Violation of Bell's Inequality with Continuous Variables and Fractional Fourier Transforms

Bell's inequality for Macroscopic Squeezed Vacuum

A non-dichotomic Bell-type inequality for macroscopic photon numbers in two spatially separated beams is derived from local realistic assumptions. This inequality is shown to be possible to violate with the macroscopic Bell states recently obtained in experiment.

Is Bell inequality violated once measurement independence is relaxed?

It has been believed that statistical inequality such as Bell inequality should be modified once measurement independence (MI), the assumption that observers can freely choose measurement settings without changing the probability distribution of hidden variables, is relaxed. However, we show that there exists the possibility that Bell inequality is still valid even if MI is relaxed. MI is only a sufficient condition to derive Bell inequality when both determinism and setting independence, usually called as local realism, are satisfied. We thus propose a new condition necessary and sufficient f…

Multipartite bound entangled states that violate Bell's inequality

We study the relation between distillability of multipartite states and violation of Bell's inequality. We prove that there exist multipartite bound entangled states (i.e. non-separable, non-distillable states) that violate a multipartite Bell inequality. This implies that (i) violation of Bell's inequality is not a sufficient condition for distillability and (ii) some bound entangled states cannot be described by a local hidden variable model.

Explanation of the violation of Bell's inequality by maintaining Einstein separability

A new interpretation offers a consistent conceptual basis for nonrelativistic quantum mechanics. The violation of Bell's inequality is explained by maintaining realism, inductive inference and Einstein separability.

Bell's inequality violation due to misidentification of spatially non stationary random processes

Correlations for the Bell gedankenexperiment are constructed using probabilities given by quantum mechanics, and nonlocal information. They satisfy Bell's inequality and exhibit spatial non stationarity in angle. Correlations for three successive local spin measurements on one particle are computed as well. These correlations also exhibit non stationarity, and satisfy the Bell inequality. In both cases, the mistaken assumption that the underlying process is wide-sense-stationary in angle results in violation of Bell's inequality. These results directly challenge the wide-spread belief that vio…

What exactly is proved by the violation of Bell's inequality?

Bell's inequality has been derived several times from quite different basic assumptions, which imply different conclusions. This resulted into widespread confusion regarding the exact implications of the experimental violations of the inequality. In this article, the structures of Bell's and of Peres' derivations are analyzed, and the title question is explicitly answered.

Bell Inequality Based on Peres-Horodecki Criterion

We established a physically utilizable Bell inequality based on the Peres-Horodecki criterion. The new quadratic probabilistic Bell inequality naturally provides us a necessary and sufficient way to test all entangled two-qubit or qubit-qutrit states including the Werner states and the maximally entangled mixed states.

Bell inequality, Bell states and maximally entangled states for n qubits

First, we present a Bell type inequality for n qubits, assuming that m out of the n qubits are independent. Quantum mechanics violates this inequality by a ratio that increases exponentially with m. Hence an experiment on n qubits violating of this inequality sets a lower bound on the number m of entangled qubits. Next, we propose a definition of maximally entangled states of n qubits. For this purpose we study 5 different criteria. Four of these criteria are found compatible. For any number n of qubits, they determine an orthogonal basis consisting of maximally entangled states generalizing t…

Remarks on the violation of Bell's inequality in Josephson phase qubits

The consistency test and error estimation for the data concerning recently reported violation of Bell inequality for Josephson phase qubits are presented in details. It is pointed out that the deviation of the Bell signal from the classical limit might still be challenged.

Bipartite Bell Inequality and Maximal Violation

We present new bell inequalities for arbitrary dimensional bipartite quantum systems. The maximal violation of the inequalities is computed. The Bell inequality is capable of detecting quantum entanglement of both pure and mixed quantum states more effectively.

Bell's inequality with Dirac particles

We study Bell's inequality using the Bell states constructed from four component Dirac spinors. Spin operator is related to the Pauli-Lubanski pseudo vector which is relativistic invariant operator. By using Lorentz transformation, in both Bell states and spin operator, we obtain an observer independent Bell's inequality, so that it is maximally violated as long as it is violated maximally in the rest frame.

