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.
Bell inequalities are certain probabilistic inequalities that should hold in the context of quantum measurement under assumption of a local hidden variable model. These inequalities can be violated according to the theory of quantum mechanics and have also been violated experimentally. Bell inequalities were therefore historically used to disprove local hidden variable models as an interpretation of quantum mechanics. An interesting question is to what degree Bell inequalities can be violated according to the theory of quantum mechanics. The degree to which a particular Bell inequality can be violated is quantified by the largest violation of the Bell inequality. A central question we address in this thesis is how large the largest violation can become when considering all possible Bell inequalities. In particular the CHSH-inequality, a well-known Bell inequality, has a largest violation equal to the square root of 2 and we want to find a Bell inequality with largest violation exceeding the square root of 2. This thesis consists of two main parts. In the first part we formally define Bell inequalities and the largest violation and prove several theorems about Bell inequalities. In the second part we search for Bell inequalities with largest violation exceeding the square root of 2. We do this by first approximating the largest violation of a given Bell inequality using numerical optimization. Next, using this approximation, we use numerical optimization to maximize the largest violation. Due to run-time restrictions we were only able to consider Bell inequalities with a relatively small number of terms and were unable to find any among them with largest violation exceeding the square root of 2.