The hierarchies of "witnesses" and the properties and characterization of entangled states as super entanglement witnesses

Abstract: Quantum entanglement lies at the heart of quantum mechanical and quantum information processing. Following the question who \emph{witnesses} entanglement witnesses, we show entangled states play as the role of super entanglement witnesses. We show separable states play the role of "super super entanglement witnesses" and "witness" other observables than entanglement witnesses. We show that there exists a hierarchy structure of witnesses and there exist "witnesses" everywhere. Furthermore, we show the properties and characterization of entangled states as super entanglement witnesses. By the role of super witnesses of entangled states, we immediately find the question when different entanglement witnesses can detect the same entangled states [{\it Phys. Lett. A }{\bf 356} 402 (2006)] is the same as the question when different entangled states can be detected by the same entanglement witnesses [{\it Phys. Rev. A} {\bf 75} 052333 (2007)]. By the role of "witnesses", we define finer entangled states and optimal entangled states. The definition gives a nonnumeric measurement of entanglement and an unambiguous discrimination of entangled states, and the procedure of optimization for a general entangled state $\rho$ is just finding the best separable approximation (BSA) to $\rho$ in [ {\it Phys. Rev. Lett.} {\bf 80} 2261 (1998)].}