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Bell's Theorem (Stanford Encyclopedia of Philosophy) Bell’s Theorem First published Wed Jul 21, 2004; substantive revision Wed Mar 13, 2019 Bell’s Theorem is the collective name for a family of results, all of which involve the derivation, from a condition on probability distributions inspired by considerations of local causality, together with auxiliary assumptions usually thought of as mild side-assumptions, of probabilistic predictions about the results of spatially separated experiments that conflict, for appropriate choices of quantum states and experiments, with quantum mechanical predictions. These probabilistic predictions take the form of inequalities that must be satisfied by correlations derived from any theory satisfying the conditions of the proof, but which are violated, under certain circumstances, by correlations calculated from quantum mechanics. Inequalities of this type are known as Bell inequalities, or sometimes, Bell-type inequalities. Bell’s theorem shows that no theory that satisfies the conditions imposed can reproduce the probabilistic predictions of quantum mechanics under all circumstances. The principal condition used to derive Bell inequalities is a condition that may be called Bell locality, or factorizability. It is, roughly, the condition that any correlations between distant events be explicable in local terms, as due to states of affairs at the common source of the particles upon which the experiments are performed. See section 3.1 for a more careful statement. The incompatibility of theories satisfying the conditions that entail Bell inequalities with the predictions of quantum mechanics permits an experimental adjudication between the class of theories satisfying those conditions and the class, which includes quantum mechanics, of theories that violate those conditions. At the time that Bell formulated his theorem, it was an open question whether, under the circumstances considered, the Bell inequality-violating correlations predicted by quantum mechanics were realized in nature. Beginning in the 1970s, there has been a series of experiments of increasing sophistication to test whether the Bell inequalities are satisfied. With few exceptions, the results of these experiments have confirmed the quantum mechanical predictions, violating the relevant Bell Inequalities. Until recently, however, each of these experiments was vulnerable to at least one of two loopholes, referred to as the communication, or locality loophole, and the detection loophole (see section 5). Finally, in 2015, experiments were performed that demonstrated violation of Bell inequalities with these loopholes blocked. This has consequences for our physical worldview; the conditions that entail Bell inequalities are, arguably, an integral part of the physical worldview that was accepted prior to the advent of quantum mechanics. If one accepts the lessons of the experimental results, then some one or other of these conditions must be rejected. For much of the interval between the original publication of Bell’s theorem and the experiments of Aspect and his collaborators, interest in Bell’s theorem was confined to a handful of physicists and philosophers. During that period, much of the discussions on the foundations of physics occurred in a mimeographed publication entitled Epistemological Letters. In the wake of the Aspect experiments (Aspect, Grangier, and Roger, 1982; Aspect, Dalibard, and Roger 1982), there was considerable philosophical discussion of the implications of Bell’s theorem; see Cushing and McMullin, eds. (1989), for a snapshot of the philosophical discussions of the time. Interest was also stimulated by the publication of a collection of Bell’s papers on the foundations of quantum mechanics (Bell 1987b). The rise of quantum information theory, which, among other things, explores the ways in which quantum entanglement can be used to perform tasks that would not be feasible classically, also contributed to raising awareness of the significance of Bell’s theorem, which throws into sharp relief the difference between quantum entanglement-based correlations and classical correlations. The year 2014 was the 50th anniversary of the original publication of Bell’s theorem, and was marked by a special issue of Physical Review A (47, number 42, 24 October 2014), a collection of essays (Bell and Gao, eds., 2016), and a large conference comprising over 400 attendees (see Bertlmann and Zeilinger, eds., 2017). The interested reader is urged to consult these collections for an overview of current discussions on topics surrounding Bell’s theorem. • 1. Introduction • 2. Proof of a Theorem of Bell’s Type • 3. The Assumptions of the Proof o 3.1. Locality and causality assumptions o 3.2. Supplementary assumptions o 3.3. On “local realism” • 