In the analysis of experiments designed to reveal violation of Bell-type inequalities, it is usually assumed that any hidden variables associated with the nth particle pair would be independent of measurement choices and outcomes for the first $(n-1)$ pairs. Models which violate this assumption exploit what we call the {\it memory loophole}. We focus on the strongest type of violation, which uses the {\it 2-sided} memory loophole, in which the hidden variables for pair $n$ can depend on the previous measurement choices and outcomes in both wings of the experiment. We show that the 2-sided memory loophole allows a systematic violation of the CHSH inequality when the data are analysed in the standard way, but cannot produce a violation if a CHSH expression depending linearly on the data is used. In the first case, the maximal CHSH violation becomes small as the number of particle pairs tested becomes large. Hence, although in principle the memory loophole implies a slight flaw in existing analyses of Bell experiments, the data still strongly confirm quantum mechanics against local hidden variables.
We consider also a related loophole, the {\it simultaneous measurement loophole}, which applies if all measurements on each side are carried out simultaneously. We show that this can increase the probability of violating the linearised CHSH inequality as well as other Bell-type inequalities.
Quantum nonlocality has recently been classified into three distinct types: quantum entanglement, Einstein-Podolsky-Rosen (EPR) steering, and Bell's nonlocality. Experimentally Bell's nonlocality is usually tested by quantum violation of the Clause-Horne-Shimony-Holt (CHSH) inequality in the two-qubit system. Bell's nonlocality is the strongest type of nonlocality, also due this reason Bell-test experiments have encountered both the locality loophole and the detection loophole for a very long time. As a weaker nonlocality, EPR steering naturally escapes from the locality loophole and is corres…
Recently much interest has been directed towards designing setups that achieve realistic loss thresholds for decisive tests of local realism, in particular in the optical regime. We analyse the feasibility of such Bell tests based on a W-state shared between multiple parties, which can be realised for example by a single photon shared between spatial modes. We develop a general error model to obtain thresholds on the efficiencies required to violate local realism, and also consider two concrete optical measurement schemes.
We present a source of entangled photons that violates a Bell inequality free of the "fair-sampling" assumption, by over 7 standard deviations. This violation is the first experiment with photons to close the detection loophole, and we demonstrate enough "efficiency" overhead to eventually perform a fully loophole-free test of local realism. The entanglement quality is verified by maximally violating additional Bell tests, testing the upper limit of quantum correlations. Finally, we use the source to generate secure private quantum random numbers at rates over 4 orders of magnitude beyond prev…
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.
In a recent paper published last october 2015 by B.Hensen et al. [1] and in two companion papers published last december 2015 by B.Hensen et al. [2] and by L. Shalm et al. [3], the authors describe beautiful and complex experiments aimed at testing the theorem of J. Bell (1965) [4] with measurements free of the detection-loophole and the locality loophole that had never been closed unquestionably in a single experiment. According to their authors, each experiment, closing both loopholes, gives conclusive evidence in favor of nonlocality. Here I will argue that this claim may be questionable, a…
In a recent article (Found Sci (2020) https://doi.org/10.1007/s10699-020-09666-0) Marek Czachor claims that the Bell inequality cannot be proved because variables of complementary measurements cannot be added or multiplied. Even though he has correctly identified the problems existing with the orthodox interpretation of the Bell inequality and dealt with them in an original way, the interpretation he addresses do not pertain to the original formulation given by John Stewart Bell.
Binarisation-loophole-free observation of high-dimensional quantum nonlocality
University of Science and Technology of China（中国科学技术大学）；CAS Center For Excellence in Quantum Information and Quantum Physics（中国科学院量子信息与量子物理前沿卓越中心）；Hefei National Laboratory（合肥国家实验室）；Lund University（隆德大学）
Contrary to what Lambare [arXiv:2008.00369] assumes, in non-Newtonian calculus (a calculus based on non-Diophantine arithmetic) an integral is typically given by a nonlinear map. This is the technical reason why all the standard proofs of Bell-type inequalities fail if non-Newtonian hidden variables are taken into account. From the non-Newtonian perspective, Bell's inequality is a property of a limited and unphysical class of hidden-variable models. An explicit counterexample to Bell's theorem can be easily constructed.
The inequalities proposed in [Phys. Rev. Lett. 105, 170404 (2010)] are not Bell's inequalities in the usual sense: there are local classical models that violate these inequalities. Thus, their violation demonstrates nonlocality only under assumptions about the physical implementation and moreover they cannot be used as a device-independent test of nonlocality.
