Violating Bell's Inequality Beyond Cirel'son's Bound

Cirel’son inequality states that the absolute value of the combination of quantum correlations appearing in the Clauser-Horne-Shimony-Holt (CHSH) inequality is bound by $2\sqrt{2}$. It is shown that the correlations of two qubits belonging to a three-qubit system can violate the CHSH inequality beyond $2\sqrt{2}$. Such a violation is not in conflict with Cirel’son’s inequality because it is based on postselected systems. The maximum allowed violation of the CHSH inequality, 4, can be achieved using a Greenberger-Horne-Zeilinger state.