Exclusivity principle and the quantum bound of the Bell inequality

We show that, for general probabilistic theories admitting sharp measurements, the exclusivity principle together with two assumptions exactly singles out the Tsirelson bound of the Clauser-Horne-Shimony-Holt Bell inequality.

Bell-type inequality in quantum coherence theory as an entanglement witness

Bell inequality is a mathematical inequality derived using the assumptions of locality and realism. Its violation guarantees the existence of quantum correlations in a quantum state. Bell inequality acts as an entanglement witness in the sense that a pure bipartite quantum state, having nonvanishing entanglement, always violates a Bell inequality. We construct Bell-type inequalities for product states in quantum coherence theory for different measures of coherence, and find that the maximally entangled states violate these inequalities. We further show that Bell-type inequalities for relative …

The Bell Inequality and Entanglement

Entanglement is a critical resource used in many current quantum information schemes. As such entanglement has been extensively studied in two qubit systems and its entanglement nature has been exhibited by violations of the Bell inequality. Can the amount of violation of the Bell inequality be used to quantify the degree of entanglement. What do Bell inequalities indicate about the nature of entanglement?

Violation of Bell's inequality: criterion for quantum communication complexity advantage

We prove that for every Bell's inequality and for a broad class of protocols, there always exists a multi-party communication complexity problem, for which the protocol assisted by states which violate the inequality is more efficient than any classical protocol. Moreover, for that advantage Bell's inequality violation is a necessary and sufficient criterion. Thus, violation of Bell's inequalities has a significance beyond that of a non-optimal-witness of non-separability.

Multipartite omnidirectional generalized Bell inequality

We derive a multipartite generalized Bell inequality which involves the entire range of settings for each of the local observers. Especially, it is applied to show non-local behavior of a six-qubit mixture of Greenberger-Horne-Zeilinger correlations stronger than previous Bell inequalities. For certain noise admixture to the correlations an explicit local realistic model exists in the case of a standard Bell experiment. Bell experiments with many local settings reveal the non-locality of the state. It turns out that the new inequality is more stringent than many other Bell inequalities in the …

Bell inequality tests using asymmetric entangled coherent states in asymmetric lossy environments

We study an asymmetric form of two-mode entangled coherent state (ECS), where the two local amplitudes have different values, for testing the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. We find that the asymmetric ECSs have obvious advantages over the symmetric form of ECSs in testing the Bell-CHSH inequality. We further study an asymmetric strategy in distributing an ECS over a lossy environment and find that such a scheme can significantly increase violation of the inequality.

On the solution of the EPR paradox and the explanation of the violation of Bell's inequality

A new interpretation offers a consistent conceptual basis for nonrelativistic quantum mechanics. The Einstein-Podolsky-Rosen (EPR) paradox is solved and the violation of Bell's inequality is explained by maintaining realism, inductive inference and Einstein separability.

Experimental Bell inequality violation without the postselection loophole

We report on an experimental violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality using energy-time entangled photons. The experiment is not free of the locality and detection loopholes, but is the first violation of the Bell-CHSH inequality using energy-time entangled photons which is free of the postselection loophole described by Aerts et al. [Phys. Rev. Lett. 83, 2872 (1999)].

Correlations for a new Bell's inequality experiment

Proofs of Bell's theorem and the data analysis used to show its violation have commonly assumed a spatially stationary underlying process. However, it has been shown recently that the appropriate Bell's inequality holds identically for cross correlations of three or four lists of + or - 1's, independently of statistical assumptions. When data consistent with its derivation are analyzed without imposition of the stationarity assumption, the resulting correlations satisfy the Bell inequality.

Bell inequality for arbitrary many settings of the analyzers

A generalization of the CHSH-Bell inequality to arbitrary many settings is presented. The singlet state of two spin $\half$ violates this inequality for all numbers of setting. In the limit of arbitrarily large number of settings, the violation tends to the finite ratio $4/π\approx 1.27$.

Mermin's n-particle Bell inequality and operators' noncommutativity

The relationship between the noncommutativity of operators and the violation of the Bell inequality is exhibited in the light of the n-particle Bell-type inequality discovered by Mermin [PRL 65, 1838 (1990)]. It is shown, in particular, that the maximal amount of violation of Mermin's inequality predicted by quantum mechanics decreases exponentially by a factor of 2^{-m/2} whenever any m among the n single-particle commutators happen to vanish.

