It is shown that Bell's counterfactuals admit joint quasiprobability distributions (i.e. joint distributions exist, but may not be non-negative). A necessary and sufficient condition for the existence among them of a true probability distribution (i.e. no-nnegative) is Bell's inequalities. This, in turn, is a necessary condition for the existence of local hidden variables. The treatment is amenable to generalization to examples of 'nonlocality without inequalities'.
It is shown that Bell's counterfactuals admit joint quasiprobability distributions (i.e. joint distributions exist, but may not be non-negative). A necessary and sufficient condition for the existence among them of a true probability distribution (i.e. no-nnegative) is Bell's inequalities. This, in turn, is a necessary condition for the existence of local hidden variables. The treatment is amenable to generalization to examples of 'nonlocality without inequalities'.
INTRODUCTION Fine (1982) proved that Bell's (1964) inequalities are sufficient, as well as necessary, to guarantee the existence of a joint probability distribution of the random variables in question, e.g., those representing certain spin correlation experiments. Garg and Mermin (1982) gave an example of eight random variables satisfying Bell's inequalities but such that no joint distribution exists. The purpose of this paper is to extend Bell's inequalities to obtain some general necessary conditions for the existence of a joint probability distribution for any finite collection of Bell-type random variables. Our results show that, for N > 4, many new elementary inequalities beyond those of Bell must be satisfied by any hidden-variable theory. Since these additional inequalities are violated by quantum mechanical predictions for appropriate choice of measurement arrangements, they serve to increase the conceptual distance between what may be called Einstein locality, after the EPR paradox, and quantum mechanics. To give a concrete sense of the nature of our results, we exhibit one of the new inequalities violated by Garg and Mermin's example. For theoretical purposes it will suffice to introduce Bell-type random variables X 1 ,…, X N , N ≥ 4, having possible values ±1, with means E ( X i ) = 0 and with what we term Bell covariances E ( X i X j ) relative to the index i 0 , with 1 ≤ i ≤ i 0 , N , 2 ≤ i 0 N – 2. Notice that, when N = 4, we must, as is familiar, have i 0 = 2.
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.
We provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no-hidden-variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem.
Violation of Bell inequalities is widely regarded as a definitive test for non-locality. However, Bell correlational inequalities must always be satisfied by all jointly present, cross-correlated data. The correlations of variable pairs obtained in repeated runs are not cross-correlated in this way and are not required to satisfy the Bell inequality. In addition, by using information regarded as non-local, proper joint correlations may be computed among counterfactual and measured variables. These correlations satisfy the Bell inequality, but are spatially non-stationary in angle. By using a simple symmetry condition, such considerations may be extended to inequalities in probabilities. The latter may be derived directly from correlational inequalities developed by Clauser, Horne, Shimony and Holt (CHSH). Violation of either correlational or probabilistic Bell inequalities then implies that the Bell correlation cannot be accounted for by a stochastic process that is spatially stationary in angle coordinates. However, other processes may still be allowed.
In the derivation of Bell's inequalities, probability distribution is supposed to be a function of only hidden variable. We point out that the true implication of the probability distribution of Bell's correlation function is the distribution of the joint measurement outcomes on the two sides. So it is a function of both hidden variable and settings. In this case, Bell's inequalities fail. Our further analysis shows that Bell's locality holds neither for dependent events nor for independent events. We think that the measurements of EPR pairs are dependent events, thus violation of Bell's inequalities cannot rule out the existence of local hidden variable. In order to explain the results of EPR-type experiments, we suppose that polarization entangled photon pair can be composed of two circularly or linearly polarized photons with correlated hidden variables, and a couple of experiments of quantum measurement are proposed. The first uses delayed measurement on one photon of the EPR pair to demonstrate directly whether measurement on the other could have any non-local influence on it. Then several experiments are suggested to reveal the components of polarization entangled photon pair. The last one uses successive polarization measurements on a pair of EPR photons to show that two photons with a same quantum state will behave in the same way under the same measuring condition.
A hidden-variable theory is the traditional, but not unique, basis for constructing various types of Bell's theorem. The starting point may also be a recognition of the existence of a positive-definite probability distribution function. This assumption alone is used to formulate and prove Bell's paradoxes of different types. A specific example is used to show that a formal quantum calculation can sometimes give negative values of the joint probabilities that are used in the proof. An attempt is made to identify the physical meaning of this result and an algorithm for determination of negative joint probabilities of this type is proposed.