We extend the electromagnetic mode competition framework to entangled photon pairs. The framework derives the Born rule from classical electromagnetism plus a single empirical input: each emission produces exactly one detection (single-event exclusivity). Here we show that the same postulates, applied to the joint mode structure of spontaneous parametric down-conversion, reproduce the quantum predictions for polarisation-entangled pairs: the sin²(θ₁ − θ₂) coincidence probability and the maximal CHSH violation S = 2√2. We also derive GHZ correlations and the Hong–Ou–Mandel dip. The framework dissolves the mystery of Bell correlations: there are not two photons coordinating across space, but one extended electromagnetic mode undergoing one detection event. Alice and Bob are endpoints of a single structure; their correlation is built in from the moment of emission and enforced by energy conservation. No action at a distance occurs—one detector absorbs the energy, and the shared pool is depleted. Bell's inequality is violated not because of nonlocality, but because Bell's factorisation assumption does not apply to measurements on a single extended object.
We extend the electromagnetic mode competition framework to entangled photon pairs. The framework derives the Born rule from classical electromagnetism plus a single empirical input: each emission produces exactly one detection (single-event exclusivity). Here we show that the same postulates, applied to the joint mode structure of spontaneous parametric down-conversion, reproduce the quantum predictions for polarisation-entangled pairs: the sin²(θ₁ − θ₂) coincidence probability and the maximal CHSH violation S = 2√2. We also derive GHZ correlations and the Hong–Ou–Mandel dip. The framework dissolves the mystery of Bell correlations: there are not two photons coordinating across space, but one extended electromagnetic mode undergoing one detection event. Alice and Bob are endpoints of a single structure; their correlation is built in from the moment of emission and enforced by energy conservation. No action at a distance occurs—one detector absorbs the energy, and the shared pool is depleted. Bell's inequality is violated not because of nonlocality, but because Bell's factorisation assumption does not apply to measurements on a single extended object.
Single-event exclusivity—the fact that one photon emission produces exactly one detection—is confirmed by anti-bunching experiments but lacks a dynamical explanation. Standard quantum mechanics postulates it; we derive it. The source is modeled as a bistable system: excited and ground configurations separated by a potential barrier, coupled bidirectionally to the electromagnetic mode. The mode exerts a force on the source that stabilizes the excited configuration. Absorbers compete stochastically; when one crosses threshold and switches to strong coupling, it rapidly drains the mode. The vanishing mode amplitude removes the force stabilizing the source, triggering its snap transition and delivering all of the source's energy ΔE to the winner. The framework extends to entangled photon pairs by treating the two-photon state as a joint mode in configuration space. The joint mode drains globally when any absorber pair triggers, producing correlations that violate Bell inequalities. The derivation uses bistable dynamics and threshold physics; nonlocality enters through the configuration-space mode structure, not through any modification of the dynamics.
In this paper we study photon entanglement in the framework of Maxwell–Schwartz field theory. The ambient state space is the complex Maxwellian distribution space W=S′(M4,C3), whose elements are fields of the form F=E→+icB→. Polarization is realized as a two-dimensional complex subspace of W, generated by suitable linearly polarized Maxwellian solutions associated with opposite propagation directions. This yields canonical polarization sectors PA and PB, each naturally isomorphic to C2. Within this setting, the Bell singlet state is represented by a non-factorizable tensorial Maxwellian field in PA⊗PB⊂W⊗W. By means of the induced rotated polarization bases, the standard joint probabilities of the photon polarization experiment are recovered exactly, and the correlation law E(a,b)=−cos(2(a−b)) is obtained. Consequently, the usual CHSH value 22 is reproduced in the Maxwell–Schwartz framework. To clarify the meaning of this violation, we first formulate the CHSH inequality in a purely measure-theoretic form, as a theorem about four correlators represented on a single probability space by bounded measurable functions. We then show that the correlators produced by the intrinsic Maxwellian Bell state do not admit such a common representation. The obstruction is structural: the ontic state is a global non-product field configuration, and the four correlations arise from different polarization resolutions of the same tensorial Maxwellian state. A second main result concerns the Born rule. For L2 scalar quantum states in the domain of the Maxwellian correspondence, we prove that the squared Hilbert norm, times the constant ε0, coincides with the electromagnetic energy of the associated field. This leads to an energy interpretation of the Born rule: the Born probability density is identified with the normalized electromagnetic energy density up to an interference term depending on the chosen Maxwell–Schwartz isomorphism, which assumes the role of a quantum context. In the context of the Aspect and collaborators’ experiment, we prove that, on the other hand, the polarization probabilities become energy contributions of the corresponding field components. These results show that photon entanglement, Bell inequality violation, and the Born rule admit a coherent interpretation within Maxwell–Schwartz field theory, where the basic ontological objects are electromagnetic-like fields rather than abstract state vectors.
