This paper investigates whether Bell-like and CHSH-type correlations can arise within a fully classical stochastic framework. The study develops an event-based model using competing Poisson processes, exponential waiting-time races, intensity-to-probability mappings, and Monte Carlo simulation to generate binary measurement outcomes and joint correlations. The model examines how contextual measurement settings, classical interference, and sinusoidal intensity relationships can produce correlation structures commonly associated with quantum-mechanical experiments. Particular attention is given to singlet-like cosine correlations, joint probability distributions, and the statistical conditions under which CHSH correlations emerge. Rather than assuming quantum probability rules at the outset, the analysis constructs the correlations from underlying stochastic events and evaluates the resulting synthetic data statistically. The work is intended as an exploratory contribution to the foundations of quantum mechanics, especially the interpretation of Bell inequalities, contextuality, and the distinction between quantum and classical stochastic descriptions.
This paper investigates whether Bell-like and CHSH-type correlations can arise within a fully classical stochastic framework. The study develops an event-based model using competing Poisson processes, exponential waiting-time races, intensity-to-probability mappings, and Monte Carlo simulation to generate binary measurement outcomes and joint correlations. The model examines how contextual measurement settings, classical interference, and sinusoidal intensity relationships can produce correlation structures commonly associated with quantum-mechanical experiments. Particular attention is given to singlet-like cosine correlations, joint probability distributions, and the statistical conditions under which CHSH correlations emerge. Rather than assuming quantum probability rules at the outset, the analysis constructs the correlations from underlying stochastic events and evaluates the resulting synthetic data statistically. The work is intended as an exploratory contribution to the foundations of quantum mechanics, especially the interpretation of Bell inequalities, contextuality, and the distinction between quantum and classical stochastic descriptions.
This preprint presents an independent investigation of Bell-type correlations and the physical assumptions underlying Bell/CHSH analyses. It develops a phase-based model intended to examine whether observed quantum correlations can be reproduced through a different underlying physical mechanism than is usually assumed in standard interpretations. The manuscript focuses on the mathematical structure of the proposed model, its relationship to Bell-type inequalities, and the conditions under which its predictions may differ from standard quantum-mechanical expectations. This is version 1.2 of the preprint and has not yet undergone formal peer review.
Abstract In classical stochastic theory, the joint probability distributions of a stochastic process obey by definition the Kolmogorov consistency conditions. Interpreting such a process as a sequence of physical measurements with probabilistic outcomes, these conditions reflect that the measurements do not alter the state of the underlying physical system. Prominently, this assumption has to be abandoned in the context of quantum mechanics, yet there are also classical processes in which measurements influence the measured system. Here, conditions that characterize uniquely classical processes that are probed by a reasonable class of such invasive measurements are derived. We then analyze under what circumstances such classical processes can simulate the statistics arising from quantum processes associated with informationally‐complete measurements. It is expected that this investigation will help build a bridge between two fundamental traits of non‐classicality, namely, coherence and contextuality.
A novel approach for analyzing "classical" alternatives to quantum mechanics for explaining the statistical results of an EPRB-like experiment is proposed. This perspective is top-down instead of bottom-up. Rather than beginning with an inequality derivation, a hierarchy of model types is constructed, each distinguished by appropriately parameterized conditional probabilities. This hierarchy ranks the "classical" model types in terms of their ability to reproduce QM statistics or not. The analysis goes beyond the usual consideration of model types that "fall short" (i.e., satisfy all of the CHSH inequalities) to ones that are "excessive" (i.e., not only violate CHSH but even exceed a Tsirelson bound). This approach clearly shows that noncontextuality is the most general property of an operational model that blocks replication of at least some QM statistical predictions. Factorizability is naturally revealed to be a special case of noncontextuality. The same is true for the combination of remote context independence and outcome determinism (RCI+OD). It is noncontextuality that determines the dividing line between "classical" model instances that satisfy the CHSH inequalities and those that don't. Outcome deterministic operational models are revealed to be the "building blocks" of all the rest, including quantum mechanical, noncontextual, and contextual ones. The set of noncontextual model instances is exactly the convex hull of all 16 RCI+OD model instances, and furthermore, the set of all model instances, including all QM ones, is equal to the convex hull of the 256 OD model instances. It is shown that, under a mild assumption, the construction of convex hulls of finite ensembles of OD model instances is (mathematically) equivalent to the traditional hidden variables approach. Plots and figures provide visual affirmation of many of the results.
Violations of Bell inequalities are commonly interpreted as evidence for either nonlocal dynamics or the failure of classical realism. In this work, we show that Bell-type correlations can arise without invoking superluminal influences, retrocausality, or local hidden variables, provided that the mapping between underlying configurations and observable outcomes is non-injective. We study a class of effective models in which multiple underlying states correspond to the same observable description. Such non-injective projections generically induce non-factorizable joint probability structures at the level of observables, even when the underlying description itself is fully local and deterministic. As a result, the statistical assumptions required to derive Bell inequalities are violated at the effective level, while locality is preserved at the underlying level. Within this framework, quantum correlations emerge as structural features of projected descriptions rather than as signatures of nonlocal causation. We clarify the precise assumption of Bell’s theorem that fails in this setting and show how classical behavior is recovered in regimes where the effective projection becomes approximately injective, suppressing nonlocal correlations through decoherence and environmental coarse-graining. Our results suggest that Bell inequality violations need not reflect a fundamental nonlocality of nature, but may instead indicate intrinsic information loss induced by non-injective mappings between underlying configurations and observable descriptions.
Bell nonlocality is the experimentally testable incompatibility between quantum correlations and explanations based on local hidden variables. Its significance now extends beyond the foundations of quantum mechanics: a Bell-inequality violation can certify entanglement, randomness, measurement incompatibility, and cryptographic security without requiring a detailed model of the devices producing the observed data. This article develops the mathematical structure of Bell experiments and explains its role in device-independent quantum information. Conditional probability distributions are classified into local, quantum, and no-signalling sets; the Clauser-Horne-Shimony-Holt inequality is derived, and its classical, quantum, and algebraic limits are distinguished. Tsirelson’s bound is obtained from the CHSH operator, while Popescu-Rohrlich correlations demonstrate that no-signalling alone does not identify the quantum set. The relation between entanglement and nonlocality is examined through local entangled states, hidden nonlocality, and activation. Nonlocal games, semidefinite relaxations, self-testing, certified randomness, and device-independent quantum key distribution are treated as operational consequences of Bell violation. Particular attention is given to entropy accumulation, finite-data security, loophole-free experiments, and the limitations of device independence. Bell nonlocality emerges as both a foundational separation between classical and quantum causal models and a quantitative resource for cryptographic and informational tasks.
We show that by taking into account randomness of realization of experimental contexts it is possible to construct common Kolmogorov space for data collected for these contexts, although they can be incompatible. We call such a construction "Kolmogorovization" of contextuality. This construction of common probability space is applied to Bell's inequality. It is well known that its violation is a consequence of collecting statistical data in a few incompatible experiments. In experiments performed in quantum optics contexts are determined by selections of pairs of angles (θi,θ'j) fixing orientations of polarization beam splitters. Opposite to the common opinion, we show that statistical data corresponding to measurements of polarizations of photons in the singlet state, e.g., in the form of correlations, can be described in the classical probabilistic framework. The crucial point is that in constructing the common probability space one has to take into account not only randomness of the source (as Bell did), but also randomness of context-realizations (in particular, realizations of pairs of angles (θi, θ'j)). One may (but need not) say that randomness of "free will" has to be accounted for.