  One of the important approaches to detect quantum entanglement is using
linear entanglement witnesses EWs. In this paper, by determining the envelope
of the boundary hyper-planes defined by a family of linear EWs, a set of
powerful nonlinear optimal EWs is manipulated. These EWs enable us to detect
some three qubits bound MUB (mutually unbiased bases) diagonal entangled
states, i.e., the PPT (positive partial transpose) entangled states. Also, in
some particular cases, the introduced nonlinear optimal EWs are powerful enough
to separate the bound entangled regions from the separable ones. Finally, we
present numerical examples to demonstrate the practical accessibility of this
approach.
  Keywords :nonlinear optimal entanglement witnesses, mutually unbiased bases,
MUB diagonal states