Motivated by very recent experiments, we consider a scenario ``\`a la Bell'' in which two protagonists test the Clauser-Horne-Shimony-Holt (CHSH) inequality using a photon-pair source based on spontaneous parametric down conversion and imperfect photon detectors. The conventional wisdom says that (i) if the detectors have unit efficiency, the CHSH violation can reach its maximum quantum value of 22. To obtain the maximal possible violation, it suffices that the source emits (ii) maximally entangled photon pairs (iii) in two well-defined single modes. Through a nonperturabive calculation of nonlocal correlations, we show that none of these statements are true. By providing the optimal pump parameters, measurement settings and state structure for any detection efficiency and dark count probability, our results give the recipe to close all the loopholes in a Bell test using photon pairs.
Abstract The dipole-wave framework proposes that all fundamental particles are point-like entities with oscillating dipole moments that radiate real guiding waves into three-dimensional space. Previous work has shown that this framework reduces to the Schrödinger equation for single massive particles and provides a deterministic account of single-photon interference. In this paper, we extend the dipole-wave framework to two-photon Bell-type correlations. We derive the joint detection probability for two photons with orthogonal dipoles and show that the quantum mechanical Bell curve—P = ½sin²(Δ)—is the interference-only, background-subtracted, normalized component of the dipole-wave prediction. The dipole-wave theory yields a raw coincidence rate proportional to 1 + 2sin²(Δ), which contains the Bell curve after background subtraction and normalization. The π/2 shift that relates this to the aligned-dipole case is derived from the orthogonality of the dipoles produced in Type-II parametric down-conversion. We further show that the four outcome probabilities follow from the geometry of the orthogonal dipoles and lead to the standard Bell correlation function E(a,b) = −cos(2Δ) and the CHSH violation |S| = 2√2. We also compare the raw prediction with published raw coincidence data and find that neither the simple raw expression nor the raw Bell curve matches the observed raw rates. This is expected, because the Bell curve is a derived quantity that emerges only after background subtraction and normalization. The central result remains: the dipole-wave framework reproduces the normalized Bell curve, the full correlation function, and the CHSH violation.
Quantum physics can provide sources of randomness that can be certified as being uncorrelated to any outside process or variable, i.e. sources of private randomness, based on a violation of a Bell inequality. Initial experimental realizations of such sources of certified randomness are based on atomic or atomic-like systems, but suffer from impractically low generation rates for most applications. High efficiency infrared photodetectors and photon pair sources permitted experimental demonstrations of loophole free violation of the Bell inequality using photons. The random bit generation rate for these setups was on the order of tens per second, where the main limitation is the fixed repetition rate of the photon pair source combined with the small violation observed. In our experiment, we close the detection loophole with a system efficiency over 82%. The source of entangled photon pairs is based on continuously pumped spontaneous parametric down conversion. We estimate a collection efficiency of ≈90% into single mode fibers and detect photons with a transition edge sensors. Detection events are time-tagged and organized into time bins, for which we consider four possible outcomes: one or more detections at Alice’s side, one or more detections at Bob’s side, one or more detections at both Alice’s and Bob’s, and no detection in either channel. These events eventually lead to a CHSH-type Bell inequality that is violated for a range of time bin widths. With such an arrangement, we reach asymptotic device-indepentnet random bit generation rates on the order of 1000 bits per second.
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.