an EW, $W$, common to both of them exists if and only if for all $0\ensuremath{\leqslant}\ensuremath{\lambda}\ensuremath{\leqslant}1$, $\ensuremath{\lambda}{\ensuremath{\rho}}_{1}+(1\ensuremath{-}\ensuremath{\lambda}){\ensuremath{\rho}}_{2}$ is an entangled state. Based on this result, we find a sufficient condition for the presence of entanglement. We then show that an EW which can detect all bell states simultaneously does not exist.