Towards Loophole-Free Optical Bell Test of CHSH Inequality

Bell test had been suggested to end the long-standing debate on the EPR paradox, while the imperfections of experimental devices induce some loopholes in Bell test experiments and hence the assumption of local reality by EPR cannot be excluded with current experimental results. In optical Bell test experiments, the locality loophole can be closed easily, while the attempt of closing detection loophole requires very high efficiency of single photon detectors. Previous studies showed that the violation of Clauser-Horne-Shimony-Holt (CHSH) inequality with maximally entangled states requires the detection efficiency to be higher than 82.8 %. In this paper, we raise a modified CHSH inequality that covers all measurement events including the efficient and inefficient detections in the Bell test and prove that all local hidden models can be excluded when the inequality is violated. We find that, when non-maximally entangled states are applied to the Bell test, the lowest detection efficiency for violation of the present inequality is 66.7 %. This makes it feasible to close the detection loophole and the locality loophole simultaneously in optical Bell test of CHSH inequality.     

Violation of Clauser-Horne-Shimony-Holt inequality for states resulting from entanglement swapping

We consider violation of CHSH inequality for states before and after entanglement swapping. We present a pair of initial states which do not violate CHSH inequality however the final state violates CHSH inequality for some results of Bell measurement performed in order to swap entanglement.     

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.     

Dendrogramic Representation of Data: CHSH Violation vs. Nonergodicity

This paper is devoted to the foundational problems of dendrogramic holographic theory (DH theory). We used the ontic–epistemic (implicate–explicate order) methodology. The epistemic counterpart is based on the representation of data by dendrograms constructed with hierarchic clustering algorithms. The ontic universe is described as a p-adic tree; it is zero-dimensional, totally disconnected, disordered, and bounded (in p-adic ultrametric spaces). Classical–quantum interrelations lose their sharpness; generally, simple dendrograms are “more quantum” than complex ones. We used the CHSH inequality as a measure of quantum-likeness. We demonstrate that it can be violated by classical experimental data represented by dendrograms. The seed of this violation is neither nonlocality nor a rejection of realism, but the nonergodicity of dendrogramic time series. Generally, the violation of ergodicity is one of the basic features of DH theory. The dendrogramic representation leads to the local realistic model that violates the CHSH inequality. We also considered DH theory for Minkowski geometry and monitored the dependence of CHSH violation and nonergodicity on geometry, as well as a Lorentz transformation of data.     

All bipartite entangled States display some hidden nonlocality

One of the most significant and well-known properties of entangled states is that they may lead to violations of Bell inequalities and are thus inconsistent with any local-realistic theory. However, there are entangled states that cannot violate any Bell inequality, and in general the precise relationship between entanglement and observable nonlocality is not well understood. We demonstrate that a violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality can be demonstrated in a certain kind of Bell experiment for all entangled states. Our proof of the result consists of two main steps. We first provide a simple characterization of the set of states that do not violate the CHSH inequality even after general local operations and classical communication. Second, we prove that for each entangled state sigma, there exists another state rho not violating the CHSH inequality, such that rhomultiply sign in circlesigma violates the CHSH inequality.     

Test of the no‐signaling principle in the Hensen loophole‐free CHSH experiment

We analyze the data from the loophole‐free CHSH experiment performed by Hensen et al ., and show that it is actually not exempt of an important loophole. By increasing the size of the sample of event‐ready detections, one can exhibit in the experimental data a violation of the no‐signaling principle with a statistical significance at least similar to that of the reported violation of the CHSH inequality, if not stronger.     

Separability criterion for quantum effects

Entanglement of quantum states is absolutely essential for modern quantum sciences and technologies. It is natural to extend the notion of entanglement to quantum observables dual to quantum states. For quantum states, various separability criteria have been proposed to determine whether a given state is entangled. In this Letter, we propose a separability criterion for specific quantum effects (binary observables) that can be regarded as a dual version of the Bell–Clauser–Horne–Shimony–Holt (Bell–CHSH) inequality for quantum states. The violation of the dual version of the Bell–CHSH inequality is confirmed by using IBM's cloud quantum computer. As a consequence, the violation of our inequality rules out the maximal tensor product state space, that satisfies information causality and local tomography. As an application, we show that an entangled observable which violates our inequality is useful for quantum teleportation.     

Nonlocality of Two-Qubit and Three-Qubit Schmidt-Correlated States

We investigate the nonlocality of Schmidt-correlated (SC) states, and present analytical expressions of the maximum violation value of Bell inequalities. It is shown that the violation of Clauser-Horne-Shimony-Holt (CHSH) inequality is necessary and sufficient for the nonlocality of two-qubit SC states, whereas the violation of the Svetlichny inequality is only a sufficient condition for the genuine nonlocality of three-qubit SC states. Furthermore, the relations among the maximum violation values, concurrence, and relative entropy entanglement are discussed.     

