When different entanglement witnesses can detect the same entangled states

Entanglement witnesses (EW) are non-positive Hermitian operators which can detect the presence of entanglement. Let DWDW denote the set of entangled states detected by the entanglement witness W  . In [M. Lewenstein, B. Kraus, J.I. Cirac, P. Horodecki, Phys. Rev. A 62 (2001) 052310], authors showed that DW2⊂DW1DW2⊂DW1 if and only if there exists a positive operator P   and 0⩽ε<10⩽ε<1 such that W2=(1−ε)W1+εPW2=(1−ε)W1+εP. In this Letter, we consider the following problem that if there exists no inclusion relation between DW1DW1 and DW2DW2, then how to determine whether their intersection is an empty set or not. We show that there exists a state ρ   detected by W1W1 and W2W2 if and only if for any λ∈[0,1]λ∈[0,1], W=λW1+(1−λ)W2W=λW1+(1−λ)W2 is not a positive operator. And this result can be generalized to the finite number of EWs.