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Recently the method based on irreducible representations of finite groups has been proposed as a tool for investigating the more sophisticated versions of Bell inequalities (V. Ugǔr Gűney, M. Hillery, Phys. Rev. A90, 062121 ([2014]) and Phys. Rev. A91, 052110 ([2015])). In the present paper an example based on the symmetry group S 4 is considered. The Bell inequality violation due to the symmetry properties of regular tetrahedron is described. A nonlocal game based on the inequalities derived is described and it is shown that the violation of Bell inequality implies that the quantum strategies outperform their classical counterparts.

Bell inequalities provide a specific setting for investigating the physics of entanglement in quantum mechanics. They give a basis for providing an experimental realization of these kinds of quantum phenomena and exhibit some of its more unusual consequences. Some useful ways to look at entanglement quantitatively are presented. It is intended that the presentation and results will provide insights which make effective experimental observation easier.

A new kind of quantum state is introduced, which can be generated by the superposition of high-order (N) photon addition to two-mode thermal state (SPA-TMTS). By using the P-representation of thermal state, the normal ordering form of SPA-TMTS is obtained. Based on this, some analytical expressions, such as the normalization coefficient, Q-function, Wigner function (WF) are presented. It is shown that the WF can possess negative value at some region. In addition, the nonlocality of the SPA-TMTS is discussed in terms of the CH and CHSH inequalities. It is found that for a given small $\bar{n}$ , the CH inequality is violated for some small N and the violation decrease with the increasing N; while the CHSH inequality is violated only for N=1.

A set of Bell inequalities classifying the quantum entanglement of arbitrary dimensional tripartite systems is presented. These inequalities can characterize full separable and bi-separable quantum states. In addition, for 3⊗3⊗3 systems, we present two kinds of different Bell inequalities to classify quantum entangled states based on the generators of SU(3) and the generalized Pauli operators.

In this paper, a Bell inequality is given to study the nonlocality of four-qubit states. It is shown that the inequality is maximally violated by the four-qubit state |ψ〉. Some other states, such as the state |χ〉, the cluster state, violate this inequality, too, but not maximally. And it is not violated by the four-qubit GHZ state. In addition, the state |ψ〉 also violates other bell inequalities, but not so optimally. Therefore the inequality can detect the entanglement of |ψ〉 well, experimentally.

We review some well-known Bell inequalities, the relations between the Bell inequality and quantum separability, and the entanglement distillation of quantum states. Bell inequalities with pseudo Hermitian operators are also discussed.

We derive a generalized Bell inequality for N qubits which involves an arbitrary number of settings for each of the local measuring apparatuses. The inequality forms a necessary condition for the existence of a local realistic model for the values of a correlation function, given in a n-setting Bell experiment. We show that a local realistic model for the values of a correlation function, given in a two-setting Bell experiment, cannot construct a local realistic model for the values of a correlation function, given in an arbitrary number of n-setting Bell experiment, even though there exist two-setting models for the n measurement directions chosen in the given n-setting experiment. Therefore, the property of two-setting model is different from the property of n-setting model. We discuss classification of local realistic theories in further detail more than the result presented in (J. Phys. A: Math. Theor. 41:155308, 2008). The generalized Bell inequality covers the previous results correctly.

A class of the single-mode excited GHZ-type entangled coherent states (EGHZECSs) are presented. we exhibit the remarkable properties of the single-mode EGHZECSs, depended on the excitation photon number, such as entanglement and nonlocality via investigating their concurrence of entanglement and examining their violation of CHSH inequality. Finally, we propose how to generate the EGHZECSs by using cavity QED and quantum measurement and by using BBO crystal and single-photon detection technique, respectively.

In the present paper the integration region Λ with more than one hidden variable is attributed to a pair of particles in the Bell's thought experiment as the local causal events in their common lightcone. Moreover, the possibility of uncommon causal events influencing the spin measurement is not ignored. Then, with regard to the separability of the influence of the uncommon events from configuration of the setup, and by relying on local realism and coherency, each of the Bell's inequality versions is obtained by measuring spin in three and four different directions.

