The Bell Theorem as a Special Case of a Theorem of Bass

The theorem of Bell states that certain results of quantum mechanics violate inequalities that are valid for objective local random variables. We show that the inequalities of Bell are special cases of theorems found 10 years earlier by Bass and stated in full generality by Vorob’ev. This fact implies precise necessary and sufficient mathematical conditions for the validity of the Bell inequalities. We show that these precise conditions differ significantly from the definition of objective local variable spaces and as an application that the Bell inequalities may be violated even for objective local random variables.     

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.     

基于量子相干性的四体贝尔不等式构建

贝尔不等式在定域性和实在性的双重假设下,对于被分隔的粒子同时被测量时其结果的可能关联程度建立了一个严格的限制,违反贝尔不等式确保量子态存在纠缠.本文利用量子相干性的l1和相对熵测度构建了四体量子贝尔不等式,发现一般实系数Greenberger-Horne-Zeilinger纯态和簇纯态总是违反四体相对熵相干性测度贝尔不等式,因此违反四体相对熵相干性测度贝尔不等式的这些态是纠缠态.     

Locality is not violated for the singlet spin state

The use of Bell inequality for the singlet spin state is analyzed. Due to superselection rules the singlet spin state is entangled with spatial spherical wave functions. Bell inequalities are violated only for subensembles which are not pure states. Locality is not violated.     

Bell inequalities for graph states

We investigate the nonlocal properties of graph states. To this aim, we derive a family of Bell inequalities which require three measurement settings for each party and are maximally violated by graph states. In turn, for each graph state there is an inequality maximally violated only by that state. We show that for certain types of graph states the violation of these inequalities increases exponentially with the number of qubits. We also discuss connections to other entanglement properties such as the positivity of the partial transpose or the geometric measure of entanglement.     

Testing the structure of multipartite entanglement with Bell inequalities

We show that the rich structure of multipartite entanglement can be tested following a device-independent approach. Specifically we present Bell inequalities for distinguishing between different types of multipartite entanglement, without placing any assumptions on the measurement devices used in the protocol, in contrast with usual entanglement witnesses. We first address the case of three qubits and present Bell inequalities that can be violated by W states but not by Greenberger-Horne-Zeilinger states, and vice versa. Next, we devise 'subcorrelation Bell inequalities' for any number of parties, which can provably not be violated by a broad class of multipartite entangled states (generalizations of Greenberger-Horne-Zeilinger states), but for which violations can be obtained for W states. Our results give insight into the nonlocality of W states. The simplicity and robustness of our tests make them appealing for experiments.     

Multipartite entanglement inequalities via spin vector geometry

We introduce inequalities for multipartite entanglement, derived from the geometry of spin vectors. The criteria are constructed iteratively from cross and dot products between the spins of individual subsystems, each of which may have arbitrary dimension. For qubit ensembles the maximum violation for our inequalities is larger than that for the Mermin-Klyshko Bell inequalities, and the maximally violating states are different from Greenberger-Horne-Zeilinger states. Our inequalities are violated by certain bound entangled states for which no Bell-type violation has yet been found.     

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-type inequalities for cold heteronuclear molecules

We introduce Bell-type inequalities allowing for nonlocality and entanglement tests with two cold heteronuclear molecules. The proposed inequalities are based on correlations between each molecule spatial orientation, an observable which can be experimentally measured with present day technology. Orientation measurements are performed on each subsystem at different times. These times play the role of the polarizer angles in Bell tests realized with photons. We discuss the experimental implementations of the proposed tests, which could also be adapted to other high dimensional quantum angular momenta systems.     

Loophole-free bell test for continuous variables via wave and particle correlations

We derive two classes of multimode Bell inequalities under local realistic assumptions, which are violated only by the entangled states negative under partial transposition in accordance with the Peres conjecture. Remarkably, the failure of local realism can be manifested by exploiting wave and particle correlations of readily accessible continuous-variable states, with very large violation of inequalities insensitive to detector efficiency, which makes a strong case for a loophole-free test.     

Unsharp spin observables, non-locality and Fry, Walther and Li experiment

Recently it has been demonstrated that Bell inequalities for spin 1/2 particles must be modified if unsharp spin observables are considered, and furthermore, the modified Bell inequalities may not be violated by quantum mechanics if the observables are sufficiently unsharp. In case of massive particles there may be more imperfection than seems to appear in the photon EPR experiments. So the experiment proposed by Fry, Walther and Li can place experimental limits on the unsharpness of spin variables. It sheds new light on the much debated issues like non-local correlations in quantum mechanics.     

Quantum mechanics in an entirely deterministic universe

It is argued that, according to the suggested interpretation of quantum mechanical probabilities, (1) the Bell inequalities are not equivalent with those inequalities derived by Pitowsky and others that indicate the Kolmogorovity of a probability model, (2) the original Bell inequalities are irrelevant to both the question of whether or not quantum mechanics is a Kolmogorovian theory as well as the problem of determinism, whereas (3) the Pitowsky-type inequalities are not violated by quantum mechanics, hence (4) quantum mechanics is a Kolmogorovian probability theory, therefore, (5) it is compatible with an entirely deterministic universe.On leave from the Institute for Theoretical Physics, Eötvös University, Budapest, Hungary.     

Is the Contextuality Loophole Fatal for the Derivation of?Bell Inequalities?

