  Entanglement witness is a Hermitian operator that is useful for detecting the
genuine multipartite entanglement of mixed states. Nonlinear entanglement
witnesses have the advantage of a wider detection range in the entangled
region. We construct genuine entanglement witnesses for four qubits density
matrices in the mutually unbiased basis. First, we find the convex feasible
region with positive partial transpose states. Then, to reveal the entangled
regions, we present some appropriate linear entanglement witnesses and, we find
the envelope of this family of linear witnesses as a nonlinear witness.
Finally, we study thermal entanglement and we show for some Hamiltonians the
witnesses can detect the entanglement at all temperatures.
