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Mermin's inequality is the generalization of the Bell-CHSH inequality forthree qubit states. The violation of the Mermin inequality guarantees the factthat there exists quantum non-locality either between two or three qubits in athree qubit system. In the absence of an analytical result to this effect, inorder to check for the violation of Mermin's inequality one has to perform anumerical optimization procedure for even three qubit pure states. Here wederive an analytical formula for the maximum value of the expectation of theMermin operator in terms of eigenvalues of symmetric matrices, that gives themaximal violation of the Mermin inequality for all three qubit pure and mixedstates.

An analytical condition for the violation of Mermin's inequality by any three qubit state