Numerical simulation of Bell inequality's violation using optical transverse modes in multimode waveguides

We numerically demonstrate that "mode-entangled states" based on the transverse modes of classical optical fields in multimode waveguides violate Bell's inequality. Numerically simulating the correlation measurement scheme of Bell's inequality, we obtain the normalized correlation functions of the intensity fluctuations for the two entangled classical fields. By using the correlation functions, the maximum violations of Bell's inequality are obtained. This implies that the two classical fields in the mode-entangled states, although spatially separated, present a nonlocal correlation.

Violation of Mermin's version of a Bell inequality in a classical statistical model

We investigate a classical statistical model and show that Mermin's version of a Bell inequality is violated. We get this violation, if the measurement modifies the ensemble, a feature, which is also characteristic for measurement processes for quantum systems.

Optimal States for Bell inequality Violations using Quadrature Phase Homodyne Measurements

We identify what ideal correlated photon number states are to required to maximize the discrepancy between local realism and quantum mechanics when a quadrature homodyne phase measurement is used. Various Bell inequality tests are considered.

Bell inequality for pairs of particle-number-superselection-rule restricted states

Proposals for Bell inequality tests on systems restricted by superselection rules often require operations that are difficult to implement in practice. In this paper, we derive a new Bell inequality, where pairs of states are used to by-pass the superselection rule. In particular, we focus on mode entanglement of an arbitrary number of massive particles and show that our Bell inequality detects the entanglement in the pair when other inequalities fail. However, as the number of particles in the system increases, the violation of our Bell inequality decreases due to the restriction in the measu…

Locality and Bell's inequality

We prove that the locality condition is irrelevant to Bell in equality. We check that the real origin of the Bell's inequality is the assumption of applicability of classical (Kolmogorovian) probability theory to quantum mechanics. We describe the chameleon effect which allows to construct an experiment realizing a local, realistic, classical, deterministic and macroscopic violation of the Bell inequalities.

On "Bell's inequality without alternative settings"

Cabello has recently (in quant-ph/0210081) observed that ``...an EPR-experiment with a fixed POVM on each particle provides a violation of Bell's inequality without requiring local observers to choose between the alternatives.'' In this note I discuss the implications of this observation for tests of locality.

Bell's Inequality: More Tests Are Needed

According to recent reports, the last loopholes in testing Bell's inequality are closed. It is argued that the really important task in this field has not been tackled yet and that the leading experiments claiming to close locality and detection efficiency loopholes, although making a very significant progress, have conceptual drawbacks.

Information-theoretic temporal Bell inequality and quantum computation

An information-theoretic temporal Bell inequality is formulated to contrast classical and quantum computations. Any classical algorithm satisfies the inequality, while quantum ones can violate it. Therefore, the violation of the inequality is an immediate consequence of the quantumness in the computation. Furthermore, this approach suggests a notion of temporal nonlocality in quantum computation.

Characterization of maximally entangled two-qubit states via the Bell-Clauser-Horne-Shimony-Holt inequality

Maximally entangled states should maximally violate the Bell inequality. In this paper, it is proved that all two-qubit states that maximally violate the Bell-Clauser-Horne-Shimony-Holt inequality are exactly Bell states and the states obtained from them by local unitary transformations. The proof is obtained by using the certain algebraic properties that Pauli's matrices satisfy. The argument is extended to the three-qubit system. Since all states obtained by local unitary transformations of a maximally entangled state are equally valid entangled states, we thus give the characterizations of …

On Determinism and Bell Test

Many Bell test results violate Bell's inequality. The premise of Bell's inequality is local determinism. We propose that, it can't be proved that something's mechanism isn't deterministic; if loopholes are not the reason of violation of Bell's inequality, violation of Bell's inequality illustrates that the mechanism isn't local. We propose a new explanation of Bell test, and the explanation illustrates the locality loophole is not closed by Weihs' experiment.

A two-qubit Bell inequality for which POVM measurements are relevant

A bipartite Bell inequality is derived which is maximally violated on the two-qubit state space if measurements describable by positive operator valued measure (POVM) elements are allowed rather than restricting the possible measurements to projective ones. In particular, the presented Bell inequality requires POVMs in order to be maximally violated by a maximally entangled two-qubit state. This answers a question raised by N. Gisin.

