The witness operator is defined as $W=P_M+Q_M^{T_M}$ for all bipartitions $M|\bar{M}$, where $P_M$, and $Q_M$ are positive semidefinite operators, and $T_M$ is the partial transpose with respect to $M$. This observable $W$ is positive on all PPT mixtures, but has a negative expectation value on at least one entangled state. To find a fully decomposable witness for a given state $\rho$, the convex optimization technique SDP becomes important, since it allows us to optimize over all fully decomposable witnesses. Hence, a state $\rho$ is a PPT mixture only if the optimization problem [@novo], $$\label{minm}
\textrm{minimize:} {}{} \mathtt{Tr}(W\rho).$$ has a positive solution. If the minimum in Eq. ( \[minm\]) is negative, $\rho$ is not a PPT mixture and hence is genuinely multipartite entangled. We denote the absolute value of the above minimization as $E(\rho)$. For solving the SDP we use the programs YALMIP and SDPT3 [@yalmip; @program], which are freely available.

Now we discuss the dynamics of the GME for the three-qubit entanglement. The initial entangled three-qubit state is chosen as the W state with only one excitation $$|W\rangle =\frac{1}{\sqrt{3}}(|100\rangle +|010\rangle +|001\rangle ),
\label{initial state}$$which corresponds to the Dicke state $|D_{3}\rangle =|-\frac{1}{2}%
\rangle $. For the Hamiltonian ( \[Ham\]) with respect 