In strict accordance with special relativity, locality is defined in terms of observed results. In a non-deterministic "hidden-parameter" theory, admitting "statistical independence" of the outcomes for a single pair, locality and a restriction ("primary locality") at the level of "primary probabilities", Bell's inequalities are recollected. Their violation is traced to the assumption of an algebraic structure defining relations between single-constituent "probabilistic" descriptions that cannot be interpreted as distinctive properties of the components. It is then concluded that the experimental violation of Bell's inequalities cannot be a test of non-locality, even if locality is interpreted at the unobservable level, in terms of "primary probabilities".
In strict accordance with special relativity, locality is defined in terms of observed results. In a non-deterministic "hidden-parameter" theory, admitting "statistical independence" of the outcomes for a single pair, locality and a restriction ("primary locality") at the level of "primary probabilities", Bell's inequalities are recollected. Their violation is traced to the assumption of an algebraic structure defining relations between single-constituent "probabilistic" descriptions that cannot be interpreted as distinctive properties of the components. It is then concluded that the experimental violation of Bell's inequalities cannot be a test of non-locality, even if locality is interpreted at the unobservable level, in terms of "primary probabilities".
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.
Bell introduced the well known locality assumption, stating that the detector outcome of particle-1 could not influence the outcome of particle-2, and vice versa, when a singlet pair of particles was subjected to simultaneous experimental tests. Simple cases of violation of the inequalities were taken as proof of inapplicability of the given assumption of locality or hidden variables concept. To this date no one examined Bell's type inequalities for logical consistency from rigorous mathematical point of view. All the terms appearing in the inequalities deal with conditional probabilities given the detector orientations fixed. In brief, conditional probabilities and conditional expectations can be added, subtracted, or compared only when they are defined on the same conditional subensembles. Careful analyses of this paper show that this requirement is not met.
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.
In the usual derivation of Bell's inequalities, Locality is assumed to justify statistical independence of the outcomes of observations performed in remote stations; as the inequalities are observed to be violated, it is concluded that these outcomes are statistically dependent, and hence, that non-Local effects exist. Here it is shown that the simplest description of Bell's experiment according to the canon of von Neumann's theory of measurement assumes statistical independence of those outcomes. In other words, the straightforward description of Bell's experiment according to Quantum Mechanics (which, of course, violates Bell's inequalities) does not assume non-Locality. Summing up this and other related results it is concluded that, within the framework of Quantum Mechanics, there is no reason to believe in the existence of non-Local effects. In addition to its foundational significance, this conclusion has practical impact in the fields of quantum-certified and device-independent randomness generation and on the vulnerability of Quantum Key Distribution schemes, which are based on the existence of quantum non-Locality as an axiom.
It is widely accepted that the violation of Bell inequalities excludes local theories of the quantum realm. This paper presents a stronger Bell argument which even forbids certain non-local theories. Among these excluded non-local theories are those whose only non-local connection is a probabilistic (or functional) dependence between the space-like separated measurement outcomes of EPR/B experiments (a subset of outcome dependent theories). In this way, the new argument shows that the result of the received Bell argument, which requires just any kind of non-locality, is inappropriately weak. Positively, the remaining non-local theories, which can violate Bell inequalities (among them quantum theory), are characterized by the fact that at least one of the measurement outcomes in some sense probabilistically depends both on its local as well as on its distant measurement setting (probabilistic Bell contextuality). Whether an additional dependence between the outcomes holds, is irrelevant for the question whether a certain theory can violate Bell inequalities. This new concept of quantum non-locality is considerably tighter and more informative than the one following from the usual Bell argument. We prove that (given usual background assumptions) the result of the stronger Bell argument presented here is the strongest possible consequence from the violation of Bell inequalities on a qualitative probabilistic level.
I define a notion of locality LOC, closely modelled on Bell's principle of local causality, construed as the condition that single-case probabilities cannot be modified by actions at spacelike separation. The new principle, like Bell's, forces Bell inequalities, but with two loopholes: one is violation of measurement independence, known to Bell, but the other is non-uniqueness of remote outcomes, a loophole only for LOC, not for Bell's principle. I also set out a theory of physical probability, applicable to the Everett interpretation, in which the Born rule is derived, and which therefore violates Bell inequalities. I show it is consistent with LOC. Surprisingly, both loopholes are exploited. I conclude not only that physical probability in the Everett interpretation involves no action at a distance, in the sense of LOC, but that the observed violation of Bell inequalities is powerful evidence for many worlds.