  We provide an analytical tripartite-study from the generalized $R$-matrix. It
provides the upper bound of the maximum violation of Mermin's inequality. For a
generic 2-qubit pure state, the concurrence or $R$-matrix characterizes the
maximum violation of Bell's inequality. Therefore, people expect that the
maximum violation should be proper to quantify Quantum Entanglement. The
$R$-matrix gives the maximum violation of Bell's inequality. For a general
3-qubit state, we have five invariant entanglement quantities up to local
unitary transformations. We show that the five invariant quantities describe
the correlation in the generalized $R$-matrix. The violation of Mermin's
inequality is not a proper diagnosis due to the non-monotonic behavior. We then
classify 3-qubit quantum states. Each classification quantifies Quantum
Entanglement by the total concurrence. In the end, we relate the experiment
correlators to Quantum Entanglement.
