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.     

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.     

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.     

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 Four-Dimensional Measurement Scheme for Single Photon Time-Polarization

We propose an experimental scheme to realize the four-dimensional projective measurements tor a single photon. The photon polarization and time-energy provide the four-dimensional Hilbert space. Based on this scheme, we suggest an experiment to test the violation of Bell inequalities of four-dimensional systems. In addition, by virtue of a maximally entangled biphoton state, we also show that it is possible to construct a quantum key distribution channel that can provide two-bit key with one pair of entangled photons.     

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.     

Exploring inequality violations by classical hidden variables numerically

There are increasingly suggestions for computer simulations of quantum statistics which try to violate Bell type inequalities via classical, common cause correlations. The Clauser–Horne–Shimony–Holt (CHSH) inequality is very robust. However, we argue that with the Einstein–Podolsky–Rosen setup, the CHSH is inferior to the Bell inequality, although and because the latter must assume anti-correlation of entangled photon singlet states. We simulate how often quantum behavior violates both inequalities, depending on the number of photons. Violating Bell 99% of the time is argued to be an ideal benchmark. We present hidden variables that violate the Bell and CHSH inequalities with 50% probability, and ones which violate Bell 85% of the time when missing 13% anti-correlation. We discuss how to present the quantum correlations to a wide audience and conclude that, when defending against claims of hidden classicality, one should demand numerical simulations and insist on anti-correlation and the full amount of Bell violation.     

Generic Bell inequalities for multipartite arbitrary dimensional systems

We present generic Bell inequalities for multipartite arbitrary dimensional systems. The inequalities that any local realistic theory must obey are violated by quantum mechanics for even dimensional systems. A large set of variants are shown to naturally emerge from the generic Bell inequalities. We discuss particular variants of Bell inequalities that are violated for all the systems including odd dimensional systems.     

The Essence of More Nonlocality with Less Entanglement in Bell Tests

In this paper, we will explore the essence of the phenomenon that state with less entanglement may generate greater Bell violation in the two-qubit Bell tests with CH-type inequalities, i.e., more nonlocality with less entanglement. We will show that this interesting but counterintuitive phenomenon is caused by the rotational asymmetry of the nonmaximally entangled state in the measurement plane. This asymmetry allows the both-side detection probabilities and the one-side detection probabilities obtain their maximal values with nonmaximally entangled state. But the maximal Bell violation may not always happen on nonmaximally entangled state, because these probabilities will compete with each other, and the Bell violation behaves differently for various CH-type inequalities.     

A method for demonstrating violation of local realism with a two-photon downconverter without use of Bell inequalities

We show that the recent proposal by Hardy and Jordan for a test of local realism without the use of Bell inequalities can be implemented in two-photon coincidence measurements with linear polarizers, when the photon pairs are produced by parametric downconversion. If the probabilities measured with real detectors are proportional to the corresponding probabilities determined with ideal detectors, this method does not depend on the use of detectors with high or even known quantum efficiencies.Dedicated to H. Walther on the occasion of his 60th birthday     

On the implication of Bell’s probability distribution and proposed experiments of quantum measurement

In derivating of Bell’s inequalities, the probability distribution is supposed to be a function only of a hidden variable. We point out that the true implication of the probability distribution of Bell’s correlation function is the distribution of joint measurement outcomes on the two sides. It is therefore a function of both the hidden variable and the settings. In this case, Bell’s inequalities fail. Our further analysis shows that Bell’s locality holds neither for dependent events nor for independent events. We think that the measurements of EPR pairs are dependent events, and hence violation of Bell’s inequalities cannot rule out the existence of local hidden variable. To explain the results of EPR-type experiments, we suppose that a polarization-entangled photon pair can be composed of two circularly or linearly polarized photons with correlated hidden variables, and a couple of experiments of quantum measurement are proposed. The first uses delayed measurement on one photon of the EPR pair to demonstrate directly whether measurement on the other could have any nonlocal influence on it. Then several experiments are suggested to reveal the components of the polarization-entangled photon pair. The last one uses successive polarization measurements on a pair of EPR photons to show that two photons with the same quantum state behave the same under the same measuring conditions.     

三光子GHZ态中不同Bell型不等式的实验研究

实验研究了Bell不等式、Mermin不等式和Svetlichny不等式在GHZ态中对定域实在论结果的破坏. 采用目标态密度矩阵的方法给出了实验制备的GHZ态的保真度, 得到了三个不等式在GHZ态中的测量值. 实验结果表明, 三个Bell型不等式在同一个状态下对定域实在论结果破坏程度不同, 在量子密钥分配中可以选择合适的非定域性描述方式考察所用信道的安全性.     

Bell inequalities with no quantum violation and unextendable product bases

The strength of classical correlations is subject to certain constraints, commonly known as Bell inequalities. Violation of these inequalities is the manifestation of nonlocality-displayed, in particular, by quantum mechanics, meaning that quantum mechanics can outperform classical physics at tasks associated with such Bell inequalities. Interestingly, however, there exist situations in which this is not the case. We associate an intriguing class of bound entangled states, constructed from unextendable product bases with a wide family of tasks, for which (i) quantum correlations do not outperform the classical ones but (ii) there exist supraquantum nonsignaling correlations that do provide an advantage.     

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.     

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.     

Is Nonlocality Responsible for the Violation of Bell’s Inequalities?

Bell’s theorem has been widely argued to show that some of the predictions of quantum mechanics which are obtained by applying the Born’s rule to a class of entangled states, are not compatible with any local-causal statistical model, via the violation of Bell’s inequalities. On the other hand, in the previous works, we have shown that quantum dynamics and kinematics are emergent from a statistical model that is singled out uniquely by the principle of Locality. Here we shall show that the local-causal model supports entangled states and give the statistical origin of their generation. We then study the Stern-Gerlach experiment to show that the Born’s rule can also be derived as a mathematical theorem in the local-causal model. These results lead us to argue that nonlocality is not responsible for the quantum mechanical and most importantly experimental violation of Bell’s inequalities. The source(s) of violation has to be sought somewhere else.     

Nonlocality for graph states

The possibility of preparing two-photon entangled states encoding three or more qubits in each photon leads to the following problem: If n quabits were distributed between two parties, which quantum pure states and qubit distributions would allow all-versus-nothing (or Greenberger-Horne-Zeilinger-like) proofs of Bell’s theorem using only single-qubit measurements? We show a necessary and sufficient condition for the existence of these proofs and provide all existing proofs up to n = 7 qubits. On the other hand, the possibility of preparing n-photon n-qubit graph states leads to the following problem: If n qubits were distributed between n parties, which would be the optimal Bell inequalities? We show all optimal n-party Bell inequalities for the perfect correlations of any graph state of n < 6 qubits. Optimal means that the ratio between the quantum violation and the bound for local hidden-variable theories is the maximum over all possible combinations of perfect correlations. This implies that the required detection efficiencies for loophole-free Bell tests are minimal.     

Bipartite bell inequalities for hyperentangled states

We show that bipartite Bell inequalities based on the Einstein-Podolsky-Rosen criterion for elements of reality and derived from the properties of some hyperentangled states allow feasible experimental verification of the fact that quantum nonlocality grows exponentially with the size of the subsystems, and Bell loophole-free tests with currently available photodetection efficiencies.