We derive tight quadratic inequalities for all kinds of hybrid separable-inseparable n-particle density operators on an arbitrary dimensional space. This methodology enables us to derive a tight quadratic inequality as tests for full n-partite entanglement in various Bell-type correlation experiments on the systems that may not be identified as a collection of qubits, e.g., those involving photons measured by incomplete detectors. It is also proved that when the two measured observables are assumed to precisely anticommute, a stronger quadratic inequality can be used as a witness of full n-partite entanglement.

We propose a generalized Bell inequality for two three-dimensional systems with three settings in each local measurement. It is shown that this inequality is maximally violated if local measurements are configured to be mutually unbiased and a composite state is maximally entangled. This feature is similar to Clauser-Horne-Shimony-Holt inequality for two qubits but is in contrast with the two types of inequalities, Collins-Gisin-Linden-Massar-Popescu and Son-Lee-Kim, for high-dimensional systems. The generalization to aribitrary prime-dimensional systems is discussed.

Rotational invariance of physical laws is a generally accepted principle. We show that it leads to an additional external constraint on local realistic models of physical phenomena involving measurements of multiparticle spin 1 2 correlations. This new constraint rules out such models even in some situations in which standard Bell inequalities allow for explicit construction of such models. The whole analysis is performed without any additional assumptions on the form of local realistic models.PACS numbers: 03.65.UdLocal realism is a foundation of classical physics [1,2,3,4]. It is a conjunction of realism, i.e. assumption that physical systems posses properties, irrespective whether these are measured or not, and locality, which is the assumption of a finite speed of influences (i.e. it is a consequence of special relativity). Quantum mechanics does not allow a local realistic interpretation. The quantum predictions violate Bell inequalities [2], which are conditions that all local realistic theories must satisfy. Many of the recent advances in quantum information theory suggest that the highly-non-classical properties of quantum states that lead to violations of Bell inequalities can be used as a resource to achieve success in some tasks, which are classically impossible. As examples can serve quantum cryptography and quantum communication complexity [5,6]. Therefore as the impossibility of existence of classical models for some processes leads to various quantum informational applications it is important to learn what are the ultimate bounds for such models.

Based on a geometrical argument introduced byŻukowski, a new multisetting Bell inequality is derived, for the scenario in which many parties make measurements on two-level systems. This generalizes and unifies some previous results. Moreover, a necessary and sufficient condition for the violation of this inequality is presented. It turns out that the class of non-separable states which do not admit local realistic description is extended when compared to the two-setting inequalities. However, supporting the conjecture of Peres, quantum states with positive partial transposes with respect to all subsystems do not violate the inequality. Additionally, we follow a general link between Bell inequalities and communication complexity problems, and present a quantum protocol linked with the inequality, which outperforms the best classical protocol.