Bell's Inequality and quantum probability trees

The relations between Bell's inequality and quantum probability trees are explained against the background offered by the concept of a quantum probability tree built in others works. It is shown that f we use a concept of probability tree it will not be necessary we set aside the principle of separability and principle of locality.

Quantum Locality, Rings a Bell?: Bell's inequality meets local reality and true determinism

By assuming a deterministic evolution of quantum systems and taking realism into account, we carefully build a hidden variable theory for Quantum Mechanics based on the notion of ontological states proposed by 't Hooft. We view these ontological states as the ones embedded with realism and compare them to the (usual) quantum states that represent superpositions, viewing the latter as mere information of the system they describe. Such a deterministic model puts forward conditions for the applicability of Bell's inequality: the usual inequality cannot be applied to the usual experiments. We buil…

Equivalence between Bell's inequality and a constraint on stochastic field theories for EPR states

A generalized form of EPR state is defined, embracing both classical and nonclassical states. It is shown that for such states, Bell's inequality is equivalent to a constraint on stochastic field theories. Thus, violation of Bell's inequality can be observed also for weak violation of stochastic field theories. The Schrodinger cat state is shown to be an example of this.

Better Bell Inequality Violation by Collective Measurements

The standard Bell inequality experiments test for violation of local realism by repeatedly making local measurements on individual copies of an entangled quantum state. Here we investigate the possibility of increasing the violation of a Bell inequality by making collective measurements. We show that nonlocality of bipartite pure entangled states, quantified by their maximal violation of the Bell-Clauser-Horne inequality, can always be enhanced by collective measurements, even without communication between the parties. For mixed states we also show that collective measurements can increase the…

Bell's Inequalities and the Accardi-Gustafson Inequality

Many issues combine for consideration when speaking of Bell's Inequalities: nonlocality, realism, hidden variables, incompatible measures, wave function collapse, other. Each of these issues then may be viewed from several viewpoints: historical, theoretical, physical, experimental, statistical, communicational, cryptographical, and mathematical. From the mathematical viewpoint, much of the Bell theory is ``just geometry''.

Bell inequality and the locality loophole: Active versus passive switches

All experimental tests of the violation of Bell's inequality suffer from some loopholes. We show that the locality loophole is not independent of the detection loophole: in experiments using low efficient detectors, the locality loophole can be closed equivalently using active or passive switches.

Single-particle Bell-type Inequality

It is generally believed that Bell's inequality holds for the case of entangled states, including two correlated particles or special states of a single particle. Here, we derive a single-particle Bell's inequality for two correlated spin states at two successive times, appealing to the statistical independence condition in an ideal experiment, for a locally causal hidden variables theory. We show that regardless of the locality assumption, the inequality can be violated by some quantum predictions.

Relevant multi-setting tight Bell inequalities for qubits and qutrits

In the celebrated paper [J. Phys. A: Math. Gen. 37, 1775 (2004)], D. Collins and N. 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 [Phys.Rev. A 79, 012115 (2009)] to construct Bell inequality for qubits to higher dimensional system. Based on the generalized…

Deletion, Bell's Inequality, Teleportation

In this letter we analyze the efficacy of the entangled output of Pati-Braunstein deletion machine [3] as a teleportation channel. We analyze the possibility of it violating the Bell's inequality. Interestingly we find that for all values of the input parameter$α$ the state does not violate the Bell's inequality but when used as a teleportation channel can give a fidelity higher than the classical optimum (i.e 2/3).

Maximal violation of Bell inequality for any given two-qubit pure state

In the case of bipartite two qubits systems, we derive the analytical expression of bound of Bell operator for any given pure state. Our result not only manifest some properties of Bell inequality, for example which may be violated by any pure entangled state and only be maximally violated for a maximally entangled state, but also give the explicit values of maximal violation for any pure state. Finally we point out that for two qubits systems there is no mixed state which can produce maximal violation of Bell inequality.

Minimal assumption derivation of a Bell-type inequality

John Bell showed that a big class of local hidden-variable models stands in conflict with quantum mechanics and experiment. Recently, there were suggestions that empirical adequate hidden-variable models might exist, which presuppose a weaker notion of local causality. We will show that a Bell-type inequality can be derived also from these weaker assumptions.

