The Bell Inequality, Inviolable by Data Used Consistently with Its Derivation, Is Satisfied by Quantum Correlations Whose Probabilities Satisfy the Wigner Inequality

It is not generally known that the inequality that Bell derived using three random variables must be identically satisfied by any three corresponding data sets of ±1’s that are writable on paper. This surprising fact is not immediately obvious from Bell’s inequality derivation based on causal random variables, but follows immediately if the same mathematical operations are applied to finite data sets. For laboratory data, the inequality is identically satisfied as a fact of pure algebra, and its satisfaction is independent of whether the processes generating the data are local, non-local, deterministic, random, or nonsensical. It follows that if predicted correlations violate the inequality, they represent no three cross-correlated data sets that can exist, or can be generated from valid probability models. Reported data that violate the inequality consist of probabilistically independent data-pairs and are thus inconsistent with inequality derivation. In the case of random variables as Bell assumed, the correlations in the inequality may be expressed in terms of the probabilities that give rise to them. A new inequality is then produced: The Wigner inequality, that must be satisfied by quantum mechanical probabilities in the case of Bell experiments. If that were not the case, predicted quantum probabilities and correlations would be inconsistent with basic algebra.

Extended Bell inequality and maximum violation

The original formula of Bell inequality（BI） in terms of two-spin singlet has to be modified for the entangled-state with parallel spin polarization. Based on classical statistics of the particle-number correlation, we prove in this paper an extended BI, which is valid for two-spin entangled states with both parallel and antiparallel polarizations. The BI and its violation can be formulated in a unified formalism based on the spin coherent-state quantum probability statistics with the statedensity operator, which is separated to the local and non-local parts. The local part gives rise to the BI, while the violation is a direct result of the non-local quantum interference between two components of entangled state. The Bell measuring outcome correlation denoted by PB is always less than or at most equal to one for the local realistic model（PB^lc≤ 1）regardless of the specific superposition coefficients of entangled state. Including the non-local quantum interference the maximum violation of BI is found as PB^max =2, which, however depends on state parameters and three measuring directions as well. Our result is suitable for entangled photon pairs.

The relation between Hardy's non-locality and violation of Bell inequality

We give an analytic quantitative relation between Hardy＇s non-locality and Bell operator. We find that Hardy＇s non-locality is a sufficient condition for the violation of Bell inequality, the upper bound of Hardy＇s non-locality allowed by information causality just corresponds to Tsirelson bound of Bell inequality and the upper bound of Hardy＇s non- locality allowed by the principle of no-signaling just corresponds to the algebraic maximum of Bell operator. Then we study the CabeUo＇s argument of Hardy＇s non-locality （a generalization of Hardy＇s argument） and find a similar relation between it and violation of Bell inequality. Finally, we give a simple derivation of the bound of Hardy＇s non-locality under the constraint of information causality with the aid of the above derived relation between Hardy＇s non-locality and Bell operator.

Conditional Independence Leads to Satisfaction of the Bell Inequality without Assuming Non-Locality or Non-Reality

The original Bell inequality was obtained in a statistical derivation assuming three mutually cross-correlated random variables (four in the later version). Given that observations destroy the particles, the physical realization of three variables from an experiment producing two particles per trial requires two separate trial runs. One assumed variable value (for particle 1) occurs at a fixed instrument setting in both trial runs while a second variable (for particle 2) occurs at alternative instrument settings in the two trial runs. Given that measurements on the two particles occurring in each trial are themselves correlated, measurements from independent realizations at mutually exclusive settings on particle 2 are conditionally independent, i.e., conditionally dependent on particle 1, through probability. This situation is realized from variables defined by Bell using entangled particle pairs. Two correlations have the form that Bell computed from entanglement, but a third correlation from conditionally independent measurements has a different form. When the correlations are computed using quantum probabilities, the Bell inequality is satisfied without recourse to assumptions of non-locality, or non-reality.

The Bell Inequality Is Satisfied by Quantum Correlations Computed Consistently with Quantum Non-Commutation

In constructing his theorem, Bell assumed that correlation functions among non-commuting variables are the same as those among commuting variables. However, in quantum mechanics, multiple data values exist simultaneously for commuting operations while for non-commuting operations data are conditional on prior outcomes, or may be predicted as alternative outcomes of the non-commuting operations. Given these qualitative differences, there is no reason why correlation functions among non-commuting variables should be the same as those among commuting variables, as assumed by Bell. When data for commuting and noncommuting operations are predicted from quantum mechanics, their correlations are different, and they now satisfy the Bell inequality.

