Optimization of entanglement witnesses

An entanglement witness (EW) is an operator that allows the detection of entangled states. We give necessary and sufficient conditions for such operators to be optimal, i.e., to detect entangled states in an optimal way. We show how to optimize general EW, and then we particularize our results to the nondecomposable ones; the latter are those that can detect positive partial transpose entangled states (PPTES’s). We also present a method to systematically construct and optimize this last class of operators based on the existence of “edge” PPTES’s, i.e., states that violate the range separability criterion [Phys. Lett. A 232, 333 (1997)] in an extreme manner. This method also permits a systematic construction of nondecomposable positive maps (PM’s). Our results lead to a sufficient condition for entanglement in terms of nondecomposable EW’s and PM’s. Finally, we illustrate our results by constructing optimal EW acting on $H=C^2\otimes C^4.$ The corresponding PM’s constitute examples of PM’s with minimal “qubit” domains, or—equivalently—minimal Hermitian conjugate codomains.