A theoretical and numerical analysis of a Bell-type experiment is presented, where linear polarization measurements on nonmaximally entangled two-qubit states are modeled through a sigmoid-based threshold mechanism that violates the fair-sampling assumption required by the CHSH inequality. Numerical simulations characterize how closely this local construction can reproduce quantum-like correlations in the postselected detected sample, for both the | Φ r 〉 and | Ψ r 〉 states over a range of entanglement parameters. For the tested configurations, the model produces CHSH-type violations only in the postselected detected sample through a setting-dependent detection mechanism, operating outside the assumptions of both the CHSH inequality and Mermin’s 82.8% efficiency bound. The model is not in conflict with loophole-free Bell experiments, which specifically close the detection loophole exploited here. The CH–Eberhard analysis in the one-detector-per-side configuration yields J < 0 throughout the full range of entanglement parameters, as required for a strictly local model. The model is further extended with a local adaptive-threshold mechanism implementing one-sided memory. A paired comparison shows that this specific memory rule produces a systematic and reproducible shift in the CH–Eberhard parameter J relative to the fixed-threshold model, while leaving the CH–Eberhard analysis qualitatively unchanged under the tested conditions. The results characterize how a local, setting-dependent postselection mechanism shapes Bell-test statistics across a range of entanglement parameters, within the specific configurations and threshold rules tested.
A theoretical and numerical analysis of a Bell-type experiment is presented, where linear polarization measurements on nonmaximally entangled two-qubit states are modeled through a sigmoid-based threshold mechanism that violates the fair-sampling assumption required by the CHSH inequality. Numerical simulations characterize how closely this local construction can reproduce quantum-like correlations in the postselected detected sample, for both the | Φ r 〉 and | Ψ r 〉 states over a range of entanglement parameters. For the tested configurations, the model produces CHSH-type violations only in the postselected detected sample through a setting-dependent detection mechanism, operating outside the assumptions of both the CHSH inequality and Mermin’s 82.8% efficiency bound. The model is not in conflict with loophole-free Bell experiments, which specifically close the detection loophole exploited here. The CH–Eberhard analysis in the one-detector-per-side configuration yields J < 0 throughout the full range of entanglement parameters, as required for a strictly local model. The model is further extended with a local adaptive-threshold mechanism implementing one-sided memory. A paired comparison shows that this specific memory rule produces a systematic and reproducible shift in the CH–Eberhard parameter J relative to the fixed-threshold model, while leaving the CH–Eberhard analysis qualitatively unchanged under the tested conditions. The results characterize how a local, setting-dependent postselection mechanism shapes Bell-test statistics across a range of entanglement parameters, within the specific configurations and threshold rules tested.
A local hidden-variable model is constructed for the postselected correlations of nonmaximally entangled photon pairs in the states |Phi_r> and |Psi_r>, continuous in r over [-1,1] from the product state to the maximally entangled state. Its root-mean-square deviations from the quantum correlations and from the coincidence probabilities, averaged over a uniform grid of polarizer angles, are 0.0322 and 0.0233, smaller than those of the sigmoid-threshold model of Ref. [14] by factors of 2.7 to 3.5. The source emits a photon pair with a probability that depends on the hidden variable and on r; each station detects with a probability that depends on the local hidden variable and on the local polarizer angle. This dependence violates the fair-sampling assumption and exploits the detection loophole. On the postselected sample the model violates the CHSH inequality for |Phi_r>, reaching 2*sqrt(2) at |r|=1, with single-detector efficiency ≈ 85%, outside the assumptions of both the CHSH inequality and Mermin's 82.8% efficiency bound. The CH--Eberhard parameter, which does not assume fair sampling, satisfies J<0 over the full entanglement range.
A local hidden-variable model for tripartite GHZ states is constructed, in which a single threshold on the local signal magnitude governs detection and violates the fair-sampling assumption. The detection probability depends on the hidden variables, so the detected sample is unrepresentative, while the single-detector efficiency is independent of the polarizer settings. On the postselected sample the model violates the Mermin and Svetlichny inequalities, reaching M3=3.85 and S3=5.66, at a single-detector efficiency of 72.49%, below the threshold eta_crit=3/4 of Cabello et al.; the full ensemble satisfies both inequalities. A necessary upper bound eta <= 74.64% for the exact reproduction of the GHZ correlations by any model in this class is derived.
Bell--CHSH is frequently paraphrased as ``local realism is impossible'' after experimental violations of CHSH. The theorem is correct, but the paraphrase hides a structural assumption: \emph{selection centrality} (fair sampling), i.e.\ that the detected sample is not a setting- and hidden-variable--dependent reweighting of the hidden prior. We give a precise measure-theoretic separation between (i) the \emph{central} sector, where Eobs=Efull and CHSH≤2 holds under measurement independence and locality, and (ii) the \emph{noncentral} sector, where the observed correlations are computed under a reweighted measure dνa,b=w(a,b,λ)dρ. We prove a sharp CHSH inflation bound Sobs≤2+4δ,δ:=supa,b∫∣w(a,b,λ)−1∣ρ(dλ), so reproducing Tsirelson's 22 by strictly local measurement-independent models via selection requires δ≥(22−2)/4≈0.2071. We provide (A) a canonical Radon--Nikod\'ym construction (piecewise constants on sign regions plus a small positive remainder), (B) an explicit local factorized detection-loophole construction that reproduces arbitrary finite correlation tables, and (C) an implementable optics protocol to estimate or upper-bound δ from time-tag data via window dithering, threshold and spectral sweeps, and auxiliary-tag binning. This paper critiques not the validity of Bell's theorem, but the common overstatement of what CHSH violations imply without an explicit, quantitative centrality audit.
