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Annalen der Physik

On the Significance of the Bell Inequalities for the Locality Problem in Different Realistic Interpretations of Quantum Mechanics

Willem M. de Muynck, J. T. P. M. van Stekelenborg

Published Jan 1, 1988• 11 citations
Publisher

Abstract

Abstract The relation between the Bell inequalities, locality and the existence of joint probability distributions is discussed in different realist interpretations of quantum mechanics. We distinguish four realist interpretations, viz., the objectivistic one, the contextualistic one, the strictly nonobjectivistic and the quasi‐objectivistic interpretation. Conclusions are differing largely in different interpretations. We also distinguish between two kinds of locality, viz, macrolocality and Einstein/Bell locality. From a classical model of stochastic measuring processes a definition of Einstein/Bell locality is derived that differs from the Bell/Clauser/Horne/Shimony factorizability condition. It is demonstrated that only in the quasi‐objectivistic interpretation the Einstein/Bell locality condition plays a role in the derivation of a Bell inequality for quantities that are experimentally relevant. It is argued that even in this interpretation it is not possible to arrive at the conclusions that the Bell inequalities stem from the locality condition because of the tacit assumption of an additional property, namely the existence of probability distributions conditionalized on the dispersionfree states of the hidden variables. Consideration of a phase space representation of the Schrödinger equation demonstrates that this latter assumption is at odds with the statistics of quantum systems.