Counterfactual Definiteness and Bell’s Inequality

Counterfactual definiteness must be used as at least one of the postulates or axioms that are necessary to derive Bell-type inequalities. It is considered by many to be a postulate that not only is commensurate with classical physics (as for example Einstein’s special relativity), but also separates and distinguishes classical physics from quantum mechanics. It is the purpose of this paper to show that Bell’s choice of mathematical functions and independent variables implicitly includes counterfactual definiteness. However, his particular choice of variables reduces the generality of his theory, as well as the physics of all Bell-type theories, so significantly that no meaningful comparison of these theories with actual Einstein-Podolsky-Rosen experiments can be made.

Strong violations of locality by testing Bell's inequality with improved entangled-photon systems

Bell’s theorem states that quantum mechanics cannot be accounted for by any local theory. One of the examples is the existence of quantum non-locality is essentially violated by the local Bell’s inequality. Therefore, the violation of Bell’s inequality(BI) has been regarded as one of the robust evidences of quantum mechanics. Until now, BI has been tested by many experiments, but the maximal violation(i.e., Cirel’son limit) has never been achieved. By improving the design of entangled sources and optimizing the measurement settings, in this work we report the stronger violations of the Clauser–Horne–Shimony–Holt(CHSH)-type Bell’s inequality. The biggest value of Bell’s function in our experiment reaches √to a significant one: S = 2.772 ± 0.063, approaching to the so-called Cirel’son limit in which the Bell function value is S = 22.Further improvement is possible by optimizing the entangled-photon sources.

Testing Bell inequality at experiments of high energy physics

Besides using the laser beam, it is very tempting to directly testify the Bell inequality at high energy experiments where the spin correlation is exactly what the original Bell inequality investigates. In this work, we follow the proposal raised in literature and use the successive decays J/ψ →γηc→ ∧∧ → pπ^- pπ^＋ to testify the Bell inequality. Our goal is twofold, namely, we first make a Monte-Carlo simulation of the processes based on the quantum field theory （QFT）. Since the underlying theory is QFT, it implies that we pre-admit the validity of quantum picture. Even though the QFT is true, we need to find how big the database should be, so that we can clearly show deviations of the correlation from the Bell inequality determined by the local hidden variable theory. There have been some critiques on the proposed method, so in the second part, we suggest some improvements which may help to remedy the ambiguities indicated by the critiques. It may be realized at an updated facility of high energy physics, such as BES III.

Quantum Violation of Bell’s Inequality: A Misunderstanding Based on a Mathematical Error of Neglect

This paper intends to show how the fabled violation of Bell’s inequality by the probabilistic specifications of quantum mechanics derives from a mathematical error, an error of neglect. I have no objection to the probabilities specified by quantum theory, nor to the inequality itself as characterized in the formulation of Clauser, Horne, Shimony, and Holt. Designed to assess consequences of Einstein’s principle of local realism, the inequality pertains to a linear combination of four polarization products on the same pair of photons arising in a gedankenexperiment. My assessment displays that in this context, the summands of the relevant CHSH quantity s(λ) inhere four symmetric functional relations which have long been neglected in analytic considerations. Its expectation E[s(λ)] is not the sum of four “marginal” expectations from a joint distribution, as quantum theory explicitly avoids such a specification. Rather, I show that E[s(λ)] has four distinct representations as the sum of three expectations of polarization products plus the expectation of a fourth which is restricted to equal a function value determined by the other three. Analysis using Bruno de Finetti’s fundamental theorem of prevision (FTP) yields only a bound for E(s) within (1.1213,2] , surely not at all as is commonly understood. I exhibit slices of the 4-dimensional polytope of joint P++ probabilities actually motivated by quantum theory at the four stipulated angle settings, as it passes through 3-dimensional space. Bell’s inequality is satisfied everywhere within the convex hull of extreme distributions cohering with quantum theoretic specifications, even while in keeping with local realism. Aspect’s proposed “estimation” of E(s) near to is based on polarization products from different photon pairs that do not have embedded within them the functional relations inhering in the relevant gedankenexperiment. When one actively embeds the restrictions into Aspect’s estimation procedure, it yields an estimate of 1.7667, although this is not and cannot be definitive. While my analysis supports the subjectivist construction of probability as clarifying issues relevant to the interpretation of quantum theory, the error resolved herein is purely mathematical. It pertains to the reconsideration of Bell violation irrespective of one’s attitude toward the meaning of probability.

