Nonlocality was discovered by John Bell in 1964, in the context of the debates about quantum theory, but is a phenomenon that can be studied in its own right. Its observation proves that measurements are not revealing pre-determined values, falsifying the idea of “local hidden variables” suggested by Einstein and others. One is then forced to make some radical choice: either nature is intrinsically statistical and individual events are unspeakable, or our familiar space-time cannot be the setting for the whole of physics. As phenomena, nonlocality and its consequences will have to be predicted by any future theory, and may possibly play the role of foundational principles in these developments. But nonlocality has found a role in applied physics too: it can be used for “device-independent” certification of the correct functioning of random number generators and other devices. After a self-contained introduction to the topic, this monograph on nonlocality presents the main tools and results following a logical, rather than a chronological, order.
Nonlocality was discovered by John Bell in 1964, in the context of the debates about quantum theory, but is a phenomenon that can be studied in its own right. Its observation proves that measurements are not revealing pre-determined values, falsifying the idea of “local hidden variables” suggested by Einstein and others. One is then forced to make some radical choice: either nature is intrinsically statistical and individual events are unspeakable, or our familiar space-time cannot be the setting for the whole of physics. As phenomena, nonlocality and its consequences will have to be predicted by any future theory, and may possibly play the role of foundational principles in these developments. But nonlocality has found a role in applied physics too: it can be used for “device-independent” certification of the correct functioning of random number generators and other devices. After a self-contained introduction to the topic, this monograph on nonlocality presents the main tools and results following a logical, rather than a chronological, order.
Abstract Nonlocality was discovered by John Bell in 1964, in the context of the debates about quantum theory, but is a phenomenon that can be studied in its own right. Its observation proves that measurements are not revealing pre-determined values, falsifying the idea of “local hidden variables” suggested by Einstein and others. One is then forced to make some radical choice: either nature is intrinsically statistical and individual events are unspeakable, or our familiar space-time cannot be the setting for the whole of physics. As phenomena, nonlocality and its consequences will have to be predicted by any future theory, and may possibly play the role of foundational principles in these developments. But nonlocality has found a role in applied physics too: it can be used for “device-independent” certification of the correct functioning of random number generators and other devices. After a self-contained introduction to the topic, this monograph on nonlocality presents the main tools and results following a logical, rather than a chronological, order.
The Bell 1964 theorem states that nonlocality is a necessary feature of hidden variable theories that reproduce the statistical predictions of quantum mechanics. In view of the no-go theorems for non-contextual hidden variable theories already existing up to 1964, and due to Gleason and Bell, one is forced to acknowledge the contextual character of the hidden variable theory which the Bell 1964 theorem refers to. Both the mathematical and the physical justifications of this contextualism are reconsidered. Consequently, the role of contextualism in recent no-hidden-variables proofs and the import of these proofs are investigated. With reference to the physical intuition underlying contextualism, the possibility is considered whether a context-dependence of individual measurement results is compatible with context-independence of the statistics of measurement results.
Various interpretations of quantum mechanics, favored (or neglected) by John Bell in the context of his nonlocality theorem, are compared and discussed. Varenna 1970 I met John Bell for the first time at the Varenna conference of 1970 [1]. I had been invited on the suggestion of Eugene P. Wigner, who had already helped me to publish my first paper on the concept that was later called decoherence – to appear in the first issue of Foundations of Physics a few months after the conference [2]. This concept arose from my conviction, based on many applications of quantum mechanics to composite systems under various conditions, that Schrödinger's wave function in configuration space, or more generally the superposition principle, is universally valid and applicable. In particular, stable narrow wavepackets can represent classical configurations, while their superpositions define novel individual properties – such as “momentum,” defined as a plane wave superposition of different positions. Superpositions of macroscopically different properties, on the other hand, are regularly irreversibly “dislocalized” (distributed over many degrees of freedom) by means of interactions described by the Schrödinger equation. The corresponding disappearance of certain local superpositions (“decoherence”) seems to explain the phenomenon of a classical world as well as the apparent occurrence of quantum jumps or stochastic “events” – see Sect. 20.4 or [3] for a historical overview of the conceptual development of quantum theory. So I had never felt any motivation to think of “hidden variables” or any other physics behind the successful wave function. Therefore, I was very surprised on my arrival in Varenna to hear everybody discuss Bell's inequality. It had been published a few years before the conference, but I had either not noticed it or not regarded it as particularly remarkable until then. As this inequality demonstrates that the predictions of quantum theory would require any conceivable reality possibly underlying the nonlocal wave function to be nonlocal itself, I simply found my conviction that the latter describes individual reality confirmed. For example, I had often discussed the conservation of total spin or angular momentum in an individual decay process, which requires nonlocal entanglement between the fragments at any distance in a form that was later called a “Bell state”.
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.
Bell's theorem proves that quantum theory is inconsistent with local physical models. It has propelled research in the foundations of quantum theory and quantum information science. As a fundamental feature of quantum theory, it impacts predictions far beyond the traditional scenario of the Einstein-Podolsky-Rosen paradox. In the last decade, the investigation of nonlocality has moved beyond Bell's theorem to consider more sophisticated experiments that involve several independent sources which distribute shares of physical systems among many parties in a network. Network scenarios, and the nonlocal correlations that they give rise to, lead to phenomena that have no counterpart in traditional Bell experiments, thus presenting a formidable conceptual and practical challenge. This review discusses the main concepts, methods, results and future challenges in the emerging topic of Bell nonlocality in networks.
Abstract This chapter is a self-contained introduction to Bell nonlocality. After a very quick historical motivation, the setting and the definition of nonlocality are given. Five classic examples of Bell tests are then introduced, including the CHSH and the GHZ tests. The remainder of the chapter is a critical reflection that contains two topics: a discussion of possible loopholes that should be avoided, and an elementary presentation of the main interpretations given to the phenomenon of nonlocality.