Despite claims that Bell's inequalities are based on the Einstein locality condition, or equivalent, all derivations make an identical mathematical assumption that local hidden-variable theories produce a set of positive-definite probabilities for detecting a particle with a given spin orientation. The standard argument is that because quantum mechanics assumes that particles are emitted in a superposition of states the theory cannot produce such a set of probabilites. We examine a paper by Eberhard, and several similar papers, which claim to show that a generalized Bell inequality, the CHSH inequality, can be derived solely on the basis of the locality condition, without recourse to hidden variables. We point out that these authors nonetheless assumes a set of positive-definite probabilities, which supports the claim that hidden variables or “locality” is not at issue here, positive-definite probabilities are. We demonstrate that quantum mechanics does predict a set of probabilities that violate the CHSH inequality; however these probabilities are not positive-definite. Nevertheless, they are physically meaningful in that they give the usual quantum-mechanical predictions in physical situations. We discuss in what sense our results are related to the Wigner distribution.

A typical sample of Bell's inequality is proved to require, besides the standard assumptions on realism and locality, the adoption of a metatheoretical classical principle for interpreting quantum laws. A new principle is proposed which is consistent with the operational philosophy of quantum physics; it is then shown that, whenever the latter principle is adopted in place of the former, realism (here intended in a purely semantical sense) and locality do not imply Bell's inequality in the form considered here, but a new inequality which is not violated in quantum physics. Thus an interpretation of quantum physics that is (semantically) realistic and local is suggested, which eliminates a number of seeming paradoxes.

In the last couple of years many important results have been derived showing that Bell's inequalities are nothing else but the indicator whether certain events and their probabilities can be represented or not within a Kolmogorovian probabilistic model. It has become evident that one can derive the Bell's inequalities without mentioning locality, causality hidden variable, etc. Many authors jumped to conclusion that the original content of the Bell's theorem had lost its meaning. I reconsider original problem posed by Bell and I show that the Bell's theorem is still valid.

Bell's inequalities are always derived assuming that local hidden-variable theories give a set of positive-definite probabilities for detecting a particle with a given spin orientation. The usual claim is that quantum mechanics, by its very nature, cannot produce a set of such probabilities. We show that this is not the case if one allows for generalized (nonpositive-definite) “master probability distributions.” The master distributions give the usual quantum mechanical violation of Bell's inequalities. Consequences for the interpretation of quantum mechanics are discussed.

Bell's inequality, as it is generally understood, is a relation involving average values over the readings of the apparatuses when dicotomic observables are measured, which can be derived on the basis of local hidden variables and can be violated by quantum mechanics. We derive a Bell type inequality, involving average values of physical quantities whose quantum mechanical operators need not be two-valued, under the hypothesis that composite systems are described by “proper” mixtures (i.e., statistical mixtures in the ordinary sense). This inequality is analyzed in detail for the case of aJ=M=0 state decaying into subsystems I, II with anyj and when average values of the quantity\((\overrightarrow J \cdot \hat a)_I \otimes (\overrightarrow J \cdot \hat b)_{II}\) are considered; it is then found that it cannot be violated ifj>1/2 by the quantum mechanical description of the composite system in terms of a “second-kind state.” A theorem is, however, established proving that in the general case suitable observables can be introduced for which a violation by quantum mechanics could be observed. This encourages the work in progress on more general situations and observables.

Bell's theorem applies only to a hybrid universe in which hidden variables determine only part of the outcomes of experiments. When applied to a fully causal hidden variable theory, in which detector settings as well as their interaction with particles during observation are determined by the variables, Bell's analysis must be modified. The result is that a fully causal hidden variable model can be produced for which a properly chosen spread of hidden variables gives precisely the same prediction as standard quantum theory.