Further Investigations of the Aspect/Bell Error: Maximum Entropy Assessment

This article identifies the maximum entropy distribution among those in the polytope of probability distributions cohering with quantum theoretic prescriptions pertinent to Bell’s inequality in the optical context. Perhaps surprisingly, the maxent distribution is not a uniform mixture of the extreme vertices of the convex hull of distributions agreeing with the theory. The expectation E(s) it supports equals 1.1296, within the permitted coherent interval of (1.1213,2]. The maxent mixture of the extreme agreeable vertices is compared herein with two other mixture distributions over the convex hull of those supported by quantum theory. One of these is a simple uniform mixture over the solution vectors to the appropriate linear programming problems that specify the polytope. The other is the mixture underlying simulated results of Aspect’s experiments that have been shown to estimate E(s) as 1.7678. Further computations provide examples of the types of claims that would be entailed in a unique distribution within the cohering convex hull such as maxent. These defy quantum theoretic adherence to the general uncertainty principle which proclaims an agnostic position with respect to imagined joint observation operators that do not commute. They also display questionable implications of the “many worlds” proposal which the author does not favour. The article raises questions that deserve to be discussed concerning the general proposal that the maximum entropy principle should be employed to make precise probabilistic assertions about equilibrium phenomena when specific physical theory prescribes only an interval.

Conceptual Problems in Bell’s Inequality and Quantum Entanglement

The description of the microscopic world in quantum mechanics is very different from that in classical physics, and there are some points of view that are contrary to intuition and logic. The first is the problem of reality;quantum mechanics believes the behavior of micro particles is random and jumping. The second is the loss of certainty;the conjugate physical variables of a system cannot be determined synchronously, they satisfy the Heisenberg uncertainty principle. The third is the non-local correlation. The measurement of one particle in the quantum entanglement pair will influence the state of the other entangled particle simultaneously. In this paper, some concepts related to quantum entanglement, such as EPR correlation, quantum entanglement correlation function, Bell’s inequality and so on, are analyzed in detail. Analysis shows that the mystery and confusion in quantum theory may be caused by the logical problems in its basic framework. Bell’s inequality is only a mathematical theorem, but its physical meaning is actually unclear. The Bell state of quantum entangled pair may not satisfy the dynamic equation of quantum theory, so it cannot describe the true state of microscopic particles. In this paper, the correct correlation functions of spin entanglement pair and photonic entanglement pair are strictly derived according to normal logic. Quantum theory is a more fundamental theory than classical mechanics, and they are not equal relation in logic. However, there are still some unreasonable contents in the framework of quantum theory, which need to be improved. In order to disclose the real relationship between quantum theory and classical mechanics, we propose some experiments which provide intuitionistic teaching materials for the new interpretation of quantum theory.

Einstein-Local Counter-Arguments and Counter-Examples to Bell-Type Proofs

J. S. Bell’s well-known proofs of inequalities (and related work) are shown to be invalidated by two counter-arguments (-examples) that are based on Einstein-local propositions: Bell-type inequalities of Einstein-Podolsky-Rosen experiments must include, as virtually all physical theories do, elements of physical reality and their mathematical representations that relate to continua as opposed to exclusively finite numbers. Furthermore, Bell-type inequalities must be valid for all possible experimental geometries that lead to the quantum result. Based on these propositions, violations of Bell-type inequalities are demonstrated without violating Einstein locality, without conspiracy type theories and even for the case that all known “loopholes” are closed.

Logical Conflict between Bell’s Locality and Probability Theory

Based on “locality” considerations, John Stuart Bell and his followers have derived inequalities and theorems that, when taken together with actual experiments that have been performed by Aspect and others, appear to contradict physical reality as defined by Einstein. However, their specifically applied concept of locality is in conflict with the Fundamental Model of probability theory and the set theoretic definition of conditional probabilities. Bell-type inequalities are, therefore, not adequate to decide ponderous questions regarding physical reality.

A Path Integral Generalization of Bell Local Hidden Variable Models for Unstable Particles

We discuss the problem of the generalization of Bell local hidden variable models for unstable particles as nucleons or decaying quantum bound states. We propose to extend the formalism of real deterministic hidden variables in the complex domain, in analogy with the quantum Gamow ket formalism, and we introduce a time dependent classical probability density distribution by which we implement hidden time dependence in the quantum expectation values. We suggest therefore a classical framework which may recover by asymptotic temporal limits the standard Bell stationary quantum statistical averages. Endly we discuss the possible relevance of our proposal for general non-isolated quantum systems in noninertial frames and the consequent dynamic effects of vacuum instabilities on E.P.R tests and Q.M. ensemble statistical averages.

