Any proof of Bell’s theorem reduces to a derivation of one of Bell’s inequalities. The key role in such derivations is played by the condition of factorizability of the joint conditional probability, which may be obtained as a consequence of two other conditions, known as parameter independence and outcome independence. The former is a quite obvious consequence of locality, whereas the la$er is controversial. However, since it is a particularization of the condition of screening off of the principle of common cause, its undermining entails questioning the la$er condition as well. If successful, any effective proof of Bell’s theorem would demand deriving some of Bell’s inequality without making use of any particularization of screening off. A direction of search for a model breaking this condition is suggested.
Any proof of Bell’s theorem reduces to a derivation of one of Bell’s inequalities. The key role in such derivations is played by the condition of factorizability of the joint conditional probability, which may be obtained as a consequence of two other conditions, known as parameter independence and outcome independence. The former is a quite obvious consequence of locality, whereas the laeriscontroversial.However,sinceitisaparticularizationoftheconditionofscreeningoffoftheprincipleofcommoncause,itsunderminingentailsquestioningthelaer condition as well. If successful, any effective proof of Bell’s theorem would demand deriving some of Bell’s inequality without making use of any particularization of screening off. A direction of search for a model breaking this condition is suggested.
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.
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.
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.
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.
Besides well-known conditions of locality or factorisability, deriving the Bell inequalities requires assuming that the distribution of hidden variables and Alice's and Bob's measurement settings be independent of each other. We show that (analogously to violations of locality due to action at a distance) certain violations of this Measurement Independence assumption can be associated with a notion of signalling in principle, thus making them also testable in principle, and spell out the appropriate conditions. Accordingly, we show that by imposing no-signalling one can prove a version of Bell's theorem that does not require the assumption of Measurement Independence. We discuss the "Schulman model" as an example, as well as lessons for "experimental metaphysics".
While it is widely agreed that Bell's theorem is an important result in the foundations of quantum physics, there is much disagreement about what exactly Bell's theorem shows. It is agreed that Bell derived a contradiction with experimental facts from some list of assumptions, thus showing that at least one of the assumptions must be wrong; but there is disagreement about what the assumptions were that went into the argument. In this paper, I make a few points in order to help clarify the situation. Introduction Different authors have expressed very different views about what Bell's theorem shows about physics. The disagreement concerns particularly the question of which assumptions go into the argument. Since we have to give up one of the assumptions leading to the empirically violated Bell inequality, knowing what the assumptions were is crucial. For example, if author X believes that the argument requires only one assumption, A 1 , while author Y believes that it requires two, A 1 and A 2 , then X will conclude that A 1 ¬ must be abandoned, while Y will conclude that either A 1 or A 2 must abandoned, so A 1 may well be true in our world if A 2 is false. In this paper, I consider several assumptions that have been mentioned in connection with Bell's theorem, and I look into their roles in the proof of Bell's theorem. While I do not say anything here that has never been said before, I hope that my remarks can nevertheless be helpful to some readers. For the sake of definiteness, the version of the relevant experiment that I will consider involves two spin-1\2 particles, initially in the singlet spin state ψ = 2 −1/2 (|↑↓ ⟩ − |↓↑ ⟩). Two widely separated experimenters, Alice and Bob, have one particle transported to each of them. At spacelike separation, they each choose a direction in space, represented by the unit vectors a and b , and carry out a quantum measurement of the spin observable in this direction on their particle, ( a · σ ) ⊗ I and I ⊗ ( b · σ ), with I the 2 × 2 unit matrix and σ the triple of Paulimatrices. The possible outcomes, A and B , are±1 on each side. Bell's theorem is the statement that under the assumptions we are about to discuss, Bell's inequality holds; quantum theory predicts violations of Bell's inequality, and experiments have confirmed these predictions.