A method for systematic construction of Bell-like inequalities and a proposal of a new type of test

Abstract: The Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, which is proven in the context of the local hidden variable theory, has been used as a test to reveal failure of the hidden variable theory and to prove validity of the quantum theory. We note that violation of the BCHSH inequality is caused by noncommutativity of quantum observables and find a systematic method for constructing generalizations of the BCHSH inequality. This method is applied for inventing a new quantity which is defined in a two-qubit system and satisfies a new type of inequality. This provides a fair test of the quantum theory. Remaining problems are also discussed.