Thermal entanglement and teleportation in a three-qubit Heisenberg XXZ model with Dzyaloshinski-Moriya anisotropic antisymmetric interaction

This paper investigates the thermal pairwise entanglement of a three-qubit Heisenberg XXZ chain in the presence of the Dzyaioshinski-Moriya （DM） anisotropic antisymmetric interaction and quantum teleportation when using the Heisenberg chain as a channel. The entanglement dependences on the DM interaction and temperature are given in detail. It obtains the relation between the concurrence and average fidelity, and shows that the same concurrence can lead different average fidelities. Moreover, it finds the thermally entangled states which do not violate the Bell inequalities, and can still be used for quantum teleportation.

Quantum entanglement and quantum nonlocality for N-photon entangled states

Quantum entanglement and quantum nonlocality of N-photon entangled states ｜ψNm） m Cm [cos γ｜N - m） 1 ｜m）2 ＋ e^iθm sinγ｜m）1｜N- m）2] and their superpositions are studied. We point out that the relative phase θm affects the quantum nonlocality but not the quantum entanglement for the state ｜ψNm）. We show that quantum nonlocality can be controlled and manipulated by adjusting the state parameters of ｜ψNm）, superposition coefficients, and the azimuthal angles of the Bell operator. We also show that the violation of the Bell inequality can reach its maximal value under certain conditions. It is found that quantum superpositions based on ｜ψNm） can increase the amount of entanglement, and give more ways to reach the maximal violation of the Bell inequality.

Quantum Mysteries for No One

I provide a critical reassessment of David Mermin’s influential and misleading parable, “Quantum Mysteries for Anyone”, identifying its errors and resolving them with a complete analysis of the quantum experiment it is meant to portray. Accessible to popular readership and requiring no knowledge of quantum physics at all, his exposition describes the curious behaviour of a machine that is designed to parody the empirical results of quantum experiments monitoring the spins of a pair of electrons under various conditions. The mysteries are said to unfold from contradictory results produced by a signal process that is proposed to explain them. I find that these results derive from a mathematical error of neglect, coupled with a confusion of two distinct types of experiments under consideration. One of these, a gedankenexperiment, provides the context in which the fabled defiance of Bell’s inequality is thought to emerge. The errors are corrected by the recognition of functional relations embedded within the experimental conditions that have been long unnoticed. A Monte Carlo simulation of results in accord with the actual abstemious claims of quantum theory supports probability values that Mermin decries as unwarranted. However, the distribution it suggests is not definitive, in accord with the expressed agnostic position of quantum theory regarding measurements that cannot be executed. Bounding quantum probabilities are computed for the results of the gedankenexperiment relevant to Bell’s inequality which inspired the parable. The problem is embedded in a 3 × 3 design of Stern-Gerlach magnet orientations at two observation stations. Computational resolution on the basis of Bruno de Finetti’s fundamental theorem of probability requires the evaluation of a battery of three paired linear programming problems. Though technicalities are ornate, the message is clear. There are no mysteries of quantum mechanics that derive from mistaken understandings of Bell’s inequality… for anyone.

Nonclassicality of a two-variable Hermite polynomial state

The nonclassicality of the two-variable Hermite polynomial state is investigated. It is found that the two-variable Hermite polynomial state can be considered as a two-mode photon subtracted squeezed vacuum state. A compact expression for the Wigner function is also derived analytically by using the Weyl-ordered operator invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of the Wigner function. Then violations of Bell＇s inequality for the two-variable Hermite polynomial state are studied.

The geometry of violation of Bell's inequality

The purpose of this paper is to deduce an analytical expression for the violation of Bell's inequality by quantum theory and plane trigonometry, and expound the violation and maximal violation of the first, second type Bell's inequality in detail. Further, we find out the sufficient conditions for the region in which Bell's inequalities are violated.

Quantum nonlocality evolution for two entangled mesoscopic fields under decoherence

Investigation of the nonlocality evolution of entangled mesoscopic fields under decoherence not only is important for understanding the quantum–classical transition,but also has relevance to quantum communication and quantum computation protocols based on continuous variables.According to previous formulations of Bell inequalities,the system loses nonlocal features far before the disappearance of entanglement.We here construct a new version of Bell signal based on rotated and displaced on–off correlations,with which the Bell inequality is violated as long as there remains entanglement and the field state components keep quasiorthogonal.Consequently,the nonlocal character revealed by our formulation decays much slower compared with those based on previous ones.More importantly,there exists a wide regime where the Bell inequality is restored with previous formulations but remains to be violated based on our correlation operators.