Summary form only given. In a Bell experiment, one prepares pairs of entangled particles and sends them to two observers, Alice and Bob, for measurement and detection [1]. For some choices of their measurement settings, Alice and Bob may observe strong correlations between their results in accordance with the predictions of quantum mechanical theory. Conversely, any physical theory that assumes no physical influence can be faster than the speed of light and that properties of physical systems are elements of reality [2] - a local realistic theory - can predict only a limited amount of correlation between Alice's and Bob's measurement outcomes. Upon observing correlations sufficient to violate Bell's inequality, Alice and Bob must abandon local realism.It is common that in an experiment, some particles emitted by the source will not be detected. Then the subset of detected particles might display correlations that violate the Bell inequality although the entire ensemble can be described by a local realistic theory [3]. To achieve a conclusive Bell violation without assuming that the detected particles are a “fair” sample, a highly efficient setup is necessary. Due to experimental limitations, fair sampling has been assumed in most Bell experiments performed to date, and it has never been possible to avoid this assumption with photons due to the absence of efficient sources and detectors. Here we report the first Bell experiment with photons that does not rely on any fair-sampling assumption. In our experiment we employ the Eberhard inequality, a Bell inequality that by construction does not rely on the fair-sampling assumption [4]. Our source of entangled photon pairs utilizes bulk-crystal spontaneous parametric downconversion in a Sagnac configuration, which has shown to be very efficient [5, 6]. For photon detection, we employ superconducting transition-edge sensors, which not only offer high efficiency but also are intrinsically free of dark counts [7]. This combination of features is imperative for an experiment in which no correction of count rates can be tolerated. We achieve uncorrected coupling efficiencies over 73% in each arm of our source, facilitating a violation of the inequality by nearly 70 standard deviations after five minutes of measurement. Since the first experimental Bell test, a satisfactory laboratory realization of the motivating gedankenexperiment has remained an outstanding challenge. The two other main assumptions include “locality” [8, 9] and “freedom of choice” [10]. Invoking any of these renders an experiment vulnerable to explanation by a local realistic theory. The realization of an experiment that is free of all three assumptions-a so-called loopholefree Bell test-remains an important outstanding goal for the physics community with strong implications for quantum technologies. We note that with our experiment, photons are the first physical system for which each of these three assumptions has been successfully addressed, albeit in different experiments. This represents promise for practical applications like one-sided device-independent quantum key distribution [11], and in an additional test, we also demonstrate that our experimental apparatus is capable of implementing this protocol on both sides.
It is already known that one can always find a set of measurements on any two-qubit entangled state that will lead to a violation of the CHSH inequality. We provide an explicit state in terms of the angle between Alice's choice of measurements and the angle between Bob's choice of measurements, such that the CHSH inequality is always violated provided Alice's or Bob's choice of inputs are not collinear. We prove that inequalities with a corresponding Bell operator written as a linear combination of tensor products of Pauli matrices, excluding the identity, will generate the most nonlocal correlations using maximally entangled states in our experiment. From this result and a proposition from Horodecki et. al., we are able to construct the state that generates these optimal correlations. To achieve this state in a lab, one party must rotate their qubit using the orthogonal operation we provide and also rotate their Bloch sphere such that all their measurements lie in the same plane. We provide a comprehensive study of how Bell inequalities change when experiments introduce error via imperfect detection efficiency. The original cases of perfect efficiency are covered first and then a more realistic approach, when inefficient detectors are used, will follow. It is shown that less entanglement is needed to demonstrate more nonlocality in some Clasure-Horne-Shimony-Holt (CHSH) experiments when detector inefficiency is introduced. An example of this is shown for any given specific set of measurements in the CHSH Bell experiment. This occurs when one party has a detector of efficiency for each choice of input and the other party makes projective measurements. The efficiency can be pushed down to fifty percent while still violating the CHSH inequality, and for the experimental set-up illustrated, there is more nonlocality with less entanglement. Furthermore, it is shown that if the first party has an imperfect detector for only one choice of inputs rather than two, the efficiency can be brought down arbitrarily close to zero percent while still violating the CHSH inequality. Historically, nonlocality and entanglement were viewed as two equivalent resources, but recently this equality has come under question; these results further support this fundamental difference. Further more, we introduce Mermin's game in the case of relaxed conditions. The original constraints were that when the detectors in separate labs of a two-qubit experiment are in the same setting, then the results should be the same. We require that the outcomes are the same at least part of the time, given by some epsilon variable. Initially, one could find a maximum violation of one-fourth by allowing to parties to share the singlet state and have measurement settings one-hundred and twenty degrees apart from one another. By allowing some epsilon error in the perfect correlations regime, one can find a maximum violation of minus one plus the square root of two using the singlet state and measurement inputs that achieve Tsirelson's bound for the CHSH experiment. The reason is that we show Mermin's inequality is technically the CHSH inequality "in disguise", but with using constraints the CHSH experiment does not use. We derive Mermin's inequality under new conditions and give the projective measurements needed to violate maximally.
A common problem in Bell-type experiments is the well-known detection loophole: if the detection efficiencies are not perfect and if one simply postselects 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 postselected 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 postselected local polytope in the Clauser-Horne-Shimony-Holt scenario, where two parties have binary inputs and outcomes. Our approach gives interesting insights on the detection loophole problem.