纠缠薛定谔猫态、真空投影子与 Bell-CHSH 不等式

Entangled Schrödinger cat states, vacuum projector and Bell-CHSH inequality

该研究借助真空投影子与幺正位移算符构造 Hermitian 二值算符，并以此分析纠缠相干薛定谔猫态的 Bell-CHSH 不等式违背。

Tight Correlation-Function Bell Inequality for Multipartite $d$-Dimensional System

We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multipartite d-dimensional systems. All the Bell inequalities based on this generalization take the same simple form as the CHSH inequality. For small systems, numerical results show that the new inequalities are tight and we believe this is also valid for higher dimensional systems. Moreover, the new inequalities are relevant to the previous ones and for bipartite system, our inequality is equivalent to the Collins-Gisin-Linden-Masser-Popescu (CGLMP) inequality.

Proposed experiment to exclude higher-than-quantum violations of the Bell inequality

We show that a recent observation by Yan leads to a method to experimentally test whether a higher-than-quantum violation of the Clauser-Horne-Shimony-Holt Bell inequality is possible (assuming that the sum of probabilities of pairwise exclusive propositions cannot exceed 1). The test requires reaching the maximum quantum violation of a noncontextuality inequality involving sequences of three compatible measurements on a five-dimensional quantum system.

Bell inequality for quNits with binary measurements

We present a generalized Bell inequality for two entangled quNits. On one quNit the choice is between two standard von Neumann measurements, whereas for the other quNit there are $N^2$ different binary measurements. These binary measurements are related to the intermediate states known from eavesdropping in quantum cryptography. The maximum violation by $\sqrt{N}$ is reached for the maximally entangled state. Moreover, for N=2 it coincides with the familiar CHSH-inequality.

Distribution of Bell inequality violation vs. multiparty quantum correlation measures

Violation of a Bell inequality guarantees the existence of quantum correlations in a quantum state. A pure bipartite quantum state, having nonvanishing quantum correlation, always violates a Bell inequality. Such correspondence is absent for multipartite pure quantum states. For a shared multipartite quantum state, we establish a connection between the monogamy of Bell inequality violation and genuine multi-site entanglement as well as monogamy-based multiparty quantum correlation measures. We find that generalized Greenberger-Horne-Zeilinger states and another single-parameter family states w…

A closed-form necessary and sufficient condition for any two-qubit state to show hidden nonlocality w.r.t the Bell-CHSH inequality

In this note, we discuss a closed-form necessary and sufficient condition for any two-qubit state to show hidden nonlocality w.r.t the Bell-CHSH inequality. This is then used to numerically compute the relative volume of states showing hidden Bell-CHSH non-locality , among all two-qubit states with one-sided reduction maximally mixed.

Entanglement measure and quantum violation of Bell-type inequality for a family of four-qubit entangled states

By calculating entanglement measures and quantum violation of Bell-type inequality, we reveal the relationship between entanglement measure and the amount of quantum violation for a family of four-qubit entangled states. It has been demonstrated that the Bell-type inequality is completely violated by these four-qubit entangled states. The plot of entanglement measure as a function of the expectation value of Bell operator shows that entanglement measure first decreases and then increases smoothly with increasing quantum violation.

Bell Inequality and Many-Worlds Interpretation

It is argued that the lesson we should learn from Bell's inequalities is not that quantum mechanics requires some kind of action at a distance, but that it leads us to believe in parallel worlds.

Bell-inequality violation with "thermal" radiation

The model of a quantum-optical device for a conditional preparation of entangled states from input mixed states is presented. It is demonstrated that even thermal or pseudo-thermal radiation can be entangled in such a way, that Bell-inequalities are violated.

Bell-inequality violation with a triggered photon-pair source

Here we demonstrate, for the first time, violation of Bell's inequality using a triggered quantum dot photon-pair source without post-selection. Furthermore, the fidelity to the expected Bell state can be increased above 90% using temporal gating to reject photons emitted at times when collection of uncorrelated light is more probable. A direct measurement of a CHSH Bell inequality is made showing a clear violation, highlighting that a quantum dot entangled photon source is suitable for communication exploiting non-local quantum correlations.