DOI
10.1002/andp.19885000309

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Abstract The relation between the Bell inequalities, locality and the existence of joint probability distributions is discussed in different realist interpretations of quantum mechanics. We distinguish four realist interpretations, viz., the objectivistic one, the contextualistic one, the strictly nonobjectivistic and the quasi‐objectivistic interpretation. Conclusions are differing largely in different interpretations. We also distinguish between two kinds of locality, viz, macrolocality and Einstein/Bell locality. From a classical model of stochastic measuring processes a definition of Einstein/Bell locality is derived that differs from the Bell/Clauser/Horne/Shimony factorizability condition. It is demonstrated that only in the quasi‐objectivistic interpretation the Einstein/Bell locality condition plays a role in the derivation of a Bell inequality for quantities that are experimentally relevant. It is argued that even in this interpretation it is not possible to arrive at the conclusions that the Bell inequalities stem from the locality condition because of the tacit assumption of an additional property, namely the existence of probability distributions conditionalized on the dispersionfree states of the hidden variables. Consideration of a phase space representation of the Schrödinger equation demonstrates that this latter assumption is at odds with the statistics of quantum systems.
Abstract Quantum states in superposition cannot be observed, suggesting that they merely embody information on possible measurement results. Yet, we think of them as describing physical systems that evolve in time according to given mathematical equations. Furthermore, evolution takes place in physical space-time, and local causality is imposed on space-time by Special Relativity (a sequence of cause and effect that constitutes a fundamental principle by which we think about and do scientific work). It is therefore strange that there is non–locality and violation of causality embedded in Quantum Mechanics. We try to give here a first step to see whether a deterministic and local interpretation is possible. We will show that Bell’s inequality is not conclusive about the non–local nature of Quantum Mechanics, in that there exists an interpretation of reality (which we put forward) which is local and in which Bell’s inequality cannot be derived. Assuming determinism and locality a new Bell-like inequality is derived, which is always satisfied by Quantum Mechanics and which allows for local hidden variables [1].
Bell's theorem is 50 years old. Still there is a controversy about its implications. Much of it has its roots in confusion regarding the premises from which the theorem can be derived. Some claim that a derivation of Bell's inequalities requires just locality assumption, and nothing more. Violations of the inequalities are then interpreted as ``nonlocality'' or ``quantum nonlocality''. We show that such claims are unfounded and that every derivation of Bell's inequalities requires a premise -- in addition to locality and freedom of choice -- which is either assumed tacitly, or unconsciously, or is embedded in a single compound condition (like Bell's ``local causality''). The premise is equivalent to the assumption of existence of additional variables which do not appear in the quantum formalism (in form of determinism, or joint probability for outcomes of all conceivable measurements, or ``additional causes`'', or ``hidden variables'', ``complete description of the state'' or counterfactual definiteness, etc.). A certain irony is that perhaps the main message of violation of Bell's inequalities is that our notion of locality should be based on an operationally well-defined no-signalling condition, rather than on local causality.
An examination is made of the differing implications from applying the two mainstream interpretations of probability, frequentist and Bayesian, to QM (quantum mechanics) theory for the Bohm-EPR experiment. The joint probability distribution for the possible spin outcomes for two particles with coupled spins is examined. Contrasting conclusions are made: (i) Under the frequentist interpretation, the QM spin distribution implies a widely-discussed non-locality because probabilistic conditioning on a spin measurement is viewed as corresponding to a causal influence. Under the Bayesian interpretation, this conditioning is viewed as providing information relevant to the spin probabilities and the argument for non-locality loses its force. (ii) The frequentist interpretation leads to the locality condition used by John Bell in 1964 to establish conditions for the existence of hidden variables behind the spin probability distribution but this condition is not consistent with the product (Bayes) rule of probability theory. Under the Bayesian interpretation, there is no motivation for this locality condition and a new stochastic hidden-variable model is given that reproduces the spin distribution. (iii) Bell's original 1964 inequality provides a necessary condition for the existence of a third-order joint probability distribution for three spin variables that is compatible through marginalization with the second-order joint distributions for the three possible spin pairs. Bell's inequality is logically equivalent to the 1969 CHSH inequality that involves expectations of four pairs of four spin variables; the CHSH inequality must be satisfied in order for a fourth-order joint probability distribution for the four spin variables to exist. When any of these inequalities are violated, a joint probability distribution for three spin outcomes fails to be valid because some probabilities are negative.
Abstract This chapter presents a detailed discussion of Bell inequalities and their experimental tests. Starting from the Aharonov-Bohm version of the Einstein-Podolsky-Rosen (EPR) paradox, we review EPR’s criteria for realism and completeness and Bell’s locality condition, before deriving the Clauser-Horne-Shimony-Holt (CHSH) inequality. This is followed by an exposition on Bell’s original inequality and other Bell inequalities by Wigner, as well as Clauser and Horne. We then discuss the milestones in the experimental violation of Bell inequalities, from the experiment of Freedman and Clauser in the 1970s, to the efforts of Aspect et al. in the 1980s, the experiments by the teams of Zeilinger and Gisin in the 1990s, and culminating the loophole-free tests in 2015. Finally we turn to a brief discussion of the interpretations of quantum mechanics, focusing on contrasting the realist views held by, e.g., Einstein and Bell, and the informational views with proponents such as Brukner and Zeilinger
Abstract It is shown that the Bell inequalities are closely related to the triangle inequalities involving distance functions amongst pairs of random variables with values { 0 , 1 } . A hidden variables model may be defined as a mapping between a set of quantum projection operators and a set of random variables. The model is noncontextual if there is a joint probability distribution. The Bell inequalities are necessary conditions for its existence. The inequalities are most relevant when measurements are performed at space-like separation, thus showing a conflict between quantum mechanics and local realism (Bell's theorem). The relations of the Bell inequalities with contextuality, the Kochen–Specker theorem, and quantum entanglement are briefly discussed.
A number of additional assumptions is discussed playing a role in derivations of the Bell inequalities along with the locality condition. An analogy between quantum mechanics and statistical thermodynamics is exploited to demonstrate that the ex istence of conditional probabilities p(ai|,Lambda)generally assumed in stochastic hidden variables theories, may be questionable within the domain of application of quan tum mechanics. It is demonstrated how this could block derivations of the Bell inequalities and of the Kochen-Specker theorem. The analogy also provides a pos sible understanding of the nonlocality of quantum mechanics as evidenced by the Borchers-Schlieder theorem.