Generalized bell inequalities and quantum entanglement

The focus of this thesis lies on Bell inequalities. We introduce the concept of generalizations of a Bell inequality, which are Bell inequalities that by construction perform at least as well at any given task as the Bell inequality they generalize. Further, we present the cone-projection technique that we use to find such generalizations of certain Bell inequalities. Specifically, we find all 3050 symmetric generalizations of the I3322 inequality to three parties and study their quantum mechanical properties. Some of them detect nonlocality of quantum states, for which all two-setting Bell inequalities fail to do so. Moreover, we find generalizations of the Svetlichny inequality, the I4422 inequality, the Guess-Your-Neighbors-Input inequality as well as Bell inequalities that simultaneously generalize the I3322 inequality and the Clauser-Horne-Shimony-Holt inequality. We study the quantum mechanical properties of all of those inequalities.
Furthermore, we investigate different hybrid models and present Bell inequalities to test them. We numerically estimate the noise robustness for all of these Bell inequalities. We also construct a family of Bell inequalities for a particular class of hybrid models. To simplify research on Bell inequalities, we present Bellpy, which is a Python library to construct and investigate facet-defining Bell inequalities.
Besides our work on Bell inequalities, we also investigate absolute maximally entangled Werner states and show that such states only exist for systems of two qubits and three qutrits. Finally, we analyze a variation of the Bose-Marletto-Vedral experiment, where two quantummechanically described beads interact gravitationally.