Genuine multipartite nonlocality (GMN), the strongest form of multipartite nonlocality that describes fully collective nonlocal correlations among all experimental parties, can be observed when different distant parties each locally measure a particle from a shared entangled many-particle state. For the demonstration of GMN, the experimentally observed statistics are typically postselected: Events for which some parties do not detect a particle must be discarded. This coincidence postselection generally leads to the detection loophole that invalidates a proper nonlocality demonstration. In thi…
Nieuwenhuizen argued that there exists some "contextuality loophole" in Bell's theorem. This claim is unjustified. In Bell's theorem non-contextuality is not presupposed but derived from Einstein causality using the EPR argument.
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)].
We point out a loophole in the derivation of the Bell inequalities in the form proposed by Clauser, Horne, Shimony and Holt (CHSH). In this derivation it is assumed that statistical independence is a necessary consequence of locality, but this is not a logical necessity.
By combining the postulates of macrorealism with Bell locality, we derive a qualitatively different hybrid inequality that avoids two loopholes that commonly appear in Leggett-Garg and Bell inequalities. First, locally invasive measurements can be used, which avoids the "clumsiness" Leggett-Garg inequality loophole. Second, a single experimental ensemble with fixed analyzer settings is sampled, which avoids the "disjoint sampling" Bell inequality loophole. The derived hybrid inequality has the same form as the Clauser-Horne-Shimony-Holt Bell inequality; however, its quantum violation intriguin…
Experimental violations of Bell inequalities are in general vulnerable to so-called "loopholes." In this work, we analyse the characteristics of a loophole-free Bell test with photons, closing simultaneously the locality, freedom-of-choice, fair-sampling (i.e. detection), coincidence-time, and memory loopholes. We pay special attention to the effect of excess predictability in the setting choices due to non-ideal random number generators. We discuss necessary adaptations of the CH/Eberhard inequality when using such imperfect devices and -- using Hoeffding's inequality and Doob's optional stop…
Bell's theorem cannot be proved if complementary measurements have to be represented by random variables which cannot be added or multiplied. One such case occurs if their domains are not identical. The case more directly related to the Einstein-Rosen-Podolsky argument occurs if there exists an `element of reality' but nevertheless addition of complementary results is impossible because they are represented by elements from different arithmetics. A naive mixing of arithmetics leads to contradictions at a much more elementary level than the Clauser-Horne-Shimony-Holt inequality.
Assumed data streams from a delayed choice gedanken experiment must satisfy a Bell's identity independently of locality assumptions. The violation of Bell's inequality by assumed correlations of identical form among these data streams implies that they cannot all result from statistically equivalent variables of a homogeneous process. This is consistent with both the requirements of arithmetic and distinctions between commuting and noncommuting observables in quantum mechanics. Neglect of these distinctions implies a logical loophole in the conventional interpretation of Bell's inequalities.
A single photon, delocalized over two optical modes, is characterized by means of quantum homodyne tomography. The reconstructed four-dimensional density matrix extends over the entire Hilbert space and thus reveals, for the first time, complete information about the dual-rail optical quantum bit as a state of the electromagnetic field. The experimental data violate the Bell inequality albeit with a loophole similar to the detection loophole in photon counting experiments.
The problem of closing the detection loophole with asymmetric systems, such as entangled atom-photon pairs, is addressed. We show that, for the Bell inequality I_3322, a minimal detection efficiency of 43% can be tolerated for one of the particles, if the other one is always detected. We also study the influence of noise and discuss the prospects of experimental implementation.
In Bell experiments, one problem is to achieve high enough photodetection to ensure that there is no possibility of describing the results via a local hidden-variable model. Using the Clauser-Horne inequality and a two-photon non-maximally entangled state, a photodetection efficiency higher than 0.67 is necessary. Here we discuss atom-photon Bell experiments. We show that, assuming perfect detection efficiency of the atom, it is possible to perform a loophole-free atom-photon Bell experiment whenever the photodetection efficiency exceeds 0.50.
