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Formalizing Bell Nonlocality

Valerio Scarani

Published Aug 7, 2019• 1 citation
Publisher

Abstract

Abstract This chapter covers the essential mathematical tools for the study of nonlocality. It begins with the main object under study: a collection of several probability distributions usually called “behavior”. The crucial definition of locality is then given, followed by Fine’s theorem that relates local behaviors to pre-existing values and clarifies the role of local deterministic processes. In turn, one finds that local behaviors belong to a polytope, whose facets are Bell inequalities. The simplest scenario, called CHSH, is studied in detail.

DOI
10.1093/oso/9780198788416.003.0002

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Abstract This chapter covers the essential mathematical tools for the study of nonlocality. It begins with the main object under study: a collection of several probability distributions usually called “behavior”. The crucial definition of locality is then given, followed by Fine’s theorem that relates local behaviors to pre-existing values and clarifies the role of local deterministic processes. In turn, one finds that local behaviors belong to a polytope, whose facets are Bell inequalities. The simplest scenario, called CHSH, is studied in detail.
Bell nonlocality of quantum states is an important resource in quantum information and then has various applications. It is usually detected by the violation of some Bell’s inequalities and the all-versus-nothing test. In the present paper, we aim to establish some mathematical methods for proving Bell nonlocality without inequalities, inspired by the work [Phys. Rev. Lett., 89, 080402 (2002)] regarding the GHZ paradox. For self-containedness, we recall the mathematical definition of Bell nonlocality proposed in [Sci. China-Phys. Mech. Astron. 62, 030311 (2019)] and then give some basic properties on it. Then we derive some necessary conditions for a multipartite state to be Bell local and obtain some sufficient conditions for a state to be Bell nonlocal in terms of “expectations” of local observables without invoking Bell inequalities. Unlike the standard approach to nonlocality detection based on violation of Bell inequalities, the obtained criteria are formulated in terms of certain relations for expectation values of local observables that are constructed from the well-known GHZ paradoxes.
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.
Introduction Fifty years after the publication of Bell's theorem, there remains some controversy regarding what the theorem is telling us about quantum mechanics, and what the experimental violations of Bell inequalities are telling us about the world. This chapter represents my best attempt to be clear about what I think the lessons are. In brief: There is some sort of nonlocality inherent in any quantum theory, and, moreover, in any theory that reproduces, even approximately, the quantum probabilities for the outcomes of experiments. But not all forms of nonlocality are the same; there is a distinction to be made between action at a distance and other forms of nonlocality, and I will argue that the nonlocality needed to violate the Bell inequalities need not involve action at a distance. Furthermore, the distinction between forms of nonlocality makes a difference when it comes to compatibility with relativistic causal structure. The Bell locality condition is a condition which, if satisfied, renders possible a completely local account of the correlations between outcomes of spatially separated experiments. Bell's theorem tells us that the probabilities that we compute from quantum mechanics do not admit of such an account. Modulo auxiliary assumptions that, in my opinion, should be noncontroversial, this yields the conclusion that there must be something nonlocal about any theory that yields the quantum mechanical probabilities or anything close to them. One way to violate the Bell locality condition – and this is the way that any deterministic theory must do it – is via straightforward action at a distance. However, we should also consider the possibility of a stochastic theorywith probabilistic laws that involve irreducible correlations. The sort of nonlocality inherent in such a theory involves relations between events that are significantly different from causal relations as usually conceived. Perhaps surprisingly, such a theory, unlike its deterministic cousins, need not invoke a distinguished relation of distant simultaneity. This makes a difference for the situation of the two sorts of theories vis-a-vis relativity. Though a deterministic theory can achieve the appearance of being relativistic, at the level of observable phenomena, it can do so only by introducing a preferred foliation and with it a causal structure at odds with relativistic causal structures. Things are otherwise with dynamical collapse theories.
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.
But this conclusion [nonlocality] needs careful discussion in order to clarify what is going on . (Redhead 1987, p. 3) Within the foundations of physics in recent years, Bell's theorem has played the role of what Thomas Kuhn calls a ‘paradigm’: that is, an exemplary piece of work that others learn from, imitate and develop. Following a period of articulation and consolidation, the first generation of developments of the Bell theorem was initiated by Heywood and Redhead (1983). They produced a nonlocality result in the algebraic style of the Bell–Kochen–Specker theorem (Bell 1966; Kochen and Specker 1967), moving away from the probabilistic relations characteristic of the Bell theorems proper. More recently a second generation develops results by Peres (1990), Greenberger–Horne–Zeilinger (1990), and Hardy (1993). In addition to moving away from probabilities, this generation tries to dispense with the limiting inequalities of the Bell theorem to yield socalled ‘Bell theorems without inequalities’. With respect to probabilities, however, Hardy is a half-way house. It requires no inequalities but the result contradicts quantum mechanics under certain locality assumptions only if the statistical predictions of quantum mechanics hold in at least one case. I want to examine the Hardy theorem and its interpretation. Initially, I intend to ignore respects in which it dispenses with probabilities because I want to point out the interesting significance of the theorem in a probabilistic context. We will see that when probabilities are restored, so are inequalities. Then we will see what the theorem has to contribute on the topic of locality.