The question of under what conditions different witnesses (e.g., $W_1,W_2$) may detect some common entangled states [i.e., there exists some state $\rho$ so that $\mathrm{Tr}(W_1\rho )<0$ and $\mathrm{Tr}(W_2\rho )<0$] is answered for both finite-dimensional and infinite-dimensional bipartite systems. Finitely many different witnesses $W_1,W_2,\dots ,W_n$ can detect some common entangled states if and only if ${\sum }_{i=1}^n{d}_i{W}_i$ is still a witness for any nonnegative numbers $d_1,d_2,\dots ,d_n$ with ${\sum }_{i=1}^n{d}_i=1$; they cannot detect any common entangled state if and only if ${\sum }_{i=1}^n{c}_i{W}_i$ is a positive operator for some nonnegative numbers $c_1,c_2,\dots ,c_n$ with ${\sum }_{i=1}^n{c}_i=1$. For two witnesses $W_1$ and $W_2$ more can be said. First, $W_1$ and $W_2$ can detect the same set of entangled states if and only if $W_1=a{W}_2$ for some number $a>0$. Second, $W_2$ can detect more entangled states than $W_1$ can if and only if $W_1=a{W}_2+D$ for some number $a>0$ and a positive operator $D$. As an application, some characterizations of the optimal witnesses are given and some structural properties of the decomposable optimal witnesses are presented.

• Received 9 June 2010

DOI:https://doi.org/10.1103/PhysRevA.82.052301