Detecting multi-qubit bound MUB diagonal entangled states via nonlinear optimal entanglement witnesses

An easier way to detect entanglement is using linear entanglement witnesses. Here by determining the envelope of the boundary hyper-planes defined by a family of linear entanglement witnesses, a set of powerful nonlinear optimal entanglement witnesses is manipulated. These nonlinear optimal entanglement witnesses enable us to detect some multi-qubit bound MUB(mutually unbiased bases) diagonal entangled states, i.e., the positive partial transpose(PPT) entangled states and in some particular cases, they are powerful enough to separate the bound entangled regions from the separable ones.