A loophole-free violation of Bell inequalities is of fundamental importance for demonstrating quantum nonlocality and long-distance device-independent secure communication. However, transmission losses represent a fundamental limitation for photonic loophole-free Bell tests. A local precertification of the presence of the photons immediately before the local measurements may solve this problem. We show that local precertification is feasible by integrating three current technologies: (i) enhanced single-photon down-conversion to locally create a flag photon, (ii) nanowire-based superconducting…
Recent experiments have reached detection efficiencies sufficient to close the detection loophole with photons. Both experiments ran multiple successive trials in fixed measurement configurations, rather than randomly re-setting the measurement configurations before each measurement trial. This opens a new potential loophole for a local hidden variable theory. The loophole invalidates one proposed method of statistical analysis of the experimental results, as demonstrated in this note. Therefore a different analysis will be necessary to definitively assert that these experiments are subject on…
A common problem in Bell type experiments is the well-known detection loophole: if the detection efficiencies are not perfect and if one simply post-selects the conclusive events, one might observe a violation of a Bell inequality, even though a local model could have explained the experimental results. In this paper, we analyze the set of all post-selected correlations that can be explained by a local model, and show that it forms a polytope, larger than the Bell local polytope. We characterize the facets of this post-selected local polytope in the CHSH scenario, where two parties have binary…
We show that the detection efficiencies required for closing the detection loophole in Bell tests can be significantly lowered using quantum systems of dimension larger than two. We introduce a series of asymmetric Bell tests for which an efficiency arbitrarily close to 1/N can be tolerated using N-dimensional systems, and a symmetric Bell test for which the efficiency can be lowered down to 61.8% using four-dimensional systems. Experimental perspectives for our schemes look promising considering recent progress in atom-photon entanglement and in photon hyperentanglement.
The chained Bell inequalities of Braunstein and Caves involving N settings per observer have some interesting applications. Here we obtain the minimum detection efficiency required for a loophole-free violation of the Braunstein-Caves inequalities for any N > 2. We discuss both the case in which both particles are detected with the same efficiency and the case in which the particles are detected with different efficiencies.
Bell inequalities are intended to show that local realist theories cannot describe the world. A local realist theory is one where physical properties are defined prior to and independent of measurement, and no physical influence can propagate faster than the speed of light. Quantum-mechanical predictions for certain experiments violate the Bell inequality while a local realist theory cannot, and this shows that a local realist theory cannot give those quantum-mechanical predictions. However, because of unexpected circumstances or "loopholes" in available experiment tests, local realist theorie…
In a recent article Giustina et al. [Phys. Rev. Lett. 115, 250401 (2015)] report on an advanced Bell experiment, simultaneously closing loopholes for local hidden-variable theories. The authors claim that 'local realism' has been refuted, unless 'truly exotic hypotheses' are made. Here I argue that a particularly wide and natural class of local hidden-variable theories survives, for instance when the hidden variables describe a background field. Such background-based theories exploit the freedom-of-choice loophole, which cannot be closed for this type of hidden variables. The dynamics of such …
A simple classical, deterministic, local situation violating the Bell inequality is described. The detectors used in the experiment are ideal and the observers who decide which pair of measuring devices to choose for a given pair of particles have free will. The construction uses random variables which are not jointly measurable in a single run of an experiment and the hidden variables have a nonsymmetric probability density. Such random variables are complementary but still fully classical. An assumption that classical random variables cannot satisfy any form of complementarity principle is f…
We consider the entanglement between two internal states of a single atom and two photon number states describing either the vaccum or a single photon and thus containing, on average, less than one photon. We show that this intriguing entanglement can be characterized through substantial violations of a Bell inequality by performing homodyne detections on the optical mode. We present the experimental challenges that need to be overcome to pave the way towards a loophole-free Bell test in this setup.
We introduce an all-versus-nothing proof of impossibility of Einstein-Podolsky-Rosen's local elements of reality for two photons entangled both in polarization and path degrees of freedom, which leads to a Bell's inequality where the classical bound is 8 and the quantum prediction is 16. A simple estimation of the detection efficiency required to close the detection loophole using this proof gives eta > 0.69. This efficiency is lower than that required for previous proposals.
Two new formulations of Bell's theorem are given here. First, we consider a definite set of two entangled photons with only two polarization directions, for which Bell's locality assumption is violated for the case of perfect correlation. Then, using a different approach, we prove an efficient Bell-type inequality which is violated by some quantum mechanical predictions, independent of the efficiency factors.
We derive two classes of multi-mode 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.
Two trapped ions that are kilometers apart can be entangled by the joint detection of two photons, each coming from one of the ions, in a basis of entangled states. Such a detection is possible with linear optical elements. The use of two-photon interference allows entanglement distribution without interferometric sensitivity to the path length of the photons. The present method of creating entangled ions also opens up the possibility of a loophole-free test of Bell's inequalities.
