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 the failure of the hidden-variable theory and to prove the validity of the quantum theory. We note that the violation of the BCHSH inequality is caused by the noncommutativity of quantum observables and find a systematic method of constructing generalizations of the BCHSH inequality. This method is used to invent 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.
A correlation inequality is derived from local realism and a supplementary assumption. Unlike Clauser-Horne (CH) inequality [or Clauser-Horne-Shimony-Holt (CHSH) inequality] which is violated by quantum mechanics by a factor of 2, this inequality is violated by a factor of 1.5. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, unlike CH (or CHSH) inequality which requires the measurement of five detection probabilities, the present inequality requires the measurement of only two detection probabilities. This inequality can therefore be used to test locality more simply than CH or CHSH inequality.