It is explained on a physical basis how absence of contextuality allows Bell inequalities to be violated, without bringing an implication on locality or realism. Hereto we connect first to the local realistic theory Stochastic Electrodynamics, and then put the argument more broadly. Thus even if Bell Inequality Violation is demonstrated beyond reasonable doubt, it will have no say on local realism, because absence of contextuality prevents the Bell inequalities to be derived from local realistic models.     

Relevant multi-setting tight Bell inequalities for qubits and qutrits

In the celebrated paper [D. Collins, N. Gisin, J. Phys. A Math. Gen. 37 (2004) 1775], Collins and Gisin presented for the first time a three-setting Bell inequality (here we call it CG inequality for simplicity) which is relevant to the Clauser–Horne–Shimony–Holt (CHSH) inequality. Inspired by their brilliant ideas, we obtained some multi-setting tight Bell inequalities, which are relevant to the CHSH inequality and the CG inequality. Moreover, we generalized the method in the paper [J.L. Chen, D.L. Deng, Phys. Rev. A 79 (2009) 012115] to construct Bell inequality for qubits to higher dimensional system. Based on the generalized method, we present, for the first time, a three-setting tight Bell inequality for two qutrits, which is maximally violated by nonmaximally entangled states and relevant to the Collins–Gisin–Linden–Massar–Popescu inequality.     

Experimental verification of a new Bell-type inequality

Arpan Das et al. proposed a set of new Bell inequalities (Das et al., 2017 [16]) for a three-qubit system and claimed that each inequality within this set is violated by all generalized Greenberger–Horne–Zeilinger (GGHZ) states. We investigate experimentally the new inequalities in the three-photon GGHZ class states. Since the inequalities are symmetric under the identical particles system, we chose one Bell-type inequality from the set arbitrarily. The experimental data well verified the theoretical prediction. Moreover, the experimental results show that the amount of violation of the new Bell inequality against locality realism increases monotonically following the increase of the tangle of the GGHZ state. The most profound physical essence revealed by the results is that the nonlocality of GGHZ state correlate with three tangles directly.     

The bell inequalities

It is shown that the Bell inequalities are conditions that must be satisfied by the probability functions of certain three-variable systems if they are to be expressible in terms of a single, nonnegative function. The generalized Bell inequalities, or CHSH inequalities, play a similar role for four-variable systems. The physical significance of the results is discussed.     

Multiparticle interference, Greenberger-Horne-Zeilinger entanglement, and full counting statistics

We investigate the quantum transport in a generalized N-particle Hanbury Brown-Twiss setup enclosing magnetic flux, and demonstrate that the Nth-order cumulant of current cross correlations exhibits Aharonov-Bohm oscillations, while there is no such oscillation in all the lower-order cumulants. The multiparticle interference results from the orbital Greenberger-Horne-Zeilinger entanglement of N indistinguishable particles. For sufficiently strong Aharonov-Bohm oscillations the generalized Bell inequalities may be violated, proving the N-particle quantum nonlocality.     

Nonlocality in many-body quantum systems detected with two-body correlators

Contemporary understanding of correlations in quantum many-body systems and in quantum phase transitions is based to a large extent on the recent intensive studies of entanglement in many-body systems. In contrast, much less is known about the role of quantum nonlocality in these systems, mostly because the available multipartite Bell inequalities involve high-order correlations among many particles, which are hard to access theoretically, and even harder experimentally. Standard, “theorist- and experimentalist-friendly” many-body observables involve correlations among only few (one, two, rarely three...) particles. Typically, there is no multipartite Bell inequality for this scenario based on such low-order correlations. Recently, however, we have succeeded in constructing multipartite Bell inequalities that involve two- and one-body correlations only, and showed how they revealed the nonlocality in many-body systems relevant for nuclear and atomic physics [Tura et al., Science 344 (2014) 1256]. With the present contribution we continue our work on this problem. On the one hand, we present a detailed derivation of the above Bell inequalities, pertaining to permutation symmetry among the involved parties. On the other hand, we present a couple of new results concerning such Bell inequalities. First, we characterize their tightness. We then discuss maximal quantum violations of these inequalities in the general case, and their scaling with the number of parties. Moreover, we provide new classes of two-body Bell inequalities which reveal nonlocality of the Dicke states—ground states of physically relevant and experimentally realizable Hamiltonians. Finally, we shortly discuss various scenarios for nonlocality detection in mesoscopic systems of trapped ions or atoms, and by atoms trapped in the vicinity of designed nanostructures.     

Effect of non-Markovian dynamics on the violation of tripartite Bell inequalities

The effect of the non-Markovian dynamics on the violation of Bell inequalities for a generic tripartite entangled state has been investigated. Our results imply that the dynamics of Bell violation depends not only on the coupling strengths between the system and the environment but also on the rotation angles. Under both the strong and weak couplings, the Bell nonlocality can be eliminated by the decoherence induced by the environment in finite time when the rotation angles take certain values.     

Bell inequalities for continuous-variable correlations

We derive a new class of correlation Bell-type inequalities. The inequalities are valid for any number of outcomes of two observables per each of n parties, including continuous and unbounded observables. We show that there are no first-moment correlation Bell inequalities for that scenario, but such inequalities can be found if one considers at least second moments. The derivation stems from a simple variance inequality by setting local commutators to zero. We show that above a constant detector efficiency threshold, the continuous-variable Bell violation can survive even in the macroscopic limit of large n. This method can be used to derive other well-known Bell inequalities, shedding new light on the importance of non-commutativity for violations of local realism.