4. Early Experimental Tests of Bell’s Inequalities • 5. The Communication and Detection Loopholes, and their Remedies o 5.1. The Communication Loophole, and its remedy o 5.2. The Detection Loophole and its remedy o 5.3 Loophole-free tests • 6. Some Variants of Bell’s Theorem • 7. Significance for Quantum Information Theory • 8. Philosophical/Metaphysical Implications o 8.1. Options left open by experimental violation of Bell Inequalities o 8.2. Quantum mechanics and relativity • Bibliography • Academic Tools • Other Internet Resources • Related Entries 1. Introduction In 1964 John S. Bell, a native of Northern Ireland and a staff member of CERN (European Organisation for Nuclear Research) whose primary research concerned theoretical high energy physics, published a paper (Bell 1964) in the short-lived journal Physics, which eventually transformed the study of the foundation of quantum mechanics. The paper showed, under conditions that were relaxed in later work by Bell (1971, 1976) himself and by his followers (Clauser, Horne, Shimony, and Holt 1969, Clauser and Horne 1974, Aspect 1983, Mermin 1986), that, on the assumption of certain auxiliary conditions, no physical theory that satisfies a certain locality condition, which may be called Bell locality, can fully reproduce the quantum probabilities for outcomes of experiments. Since that time, variants on the theorem, with family resemblances, have been formulated. “Bell’s Theorem” is the collective name for the entire family. The theorem has roots in Bell’s investigations into the status of the hidden-variables program, and on earlier work concerning quantum entanglement. Bell presented several formulations of the theorem over the years (Bell 1964, 1971, 1976, 1990), and variants of it have been presented by others. The original derivation (1964) relied on a set-up involving perfect anticorrelation of the results of spin experiments on pairs of spin-1/2 particles prepared in the singlet state. Under this condition, the Bell locality condition entails that outcomes of experiments are predetermined by the complete specification of state, a condition which we will call outcome determinism (OD). Clauser, Horne, Shimony and Holt (1969) derived an inequality, the CHSH inequality, that does not require this assumption. Though in their proof they employed the condition OD, this condition is unnecessary for the derivation of the inequality, as shown by Bell (1971), who provided a proof of the CHSH inequality that relies on neither the assumption of perfect anticorrelation nor an assumption of outcome determinism. One line of investigation in the prehistory of Bell’s Theorem is Bell’s examination of the hidden-variables program. This program involves supplementation of the quantum mechanical state of a system by further “elements of reality”, or “hidden variables”, the incompleteness of the quantum state being the explanation for the statistical character of quantum mechanical predictions concerning the system. A pioneering version of a hidden variables theory was proposed by Louis de Broglie in 1926–7 (de Broglie 1927, 1928), and revived by David Bohm in 1952 (Bohm 1952; see also the entry on Bohmian mechanics). In a paper (Bell 1966) that was written before the one in which Bell’s theorem first appeared, but, due to an editorial mishap, was published later, Bell raises the question of the viability of a hidden-variables theory that reproduces the statistical predictions of quantum mechanics via averaging over better defined states that uniquely determine the result of any experiment that could be performed. In this paper he examines several theorems that had been presented as no-go theorems for theories of this sort, and supplements them with one of his own, a theorem that was independently formulated by Specker (1960), and published by Kochen and Specker (1967), and has come to be known as the Kochen-Specker Theorem or Bell-Kochen Specker Theorem (see entry on the Kochen-Specker Theorem for more details). In each case Bell argues that the proof contains premises that are physically unwarranted. The Bell-Kochen-Specker theorem is a corollary of Gleason’s theorem (Gleason 1957), though Bell and Kochen-Specker obtain it directly, and not via Gleason’s theorem, whose proof is considerably more intricate. The question addressed by Gleason has to do with assignments of probabilities to closed subspaces of a Hilbert space (or, equivalently, to projection operators onto such subspaces), such that the probabilities assigned to orthogonal projections are additive. Gleason proved that, in a Hilbert space of dimension 3 or greater, any such assignment of probabilities can be represented by a density operator. The BKS theorem deals with the special case in which the assignments are confined to the values 1
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