Expectation Value in Bells Theorem

We will demonstrate in this paper that Bell′s theorem (Bell′s inequality) does not really conflict with quantum mechanics, the controversy between them originates from the different definitions for the expectation value using the probability distribution in Bell′s inequality and the expectation value in quantum mechanics. We can not use quantum mechanical expectation value measured in experiments to show the violation of Bell′s inequality and then further deny the local hidden-variables theory. Considering the difference of their expectation values, a generalized Bell′s inequality is presented, which is coincided with the prediction of quantum mechanics. 03.65.Ud, 03.65.Ta. Local hidden-variables theory[1] was ever a striking point in the development of quantum mechanics, yet its validity was questioned by Bell. In 1964, Bell proposed a famous inequality[2](or theorem) and asserted that local hiddenvariables theory conflicts with quantum mechanics and can not reproduce all the prediction of the latter. This theorem was latterly improved to a new version after Clauser et al.s experiment[3] of testing local hidden-variables theory, which is suit for the entire family of deterministic and nondeterministic local hiddenvariables theory[4]. From then on, Bells inequality had been discussed widely whatever in theories or experiments. In 1982, Aspect et al.[5] proposed an experiment to test Bells inequality by use of time-varying analyzers, their experimental results coincide with the prediction of quantum mechanics and indicate Bells inequality is violated. That is a serious challenge to local hiddenvariables theory. However, the debates never stopped[6]. Jaynes ever criticized that the probabilistic reasoning in Bells theorem does not follow the rules of probability theory[7], Fine discussed the joint distributions and commtativity in Bell theorem[8], there are other viewpoints on Bells theorem[9, 10], too. Does Bells theorem really conflict with theory of quantum mechanics? We said: no. The superficial paradox comes from the confusion of definition for the expectation value using the probability distribution in Bells inequality and the expectation value in quantum mechanics. Consider two particles A and B which have spin 1/2 and are in quantum mechanical state |Ψ〉AB. a and b are vectors in ordinary three-space, A(a,λ)