  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.
