Bell-type inequalities in classical probability theory

A review of the tomographic probability representation for qudit sates is presented. Properties of related stochastic matrices are considered. Tomograms of two qubits and three qubits are used to study the Bell-type inequalities. The Bell-type inequalities in the standard classical probability theory are discussed. Joint probability distributions of classical systems with several random variables and their properties in the case of factorized distribution functions are considered.     

Formalism for the individual state of a quantum particle compatible with the bell inequalities,and a dissipative environment conjecture

We suggest a mathematical construction to define an individual state of a quantum particle which enables one to eliminate a contradiction between quantum mechanics of correlated particles and the Relativity Theory.     

Bell-type inequalities in horizontal sums of Boolean algebras

We give a necessary and sufficient condition for a Bell-type inequality to hold in a horizontal sum of finitely many finite Boolean algebras.     

Bell-type inequalities on tensor products of quantumlogics

The problem of Bell-type inequalities on tensor products of quantum logics is studied. To formulate the problem the definition of tensor product used for sum logics by Hudson and Pulmannová is generalized to arbitrary orthomodular lattices and the existence of such product for lattices describing the simplest EPR-Bohm experiment is shown. It is proved that Bell-type inequalities hold for all product states.     

Bell-type inequalities and tomographic entropies of multiqudit states

Tomographic entropies of multiqudit systems are studied. A comparison of Shannon and von Neumann entropic inequalities with analogous inequalities for tomographic entropies is presented. An attempt to associate the violation of these and Bell-type inequalities of multipartite states is done within the framework of tomographic probability theory.     

The geometrical aspects of the bell inequalities

The Bell inequalities of the metric form are introduced. The quantum-mechanical correlations of the particles with s=1/2 and photons are described using the relative measure of probability on the concave surfaces. The relation of the proposed scheme with the Bayes theorem about conditional information entropy and J. von Neumann's postulates is discussed.     

Electron entanglement near a superconductor and bell inequalities

Near the interface between a normal metal and a superconductor, Cooper pairs penetrate into the normal side, giving rise to the proximity effect. The two electrons of these pairs have entangled spin and orbital degrees of freedom. Nonlocal features of quantum mechanics can be probed by separating these two electrons. This is achieved with a fork geometry with two normal leads containing either spin- or energy-selective filters. A signature of entanglement can be detected by measuring the positive noise cross-correlations in this fork. In the case of energy filters, Bell-inequality checks constitute a definite probe of entanglement. We formulate Bell-type inequalities in terms of current-current cross-correlations associated with contacts with varying magnetization orientations. We find maximal violation (as in photons) when a superconductor is the particle source.     

Conjunctions,disjunctions, and Bell-type inequalities in orthoalgebras

A logicoalgebraic approach to the hidden variables issue is outlined. The motivation for studying Bell-type inequalities on orthoalgebras is given and the problem of conjunctions and disjunctions of compatible elements is discussed. A theorem concerning a very general form of Bell-type inequalities on orthoalgebras is proved.     

Proof of a multipole conjecture due to Geroch

A result, first conjectured by Geroch, is proved to the extent, that the multipole moments of a static space-time characterize this space-time uniquely. As an offshoot of the proof one obtains an essentially coordinate-free algorithm for explicitly writing down a geometry in terms of it's moments in a purely algebraic manner. This algorithm seems suited for symbolic manipulation on a computer.     

On an alternative interpretation of the bell inequalities

An alternative interpretation is discussed on which the Bell inequalities are not consequences of local realism and on which Einstein-Podolsky-Rosen-like experiments are no tests of the inequalities, because the Bell inequalities only need to be satisfied if all observables are measured jointly.     

Experimentally simulating the violation of Bell-type inequalities for generalized GHZ states

Using NMR techniques, we implemented the simulation of the violations of two Bell-type inequalities: Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality and Chen's inequality, for the 3-qubit generalized GHZ states. The experimental results are in good agreement with the quantum predictions and show that Chen's inequality is more efficient than MABK inequality in the case of the generalized GHZ entangled states.     

Information-theoretic bell inequalities and the relative measure of probability

The information-theoretic Bell inequalities of Braunstein and Caves are studied in relation to the concept of the relative probability measure which one allows to overcome the limitations of the considered theorem.I esteem it an honor to dedicate this paper to Sir Karl Popper on the occasion of his 90th birthday.     

Bell-type inequalities in orthomodular lattices. I. Inequalities of order 2

We study Bell-type inequalities of ordern with emphasis on the casen = 2 in the framework of the structure of an orthomodular lattice, which is a logicoalgebraic model of quantum mechanics. We give necessary and sufficient conditions for the validity of Bell-type inequalities of order 2. In particular, we study Bell-type inequalities in various structures connected with a Hilert space, and we give a characterization of Boolean algebras via the validity of certain Bell-type inequalities.     

Bell-type inequalities in orthomodular lattices. II. Inequalities of higher order

This paper is a continuation of the first part and it is devoted to the study of Bell-type inequalities of order at least 3 in orthomodular lattices. We give some necessary and sufficient conditions for the validity of Bell-type inequalities of order 3 and also, more generally, for those of ordern.     

New Bell-type inequalities forN > 4 necessary for existence of a hidden variable

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 forN > 4 many new elementary inequalities beyond those of Bell must be satisfied by any hidden variable theory.     

Proof of an entropy conjecture of Wehrl

Wehrl has proposed a new definition of classical entropy,S, in terms of coherent states and conjectured thatS1. A proof of this is given. We discuss the analogous problem for Bloch coherent spin states, but in this case the conjecture is still open. An inequality for the entropy of convolutions is also given.Work partially supported by US National Science Foundation grant MCS 75-21684 A02     

Bell-type inequalities for partial separability in N-particle systems and quantum mechanical violations

We derive N-particle Bell-type inequalities under the assumption of partial separability, i.e., that the N-particle system is composed of subsystems which may be correlated in any way (e.g., entangled) but which are uncorrelated with respect to each other. These inequalities provide, upon violation, experi-mentally accessible sufficient conditions for full N-particle entanglement, i.e., for N-particle entanglement that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. The inequalities are shown to be maximally violated by the N-particle Greenberger-Horne-Zeilinger states.     

First experimental test of bell inequalities performed using a non-maximally entangled state

We describe the realisation of a new test of Bell inequalities using a new scheme obtained by the superposition of type I parametric down conversion produced in two different non-linear crystals pumped by the same laser, but with different polarisations. This experiment is the first test of Bell inequalities using a non-maximally entangled state and thus represents an important step in the direction of eliminating the detection loophole.     

Convolution of Invariant Distributions: Proof of the Kashiwara–Vergne conjecture

Consider the Kontsevich star product on the symmetric algebra of a finite-dimensional Lie algebra g, regarded as the algebra of distributions with support 0 on g. In this Letter, we extend this star product to distributions satisfying an appropriate support condition. As a consequence, we prove a long-standing conjecture of Kashiwara–Vergne on the convolution of germs of invariant distributions on the Lie group G.