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 ${\rho }_1$ and ${\rho }_2$ an EW, $W$, common to both of them exists if and only if for all $0⩽\lambda ⩽1$, $\lambda {\rho }_1+(1-\lambda ){\rho }_2$ 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.

• Received 14 September 2006

DOI:https://doi.org/10.1103/PhysRevA.75.052333