Bell Inequalities and Group Symmetry

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 G?ney, 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.     

高能物理中的Bell不等式

评述了在高能物理中检验量子力学的完备性, 特别是检验Bell不等式的研究工作进展情况, 简略地介绍了Einstain, Podolsky和Rosen (EPR)佯谬及其相应的光子实验得到的结果. 概括地阐述了在高能物理中早期利用粒子衰变中的自旋关联以及后来用中性介子的混合形成的准自旋纠缠态检验Bell不等式的各种尝试, 给出了K⁰和B⁰系统的相关实验结果. 介绍了一种可在φ工厂中实施实验检验的, 基于非最大纠缠态的新的方案. 最后还讨论了把这一实验方案推广到τ-粲工厂的可能性.     

The Violation of Bell Inequalities in the Macroworld

We show that Bell inequalities can be violated in the macroscopic world. The macroworld violation is illustrated using an example involving connected vessels of water. We show that whether the violation of inequalities occurs in the microworld or the macroworld, it is the identification of nonidentical events that plays a crucial role. Specifically, we prove that if nonidentical events are consistently differentiated, Bell-type Pitowsky inequalities are no longer violated, even for Bohm's example of two entangled spin 1/2 quantum particles. We show how Bell inequalities can be violated in cognition, specifically in the relationship between abstract concepts and specific instances of these concepts. This supports the hypothesis that genuine quantum structure exists in the mind. We introduce a model where the amount of nonlocality and the degree of quantum uncertainty are parameterized, and demonstrate that increasing nonlocality increases the degree of violation, while increasing quantum uncertainty decreases the degree of violation.     

Separate Common Causal Explanation and the Bell Inequalities

In the paper we ask how the following two facts are related: (i) a set of correlations has a local, non-conspiratorial separate common causal explanation; (ii) the set satisfies the Bell inequalities. Our answer will be partial: we show that no set of correlations violating the Clauser–Horne inequalities can be given a local, non-conspiratorial separate common causal model if the model is deterministic.     

All Inequalities for the Relative Entropy

The relative entropy of two n-party quantum states is an important quantity exhibiting, for example, the extent to which the two states are different. The relative entropy of the states formed by reducing two n-party states to a smaller number m of parties is always less than or equal to the relative entropy of the two original n-party states. This is the monotonicity of relative entropy.Using techniques from convex geometry, we prove that monotonicity under restrictions is the only general inequality satisfied by quantum relative entropies. In doing so we make a connection to secret sharing schemes with general access structures: indeed, it turns out that the extremal rays of the cone defined by monotonicity are populated by classical secret sharing schemes.A surprising outcome is that the structure of allowed relative entropy values of subsets of multiparty states is much simpler than the structure of allowed entropy values. And the structure of allowed relative entropy values (unlike that of entropies) is the same for classical probability distributions and quantum states.     

Bell Inequalities as Constraints on Unmeasurable Correlations

The interpretation of the violation of Bell-Clauser-Horne inequalities is revisited, in relation with the notion of extension of QM predictions to unmeasurable correlations. Such extensions are compatible with QM predictions in many cases, in particular for observables with compatibility relations described by tree graphs. This implies classical representability of any set of correlations 〈A _i 〉, 〈B〉, 〈A _i B〉, and the equivalence of the Bell-Clauser-Horne inequalities to a non void intersection between the ranges of values for the unmeasurable correlation 〈A ₁ A ₂〉 associated to different choices for B. The same analysis applies to the Hardy model and to the “perfect correlations” discussed by Greenberger, Horne, Shimony and Zeilinger. In all the cases, the dependence of an unmeasurable correlation on a set of variables allowing for a classical representation is the only basis for arguments about violations of locality and causality.     

From Quantum State Targeting to Bell Inequalities

Quantum state targeting is a quantum game which results from combining traditional quantum state estimation with additional classical information. We consider a particular version of the game and show how it can be played with maximally entangled states. The optimal solution of the game is used to derive a Bell inequality for two entangled qutrits. We argue that the nice properties of the inequality are direct consequences of the method of construction.     

Hidden Variables and Bell Inequalities on Quantum Logics

In the quantum logic approach, Bell inequalities in the sense of Pitowski are related with quasi hidden variables in the sense of Deliyannis. Some properties of hidden variables on effect algebras are discussed.     

Random Constructions in Bell Inequalities: A Survey

Initially motivated by their relevance in foundations of quantum mechanics and more recently by their applications in different contexts of quantum information science, violations of Bell inequalities have been extensively studied during the last years. In particular, an important effort has been made in order to quantify such Bell violations. Probabilistic techniques have been heavily used in this context with two different purposes. First, to quantify how common the phenomenon of Bell violations is; and second, to find large Bell violations in order to better understand the possibilities and limitations of this phenomenon. However, the strong mathematical content of these results has discouraged some of the potentially interested readers. The aim of the present work is to review some of the recent results in this direction by focusing on the main ideas and removing most of the technical details, to make the previous study more accessible to a wide audience.     

Large Violation of Bell Inequalities with Low Entanglement

In this paper we obtain violations of general bipartite Bell inequalities of order \({\frac{\sqrt{n}}{\log n}}\) with n inputs, n outputs and n-dimensional Hilbert spaces. Moreover, we construct explicitly, up to a random choice of signs, all the elements involved in such violations: the coefficients of the Bell inequalities, POVMs measurements and quantum states. Analyzing this construction we find that, even though entanglement is necessary to obtain violation of Bell inequalities, the entropy of entanglement of the underlying state is essentially irrelevant in obtaining large violation. We also indicate why the maximally entangled state is a rather poor candidate in producing large violations with arbitrary coefficients. However, we also show that for Bell inequalities with positive coefficients (in particular, games) the maximally entangled state achieves the largest violation up to a logarithmic factor.     

A Note of Bell Inequalities for Graph States

International Journal of Theoretical Physics - In this paper, we construct a general Bell inequality for the graph state. Firstly, we show that the Bell inequality is maximally violated by graph...     

Bell Inequalities and Separability of Uncommon Causes from the Setup Configuration

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. PACS numbers: 03.65.Ud 03.65.Ta.     

Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory

In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order ${{\rm \Omega} \left(\frac{\sqrt{n}}{\log^2n} \right)}In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order W (\frac?nlog²n ){{\rm \Omega} \left(\frac{\sqrt{n}}{\log^2n} \right)} when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and operator space theory and, in particular, the very recent noncommutative L _p embedding theory.     

Classification of Separable States for Tripartite Qudits via Bell Inequalities

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.     

Bell’s Theorem Without Inequalities and Only Two Distant Observers

A proof of Bell's theorem without inequalities and involving only two observers is given by suitably extending a proof of the Bell-Kochen-Specker theorem due to Mermin. This proof is generalized to obtain an inequality-free proof of Bell's theorem for a set of n Bell states (with n odd) shared between two distant observers. A generalized CHSH inequality is formulated for n Bell states shared symmetrically between two observers and it is shown that quantum mechanics violates this inequality by an amount that grows exponentially with increasing n.     

Quantum Logic, Probability, and Information: The Relation with the Bell Inequalities

We study the relation between the Bell inequalities—characteristic of noncontextual hidden variables theories of quantum mechanics—with quantum logic, quantum probability, and quantum information. The emphasis is on clarity and simplicity, although sometimes this implies a lack of mathematical rigor which, I hope, could be resolved without difficulty by the reader.