Problem of hidden variables

The problem of hidden variables in quantum mechanics is formalized as follows. A general or contextual (noncontextual) hidden-variables theory is defined as a mappingf: Q×M C (f: QC) whereQ is the set of projection operators in the appropriate (quantum) Hilbert space,M is the set of maximal Boolean subalgebras ofQ andC is a (classical) Boolean algebra. It is shown that contextual (noncontextual) hidden-variables always exist (do not exist).     

POVMs and hidden variables

Recent results by Paul Busch and Adán Cabello claim to show that by appealing to POVMs, non-contextual hidden variables can be ruled out in two dimensions. While the results of Busch and Cabello are mathematically correct, interpretive problems render them problematic as no hidden variable proofs.     

Investment strategies and hidden variables

The present study shows how the information on `hidden' market variables effects optimal investment strategies. We take the point of view of two investors, one who has access to the hidden variables and one who only knows the quotes of a given asset. Following Kelly's theory on investment strategies, the Shannon information and the doubling investment rate are quantified for both investors. Thanks to his privileged knowledge, the first investor can follow a better investment strategy. Nevertheless, the second investor can extract some of the hidden information looking at the past history of the asset variable. Unfortunately, due to the complexity of his strategy, this investor will have computational difficulties when he tries to apply it. He will than follow a simplified strategy, based only on the average sign of the last l quotes of the asset. This results have been tested with some Monte Carlo simulations.     

Indistinguishable particles and hidden variables

An axiomatics for the indistinguishability of elementary particles in terms of hidden variables is presented in a manner which depart from the standard approaches usually given to hidden variables. Quantum distribution functions are also discussed and some possible related lines of work are suggested.     

Algebraic constraints on hidden variables

In the contemporary discussion of hidden variable interpretations of quantum mechanics, much attention has been paid to the no hidden variable proof contained in an important paper of Kochen and Specker. It is a little noticed fact that Bell published a proof of the same result the preceding year, in his well-known 1966 article, where it is modestly described as a corollary to Gleason's theorem. We want to bring out the great simplicity of Bell's formulation of this result and to show how it can be extended in certain respects.Work on this paper was partially supported by National Science Foundation Grants SOC 76-82113 and SOC 76-10659.     

Bayes' rule and hidden variables

We show that a quantum system admits hidden variables if and only if there is a rich set of states which satisfy a Bayesian rule. The result is proved using a relationship between Bayesian type states and dispersion-free states. Various examples are presented. In particular, it is shown that for classical systems every state is Bayesian and for traditional Hilbert space quantum systems no state is Bayesian.     

p-adic probability distributions of hidden variables

During the last years large interest was shown in p-adic quantum models (especially, in string theory). As usual, new physical models generate new mathematical methods. In our case a new type of stochastics, p-adic stochastics, was arisen inside p-adic quantum physics. We apply this stochastics to propose a justification of the Einstein-Podolsky-Rosen theory of hidden variables, which was in large contradiction with the Bell type inequality. Our main result is the following: if we consider a p-adic probability distribution of hidden variables, then there are no problems with Bell's inequality.     

Conditions excluding the existence of approximate hidden variables

The problem of approximate hidden variables is redefined in the operator algebraic framework of quantum mechanics and a theorem containing negative results is formulated.     

Proposed test for a hidden variables theory

A hidden variables model for quantum mechanics is proposed and a possible test for its validity is described.     

Existence of hidden variables having only upper probabilities

We prove the existence of hidden variables, or, what we call generalized common causes, for finite sequences of pairwise correlated random variables that do not have a joint probability distribution. The hidden variables constructed have upper probability distributions that are nonmonotonic. The theorem applies directly to quantum mechanical correlations that do not satisfy the Bell inequalities.It is a pleasure to dedicate this paper to Karl Popper in celebration of this 90th birthday. The first author has known Popper for more than three decades, and has profited much from their discussion of many different topics, among which have been the foundations of probability and the foundations of quantum mechanics, both central to the present paper.     

Simple test for hidden variables in spin-1 systems

We resolve an old problem about the existence of hidden parameters in a three-dimensional quantum system by constructing an appropriate Bell's type inequality. This reveals the nonclassical nature of most spin-1 states. We shortly discuss some physical implications and an underlying cause of this nonclassical behavior, as well as a perspective of its experimental verification.     

A search for hidden variables in the domain of high energy physics

The idea of hidden variable theories, which contests the notion of quantum mechanics being the fundamental principle of nature, is well known and seems to need no introduction. In 1966 such a theory (of a non-local character) was proposed by D. Bohm and J. Bub. We present a scheme in which measured decay processes may constitute an adequate substitute to the original test proposed in 1966 and which until now proved to be realizable only for massless particles. Finally, we consider a specific proposal concerning Tau decays. A preliminary overview of several experimental data sets is presented.     

Exploring inequality violations by classical hidden variables numerically

There are increasingly suggestions for computer simulations of quantum statistics which try to violate Bell type inequalities via classical, common cause correlations. The Clauser–Horne–Shimony–Holt (CHSH) inequality is very robust. However, we argue that with the Einstein–Podolsky–Rosen setup, the CHSH is inferior to the Bell inequality, although and because the latter must assume anti-correlation of entangled photon singlet states. We simulate how often quantum behavior violates both inequalities, depending on the number of photons. Violating Bell 99% of the time is argued to be an ideal benchmark. We present hidden variables that violate the Bell and CHSH inequalities with 50% probability, and ones which violate Bell 85% of the time when missing 13% anti-correlation. We discuss how to present the quantum correlations to a wide audience and conclude that, when defending against claims of hidden classicality, one should demand numerical simulations and insist on anti-correlation and the full amount of Bell violation.     

Identification of hidden variables through Everett’s formalism

It is well known that the conventional interpretation of quantum mechanics entails several paradoxes which have not been resolved in a complete and convincing way. To avoid these paradoxes other interpretations were suggested and advances, among which were the “hidden variables” theory (HV) of Bohm and Bub [1], and the “relative state” theory of Everett [2]. These HV have not, up to now, been identified, but if we compare these theories we can show an equivalence between them, and arrive at a possible definition of the HV.     

An algebraic approach to the restoration of images

Two different matrix algorithms are described for the restoration of blurred pictures. These are illustrated by numerical examples.