Bell's conclusion from his famous inequality was that any hidden variable theory that satisfies Local Causality is incompatible with the predictions of Quantum Mechanics (QM) for Bell's Experiment. However, Local Causality does not appear in the derivation of Bell's inequality. Instead, two other assumptions are used, namely Factorizability and Settings Independence. Therefore, in order to establish Bell's conclusion, we need to relate these two assumptions to Local Causality. The prospects for doing so turn out to depend on the assumed location of the hidden states that appear in Bell's inequality. In this paper, I consider the following two views on such states: (1) that they are states of the two-particle system at the moment of preparation, and (2) that they are states of thick slices of the past light cones of measurements. I argue that straightforward attempts to establish Bell's conclusion fail in both approaches. Then, I consider three refined attempts, which I also criticise, and I propose a new way of establishing Bell's conclusion that combines intuitions underlying several previous approaches.
Bell's conclusion from his famous inequality was that any hidden variable theory that satisfies Local Causality is incompatible with the predictions of Quantum Mechanics (QM) for Bell's Experiment. However, Local Causality does not appear in the derivation of Bell's inequality. Instead, two other assumptions are used, namely Factorizability and Settings Independence. Therefore, in order to establish Bell's conclusion, we need to relate these two assumptions to Local Causality. The prospects for doing so turn out to depend on the assumed location of the hidden states that appear in Bell's inequality. In this paper, I consider the following two views on such states: (1) that they are states of the two-particle system at the moment of preparation, and (2) that they are states of thick slices of the past light cones of measurements. I argue that straightforward attempts to establish Bell's conclusion fail in both approaches. Then, I consider three refined attempts, which I also criticise, and I propose a new way of establishing Bell's conclusion that combines intuitions underlying several previous approaches.
One of the conclusions that Bell drew from his famous inequality was that any hidden variable theory that satisfies Local Causality is incompatible with the predictions of Quantum Mechanics for Bell’s Experiment. However, Local Causality does not appear in the derivation of Bell’s inequality. Instead, two other assumptions are used, namely Factorizability and Settings Independence. Therefore, in order to establish the mentioned Bell’s conclusion, we need to relate these two assumptions to Local Causality. The prospects for doing so turn out to depend on the assumed location of the hidden states that appear in Bell’s inequality. In this paper, I consider the following two views on such states: (1) that they are states of the two-particle system at the moment of preparation, and (2) that they are states of thick slices of the past light cones of measurements. I argue that straightforward attempts to establish Bell’s conclusion fail in both approaches. Then, I consider three refined attempts, which I also criticise, and I propose a new way of establishing Bell’s conclusion that combines intuitions underlying several previous approaches.
By implicitly assuming that all possible Bell-measurements occur simultaneously, all proofs of Bell's Theorem violate Heisenberg's Uncertainty Principle. This assumption is made in the original form of Bell's inequality, in Wigner's probability inequalities, and in the ``nonlocality without inequalities'' arguments. The introduction of time into derivations of these variants of Bell's theorem results in extra terms related to the time order of the measurements used in constructing correlation coefficients. Since the same locality assumptions are made in the Heisenberg-compliant derivations of this paper, only time-independent classical local hidden variable theories are forbidden by violations of the original Bell inequalities; time-dependent quantum local hidden variable theories can satisfy this new bound and agree with experiment. We further point out that factorizable wavefunctions have been used to describe some EPR experiments and can be used to describe others. These will generate local de Broglie-Bohm trajectories in the description of the data. This second, independent, line of argument also shows that violation of Bell's inequality is only evidence that Heisenberg's Uncertainty Principle cannot be ignored.
I compare deterministic and stochastic hidden variable models of the Bell experiment, exphasising philosophical distinctions between the various ways of combining conditionals and probabilities. I make four main claims. (1) Under natural assumptions, locality as it occurs in these models is equivalent to causal independence, as analysed (in the spirit of Lewis) in terms of probabilities and conditionals. (2) Stochastic models are indeed more general than deterministic ones. (3) For factorizable stochastic models, relativity's lack of superluminal causation does not favour locality over completeness. (4) If we prohibit all superluminal causation, then the violation of the Bell inequality teaches us a lesson, besides quantum mechanics' familiar ones that quantities can lack precise values and that pairs of quantities can lack joint probabilities: namely, some pairs of events are not screened off by their common past.
While initial versions of Bell's theorem captured the notion of locality with the assumption of factorizability, in later presentations, Bell argued that factorizability could be derived from the more fundamental principle of local causality. Here we show that, contrary to what is commonly assumed, in order to derive factorizability from the principle of local causality, a non-trivial assumption, similar but strictly independent of settings independence, is required. Loosely speaking, such an extra assumption demands independence between the states of the measurement apparatuses. We conclude that it is possible to construct a model, satisfying both the principle of local causality and settings independence, but that, in virtue of violating this additional assumption--and thus factorizability--is able to break Bell's inequality.
Bell's theorem rests on the following fundamental condition for a local system: P(a,b|alpha, beta, lambda)= P(a|alpha, lambda)P(b|beta, lambda). Here a and b are the outcomes respectively for measurements alpha on one side, and beta on the other, of an experiment involving two entangled particles traveling in opposite directions from a source. The parameter lambda (the set of "hidden variables") represents a more complete description of the joint state of the two particles. Because of lambda, the joint probability of detection is now dependent only on lambda and the local measurement setting of alpha; similarly for the other side and the setting beta. From this equation John Bell derived a simple inequality that is violated by the predictions of quantum mechanics, which is generally taken to imply that quantum mechanics is a nonlocal theory. But, by combining Richard Feynman's formulation of quantum mechanics with a model of particle interaction described by David Deutsch, we develop a system (the "space of all paths," SP) that (1) is immediately seen to replicate the predictions of quantum mechanics, (2) has a single outcome for each quantum event (unlike MWI on which it is partly based), and (3) contains the set lambda of hidden variables consisting of all possible paths from the source to the detectors on each side of the two-particle experiment. However, the set lambda is nonmeasurable, and therefore the above equation is meaningless in SP.
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.