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We provide an analytical tripartite-study from the generalized $R$-matrix. Itprovides the upper bound of the maximum violation of Mermin's inequality. For ageneric 2-qubit pure state, the concurrence or $R$-matrix characterizes themaximum violation of Bell's inequality. Therefore, people expect that themaximum violation should be proper to quantify Quantum Entanglement. The$R$-matrix gives the maximum violation of Bell's inequality. For a general3-qubit state, we have five invariant entanglement quantities up to localunitary transformations. We show that the five invariant quantities describethe correlation in the generalized $R$-matrix. The violation of Mermin'sinequality is not a proper diagnosis due to the non-monotonic behavior. We thenclassify 3-qubit quantum states. Each classification quantifies QuantumEntanglement by the total concurrence. In the end, we relate the experimentcorrelators to Quantum Entanglement.

Tripartite Entanglement and Quantum Correlation