Quantum Perfect Correlations and Hardy’s Nonlocality Theorem

In this paper the failure of Hardy's nonlocality proof for the class of maximally entangled states is considered. A detailed analysis shows that the incompatibility of the Hardy equations for this class of states physically originates from the fact that the existence of quantum perfect correlations for the three pairs of two-valued observables (D ₁₁, D ₂₁), (D ₁₁, D ₂₂), and (D ₁₂, D ₂₁) [in the sense of having with certainty equal (different) readings for a joint measurement of any one of the pairs (D ₁₁, D ₂₁), (D ₁₁, D ₂₂), and (D ₁₂, D ₂₁)], necessarily entails perfect correlation for the pair of observables (D ₁₂, D ₂₂) [in the sense of having with certainty equal (different) readings for a joint measurement of the pair (D ₁₂, D ₂₂)]. Indeed, the set of these four perfect correlations is found to satisfy the CHSH inequality, and then no violations of local realism will arise for the maximally entangled state as far as the four observables D _ij, i,j = 1 or 2, are concerned. The connection between this fact and the impossibility for the quantum mechanical predictions to give the maximum possible theoretical violation of the CHSH inequality is pointed out. Moreover, it is generally proved that the fulfillment of all the Hardy nonlocality conditions necessarily entails a violation of the resulting CHSH inequality. The largest violation of this latter inequality is determined.     

A Rigorous Analysis of the Clauser–Horne–Shimony–Holt Inequality Experiment When Trials Need Not be Independent

The Clauser–Horne–Shimony–Holt (CHSH) inequality is a constraint that local hidden variable theories must obey. Quantum Mechanics predicts a violation of this inequality in certain experimental settings. Treatments of this subject frequently make simplifying assumptions about the probability spaces available to a local hidden variable theory, such as assuming the state of the system is a discrete or absolutely continuous random variable, or assuming that repeated experimental trials are independent and identically distributed. In this paper, we do two things: first, show that the CHSH inequality holds even for completely general state variables in the measure-theoretic setting, and second, demonstrate how to drop the assumption of independence of subsequent trials while still being able to perform a hypothesis test that will distinguish Quantum Mechanics from local theories. The statistical strength of such a test is computed.     

Nonlocality of High-Order Superposition Photon Addition Two-Mode Thermal State

A new kind of quantum state is introduced, which can be generated by the superposition of high-order (N) photon addition to two-mode thermal state (SPA-TMTS). By using the P-representation of thermal state, the normal ordering form of SPA-TMTS is obtained. Based on this, some analytical expressions, such as the normalization coefficient, Q-function, Wigner function (WF) are presented. It is shown that the WF can possess negative value at some region. In addition, the nonlocality of the SPA-TMTS is discussed in terms of the CH and CHSH inequalities. It is found that for a given small $\bar{n}$ , the CH inequality is violated for some small N and the violation decrease with the increasing N; while the CHSH inequality is violated only for N=1.     

Three-Slit Experiments and Quantum Nonlocality

An interesting link between two very different physical aspects of quantum mechanics is revealed; these are the absence of third-order interference and Tsirelson’s bound for the nonlocal correlations. Considering multiple-slit experiments—not only the traditional configuration with two slits, but also configurations with three and more slits—Sorkin detected that third-order (and higher-order) interference is not possible in quantum mechanics. The EPR experiments show that quantum mechanics involves nonlocal correlations which are demonstrated in a violation of the Bell or CHSH inequality, but are still limited by a bound discovered by Tsirelson. It now turns out that Tsirelson’s bound holds in a broad class of probabilistic theories provided that they rule out third-order interference. A major characteristic of this class is the existence of a reasonable calculus of conditional probability or, phrased more physically, of a reasonable model for the quantum measurement process.     

Activation of nonlocal quantum resources

We find two two-qubit bipartite states ρ1, ρ2 such that arbitrarily many copies of one or the other cannot exhibit nonlocal correlations in a two-setting-two-outcome Bell scenario. However, the bipartite state ρ1 ? ρ2 violates the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).] by an amount of 2.023. We also identify a CHSH-local state ρ such that ρ?2 is CHSH inequality-violating. The tools employed can be easily adapted to find instances of nonlocality activation in arbitrary Bell scenarios.     