We demonstrate that different kind of mesoscopic quantum states of light can be efficiently generated from a simple iterative scheme using homodyne heralding. These states exhibit strong non-classical features, and are of great interest for many applications such as quantum error-correcting codes or fundamental testings. On this basis we propose a protocol allowing a large loophole-free violation of a CHSH-type Bell inequality with a remarkable robustness to line losses.
We propose a feasible optical setup allowing for a loophole-free Bell test with efficient homodyne detection. A non-gaussian entangled state is generated from a two-mode squeezed vacuum by subtracting a single photon from each mode, using beamsplitters and standard low-efficiency single-photon detectors. A Bell violation exceeding 1% is achievable with 6-dB squeezed light and an homodyne efficiency around 95%. A detailed feasibility analysis, based upon the recent generation of single-mode non-gaussian states, confirms that this method opens a promising avenue towards a complete experimental B…
Quantum mechanics is strictly incompatible with local realism. It has been shown by Bell and others that it is possible, in principle, to experimentally differentiate between local realism and quantum mechanics. Numerous experiments have attempted to falsify local realism; however, they have consistently failed to close the detection loophole under strict locality conditions, thereby allowing local realistic explanations for their observations. In 2015, three experiments took place that tested local realism without the impediments of these significant loopholes. Between these three experiments…
The question whether Quantum Mechanics describes all aspects of both the micro-world of elementary particles and living organisms adequately is still open. A Bell inequality test without locality and detection loopholes will deliver the answer when applied to macroscopic systems. Here we show that an experimentally feasible loophole-free Bell test is possible using only multi-photon polarization entanglement and linear optics. We apply a preselection protocol which produces genuine macroscopic superpositions and entanglement, thus eliminating postselection and dark counts. Such superpositions …
So far, all experimental tests of Bell inequalities which must be satisfied by all local realistic hidden-variable theories and are violated by quantum mechanical predictions have left at least one loophole open. We propose a feasible setup allowing for a loophole-free test of the Bell inequalities. Two electron spin qubits of phosphorus donors in semiconductors in different cavities 300 m apart are entangled through a bright coherent light and postselections using homodyne measurements. The electron spins are then read out randomly and independently by Alice and Bob, respectively, with unity …
We show two experimental realizations of Hardy ladder test of quantum nonlocality using energy-time correlated photons, following the scheme proposed by A. Cabello \emph{et al.} [Phys. Rev. Lett. \textbf{102}, 040401 (2009)]. Unlike, previous energy-time Bell experiments, these tests require precise tailored nonmaximally entangled states. One of them is equivalent to the two-setting two-outcome Bell test requiring a minimum detection efficiency. The reported experiments are still affected by the locality and detection loopholes, but are free of the post-selection loophole of previous energy-ti…
Experiments over three decades have been unable to demonstrate weak nonlocality in the sense of Bell unambiguously, without loopholes. The last important loophole remaining is the detection loophole, which is being tackled by at least three experimental groups. This letter counters five common beliefs about Bell experiments, and presents alternative scenarios for future developments.
In a local realist world view, physical properties are defined prior to and independent of measurement, and no physical influence can propagate faster than the speed of light. Proper experimental violation of a Bell inequality would show that the world cannot be described within local realism. Such experiments usually require additional assumptions that make them vulnerable to a number of "loopholes." A recent experiment [Giustina et al, Nature, 2013] violated a Bell inequality without being vulnerable to the detection (or fair-sampling) loophole, therefore not needing the fair-sampling assump…
We demonstrate extraction of randomness from spontaneous-emission events less than 36 ns in the past, giving output bits with excess predictability below $10^{-5}$ and strong metrological randomness assurances. This randomness generation strategy satisfies the stringent requirements for unpredictable basis choices in current "loophole-free Bell tests" of local realism [Hensen et al., Nature (London) 526, 682 (2015); Giustina et al., Phys. Rev. Lett. 115, 250401 (2015); Shalm et al., Phys. Rev. Lett. 115, 250402 (2015)].
