When different entanglement witnesses detect the same entangled states
Jinchuan Hou, Yu Jing Guo · Physical Review A · 2010
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 $\ensuremath{\rho}$ so that $\mathrm{Tr}({W}_{1}\ensuremath{\rho})0$. 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.