Renormalization of quantum discord and Bell nonlocality in the XXZ model with Dzyaloshinskii–Moriya interaction

In this paper, we have investigated the dynamical behaviors of the two important quantum correlation witnesses, i.e. geometric quantum discord (GQD) and Bell–CHSH inequality in the XXZ model with DM interaction by employing the quantum renormalization group (QRG) method. The results have shown that the anisotropy suppresses the quantum correlations while the DM interaction can enhance them. Meanwhile, using the QRG method we have studied the quantum phase transition of GQD and obtained two saturated values, which are associated with two different phases: spin-fluid phase and the Néel phase. It is worth mentioning that the block–block correlation is not strong enough to violate the Bell–CHSH inequality in the whole iteration steps. Moreover, the nonanalytic phenomenon and scaling behavior of Bell inequality are discussed in detail. As a byproduct, the conjecture that the exact lower and upper bounds of Bell inequality versus GQD can always be established for this spin system although the given density matrix is a general X state.     

Implementation Scheme of Two-Photon Post-Quantum Correlations

The pre-and post-selection processes of the "two-state vector formalism" lead to a fair sampling loophole in Bell test, so it can be used to simulate post-quantum correlations. In this paper, we propose a physical implementation of such a correlation with the help of quantum non-demolition measurement, which is realized via the cross-Kerr nonlinear interaction between the signal photon and a probe coherent beam. The indirect measurement on the polarization state of photon is realized by the direct measurement on the phase shift of the probe coherent beam, which enhances the detection efficiency greatly and leaves the signal photon unabsorbed. The maximal violation of the CHSH inequality 4 can be achieved by pre-and post-selecting maximally entangled states. The reason why we can get the post-quantum correlation is that the selection of the results after measurement opens fair-sampling loophole. The fair-sampling loophole opened here is different from the one usually used in the currently existing simulation schemes for post-quantum correlations,which are simulated by selecting the states to be measured or enlarging the Hilbert space. So, our results present an alternative way to mimic post-quantum correlations.     

QUANTUM MECHANICAL PROBABILITIES AND GENERAL PROBABILISTIC CONSTRAINTS FOR EINSTEIN–PODOLSKY–ROSEN–BOHM EXPERIMENTS

Relativistic causality, namely, the impossibility of signaling at superluminal speeds, restricts the kinds of correlations which can occur between different parts of a composite physical system. Here we establish the basic restrictions which relativistic causality imposes on the joint probabilities involved in an experiment of the Einstein–Podolsky–Rosen–Bohm type. Quantum mechanics, on the other hand, places further restrictions beyond those required by general considerations like causality and consistency. We illustrate this fact by considering the sum of correlations involved in the CHSH inequality. Within the general framework of the CHSH inequality, we also consider the nonlocality theorem derived by Hardy, and discuss the constraints that relativistic causality, on the one hand, and quantum mechanics, on the other hand, impose on it. Finally, we derive a simple inequality which can be used to test quantum mechanics against general probabilistic theories.     

Partial list of bipartite Bell inequalities with four binary settings

We give a partial list of 26 tight Bell inequalities for the case where Alice and Bob choose among four two-outcome measurements. All tight Bell inequalities with less settings are reviewed as well. For each inequality we compute numerically the maximal quantum violation, the resistance to noise and the minimal detection efficiency required for closing the detection loophole. Surprisingly, most of these inequalities are outperformed by the CHSH inequality.     

Nonlinear Channels of Werner States

We study the nonlinear positive map of the density matrix of two-qubit Werner states, called the nonlinear channel. The map ρ → Φ(ρ) is realized by the rational function Φ. We discuss the influence of the map on the entanglement properties of the transformed density matrix. We investigate the violation of the Bell inequality (CHSH inequality) for the two-qubit state Φ(ρ). The nonlinear channels under discussion create the entangled state from a separable Werner state. We study the quantum spin tomograms of the states.     

General correlation functions of the Clauser-Horne-Shimony-Holt inequality for arbitrarily high-dimensional systems

We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to arbitrarily high-dimensional systems. Based on this generalization, we construct the general CHSH inequality for bipartite quantum systems of arbitrarily high dimensionality, which takes the same simple form as CHSH inequality for two dimensions. This inequality is optimal in the same sense as the CHSH inequality for two-dimensional systems, namely, the maximal amount by which the inequality is violated consists of the maximal resistance to noise. We also discuss the physical meaning and general definition of the correlation functions. Furthermore, by giving another specific set of the correlation functions with the same physical meaning, we realize the inequality presented by Collins et al. [Phys. Rev. Lett. 88, 040404 (2002)]].     

Entanglement versus bell violations and their behavior under local filtering operations

We discuss the relations between the violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality for systems of two qubits on the one side and entanglement of formation, local filtering operations, and the entropy and purity on the other. We calculate the extremal Bell violations for a given amount of entanglement of formation and characterize the respective states, which turn out to have extremal properties also with respect to the entropy, purity, and several entanglement monotones. The optimal local filtering operations leading to the maximal Bell violation for a given state are provided, and the special role of the resulting Bell diagonal states in the context of Bell inequalities is discussed.