Steering nonlocality in high-speed telecommunication system without detection loophole
Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）；School of Science and State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications（北京邮电大学理学院信息光子学与光通信国家重点实验室）
A recent experiment yielding results in agreement with quantum theory and violating Bell inequalities was interpreted [Nature 526 (29 Octobert 2015) p. 682 and p. 649] as ruling out any local realistic theory of nature. But quantum theory itself is both local and realistic when properly interpreted using a quantum Hilbert space rather than the classical hidden variables used to derive Bell inequalities. There is no spooky action at a distance in the real world we live in if it is governed by the laws of quantum mechanics.
In a recent note, Cavalcanti and Scarani (CS) constructed a counter local-hidden-variable model to explain the violation of our inequalities in Phys. Rev. Lett. 105, 170404 (2010). Here, we briefly discuss some issues in response to the comments raised by CS.
We present a scheme to demonstrate loophole-free Bell inequality violation where the entanglement between photon pairs is transferred to solid state (spin) qubits mediated by cavity QED interactions. As this transfer can be achieved in a heralded way, our scheme is basically insensitive to losses on the channel. This makes it appealing for the implementation of quantum information protocols based on nonlocality, such as device-independent quantum key distribution. We consider potential experimental realisations of our scheme using single atom, colour centre and quantum dot cavity systems.
We study the problem of closing the detection loophole in three-qubit Bell tests, the experimentally most relevant case beyond the usual bipartite scenario, and show that the minimal detection efficiencies required can be considerably lowered compared to the two-qubit case. The lowest reported detection efficiency thresholds for two and three qubits so far are $\sim66.7\%$ and $60\%$, respectively. Using the three-qubit W state and a 3-setting Bell inequality, we beat these thresholds and with an 8-setting Bell inequality we reach $50.13\%$. We also investigate generic three-qubit states which…
The violation of Bell inequalities where both detection and locality loopholes are closed is crucial for device independent assessments of quantum information. While of technological nature, the simultaneous closing of both loopholes still remains a challenge. In Nat. Commun. 4:2104(2013), a realistic setup to produce an atom-photon entangled state that could reach a loophole free Bell inequality violation within current experimental technology was proposed. Here we improve the analysis of this proposal by giving an analytical treatment that shows that the state proposed in Nat. Commun. 4:2104…
It has long been known that the "detection loophole", present when detector efficiencies are below a critical figure, could open the way for alternative "local realist" explanations for the violation of Bell tests. It has in recent years become common to assume the loophole can be ignored, regardless of which version of the Bell test is employed. A simple model is presented that illustrates that this may not be justified. Two of the versions -- the standard test of form -2 <= S <= 2 and the currently-popular "visibility" test -- are at grave risk of bias. Statements implying that experimental …
We apply a distance-based Bell-test analysis method [E. Knill et al., Phys. Rev. A. 91, 032105 (2015)] to three experimental data sets where conventional analyses failed or required additional assumptions. The first is produced from a new classical source exploiting a "coincidence-time loophole" for which standard analysis falsely shows a Bell violation. The second is from a source previously shown to violate a Bell inequality; the distance-based analysis agrees with the previous results but with fewer assumptions. The third data set does not show a violation with standard analysis despite the…
This paper analyzes effects of time-dependence in the Bell inequality. A generalized inequality is derived for the case when coincidence and non-coincidence [and hence whether or not a pair contributes to the actual data] is controlled by timing that depends on the detector settings. Needless to say, this inequality is violated by quantum mechanics and could be violated by experimental data provided that the loss of measurement pairs through failure of coincidence is small enough, but the quantitative bound is more restrictive in this case than in the previously analyzed "efficiency loophole."
We investigate the problem of closing the detection loophole in multipartite Bell tests, and show that the required detection efficiencies can be significantly lowered compared to the bipartite case. In particular, we present Bell tests based on n-qubit Greenberger-Horne-Zeilinger states, which can tolerate efficiencies as low as 38% for a reasonable number of parties and measurements. Even in the presence of a significant amount of noise, efficiencies below 50% can be tolerated, which is encouraging given recent experimental progress. Finally we give strong evidence that, for a sufficiently l…
The problem of closing the detection loophole in Bell tests is investigated in the presence of a limited number of efficient detectors using emblematic multipartite quantum states. To this end, a family of multipartite Bell inequalities is introduced basing on local projective measurements conducted by $N-k$ parties and applying a $k$-party Bell inequality on the remaining parties. Surprisingly, we find that most of the studied pure multipartite states involving e.g. cluster states, the Dicke states, and the Greenberger-Horne-Zeilinger states can violate our inequalities with only the use of t…
