The no-signalling principle is a fundamental assumption in Bell-inequality and quantum-steering experiments. Nonetheless, experimental imperfections can lead to apparent violations beyond those expected from finite-sample statistics. Here, we propose extensions of local hidden variable and local hidden state theories that allow for bounded, operationally quantifiable, amounts of signalling. We show how non-classicality tests can be developed for these models, both through exact methods based on the full set of observed statistics and through corrections to the standard Bell and steering inequalities. We demonstrate the applicability of these methods via two scenarios that feature apparent signalling: an IBM quantum processor and post-selected data from inefficient detectors.
The no-signalling principle is a fundamental assumption in Bell-inequality and quantum-steering experiments. Nonetheless, experimental imperfections can lead to apparent violations beyond those expected from finite-sample statistics. Here, we propose extensions of local hidden variable and local hidden state theories that allow for bounded, operationally quantifiable, amounts of signalling. We show how non-classicality tests can be developed for these models, both through exact methods based on the full set of observed statistics and through corrections to the standard Bell and steering inequalities. We demonstrate the applicability of these methods via two scenarios that feature apparent signalling: an IBM quantum processor and post-selected data from inefficient detectors.
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.
The seminal 1935 paper by Einstein, Podolsky, and Rosen (EPR) sparked a fundamental debate in quantum mechanics about the completeness of physical reality through the EPR paradox. They questioned local realism by suggesting that particles could exhibit instantaneous correlations across any distance, challenging classical physics. Bell's theorem (1964) argued that no local hidden variable theory could replicate quantum predictions, introducing Bell's inequalities and their experimental tests. These tests, particularly through violations of Bell's inequalities, confirmed quantum predictions of non-local correlations that local realism could not account for. Quantum theory, while supporting such correlations, adheres to the no-signaling principle to prevent faster-than-light communication and preserve causality. This paper presents a projection-based method for estimating the no-signaling distribution in weakly signaling data, offering a more efficient alternative to traditional methods. Our theoretical and experimental analysis demonstrates that CHSH parameters, indicative of quantum non-locality, remain unchanged even when a weakly signaling distribution is projected onto a non-signaling subspace, thereby upholding the intrinsic properties of quantum experiments.
This is the first of two papers that attempt to comprehensively analyse superdeterministic hidden-variables models of Bell correlations. We first give an overview of superdeterminism and discuss various criticisms of it raised in the literature. We argue that the most common criticism, the violation of ‘free-will’, is incorrect. We take up Bell’s intuitive criticism that these models are ‘conspiratorial’. To develop this further, we introduce non-equilibrium extensions of superdeterministic models. We show that the measurement statistics of these extended models depend on the physical system used to determine the measurement settings. This suggests a fine-tuning in order to eliminate this dependence from experimental observation. We also study the signalling properties of these extended models. We show that although they generally violate the formal no-signalling constraints, this violation cannot be equated to an actual signal. We therefore suggest that the so-called no-signalling constraints be more appropriately named the marginal-independence constraints. We discuss the mechanism by which marginal-independence is violated in superdeterministic models. Lastly, we consider a hypothetical scenario where two experimenters use the apparent-signalling of a superdeterministic model to communicate with each other. This scenario suggests another conspiratorial feature peculiar to superdeterminism. These suggestions are quantitatively developed in the second paper.
Although the original EPR paradox was formulated in terms of position and momentum, most studies of these phenomena have focused on measurement scenarios with only a discrete number of possible measurement outcomes. Here, we present a framework for studying non-locality that is agnostic to the dimension of the physical systems involved, allowing us to probe purely continuous-variable, discrete-variable, and hybrid continuous/discrete non-locality. Our approach allows us to find the optimal Bell inequality for any given measurement scenario and quantifies the amount of non-locality that is present in measurement statistics. This formalism unifies the existing literature on continuous-variable non-locality and allows us to identify new states in which Bell non-locality can be probed through homodyne detection. Notably, we find the first example of two-mode continuous-variable non-locality that cannot be mapped to a binary Bell inequality. Moreover, we provide several examples of simple hybrid DV-CV entangled states that could lead to near-term violation of Bell inequalities.
This paper demonstrates the capability to simulate EPR experiments in a realistic way (i.e. without transmission of information between the source and the detector) and confirms even more the Bell’s theorem, by demonstrating that effectively non-local variables are required to violate the Bell’s inequality. Unlike all EPR experiments, this modeling instead of simply noting the inadequacy of a local modeling proposes a new way of non-local idealization. These simulations work on spreadsheet by idealizing objects with a non-linear notion, the extended object. This allows to set up a new principle of locality (or non-locality) and a new meaning on the loopholes of EPR experiments. It sheds new light on several concepts of Quantum Mechanics (QM): the quantum states (would represent an equivalence class of many different indistinguishable objects seen as equivalent in terms of measurement), the wave function (would represent a probability on this equivalence class seen as a single set despite different probabilities of happening for the individual objects composing this equivalence class), the measurement (in our simulation, the measured object is not modified), the superposition of states (in our simulation, Schrödinger's cat is never both dead and alive). These simulations lead to several kinds of EPR experiments with violations of the Bell’s inequalities with values different from the QM, allow to realize entanglements with more than 2 objects (whatever the number of Alice and Bob) or even with more than 2 results (beyond the only “+1” or “-1” outputs). Many questions remain to be explored in the physics domain on quantum phenomena (teleportation, encryption, quantum computing, etc.…), but this theoretical approach also reveals the necessity of developing new non-linear tools in mathematics domain and seems to show that QM could finally be a limit case of a more fundamental nonlinear physics theory founded on these extended objects. This observation is strangely reminiscent of developments such as string theory.
In this work, we deal with the relaxation of two central assumptions in standard locally realistic hidden variable (LRHV) inequalities: free will in choosing measurement settings, and the presence of perfect detectors at the measurement devices. Quantum correlations violating LRHV inequalities are called quantum nonlocal correlations. In principle, in an adversarial situation, there could be a hidden variable introducing bias in the selection of measurement settings, but observers with no access to that hidden variable could be unaware of the bias. In practice, however, detectors do not have perfect efficiency. A main focus of this paper is the introduction of the framework in which given a quantum state with nonlocal behavior under constrained free will, we can determine the threshold values of detector parameters (detector inefficiency and dark counts) such that the detectors are robust enough to certify nonlocality. We also introduce a class of LRHV inequalities with constrained free will, and we discuss their implications in the testing of quantum nonlocal correlations.