The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
An investigated process can be studied in terms of the causal relations among the involved variables, representing it as a causal model. Some causal models are particularly relevant, since they can be tested through mathematical constraints between the joint probability distributions of the observables. This is a valuable tool because, if some data violates the constraints of a causal model, the implication is that the observed statistics is not compatible with that causal structure. Strikingly, when non-classical correlations come to play, a discrepancy between classical and quantum causal predictions can arise, producing a quantum violation of the classical causal constraints. The simplestscenario admitting such quantum violation is given by the instrumental causal processes. Here, we experimentally violate an instrumental test on a photonic platform and show how the quantum correlations violating the CHSH inequality can be mapped into correlations violating an instrumental test, despite the different forms of non-locality they display. Indeed, starting from a Bell-like scenario, we recover the violation of the instrumental scenario through a map between the two behaviours, which includes a post-selection of data and then we test an alternative way to violate the CHSH inequality, adopting the instrumental process platform.
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.
Nonlocality and contextuality are at the root of conceptual puzzles in quantum mechanics, and they are key resources for quantum advantage in information-processing tasks.Bell nonlocality is best understood as the incompatibility between quantum correlations and the classical theory of causality, applied to relativistic causal structure.Contextuality, on the other hand, is on a more controversial foundation.In this work, I provide a common conceptual ground between nonlocality and contextuality as violations of classical causality.First, I show that Bell inequalities can be derived solely from the assumptions of no signaling and no fine-tuning of the causal model.This removes two extra assumptions from a recent result from Wood and Spekkens and, remarkably, does not require any assumption related to independence of measurement settings-unlike all other derivations of Bell inequalities.I then introduce a formalism to represent contextuality scenarios within causal models and show that all classical causal models for violations of a Kochen-Specker inequality require fine-tuning.Thus, the quantum violation of classical causality goes beyond the case of spacelike-separated systems and already manifests in scenarios involving single systems.
Since Bell's theorem, it is known that local realism fails to explain quantum phenomena. Bell inequality violations manifestly show the incompatibility of quantum theory with classical notions of cause and effect. As recently found, however, the instrumental scenario-a pivotal tool in causal inference-allows for nonclassicality signatures going beyond this paradigm. If we are not limited to observational data and can intervene in our setup, then we can witness quantum violations of classical bounds on the causal influence among the involved variables even when no Bell-like violation is possible. That is, through interventions, the quantum behavior of a system that would seem classical can be demonstrated. Using a photonic setup-faithfully implementing the instrumental causal structure and switching between observation and intervention run by run-we experimentally witness such a nonclassicality. We also test quantum bounds for the causal influence, showing that they provide a reliable tool for quantum causal modeling.
Nonlocality and contextuality are at the root of conceptual puzzles in quantum mechanics, and are key resources for quantum advantage in information-processing tasks. Bell nonlocality is best understood as the incompatibility between quantum correlations and the classical theory of causality, applied to relativistic causal structure. Contextuality, on the other hand, is on a more controversial foundation. In this work, I provide a common conceptual ground between nonlocality and contextuality as violations of classical causality. First, I show that Bell inequalities can be derived solely from the assumptions of no-signalling and no-fine-tuning of the causal model. This removes two extra assumptions from a recent result from Wood and Spekkens, and remarkably, does not require any assumption related to independence of measurement settings -- unlike all other derivations of Bell inequalities. I then introduce a formalism to represent contextuality scenarios within causal models and show that all classical causal models for violations of a Kochen-Specker inequality require fine-tuning. Thus the quantum violation of classical causality goes beyond the case of space-like separated systems, and manifests already in scenarios involving single systems.