On Bell-type inequalities

A Bell-type inequality is defined as an inequality of the type 0L1,where L is a linear combination with real coefficients of probabilities p _i and joint probabilities p _ij ,p _ijk ,...,p _l ,...,n corresponding to n events. A general theorem on the validity of such inequalities in correspondence to physical assumptions about commutativity or noncommutativity is given. Examples and possible physical applications are discussed.     

Time and Bell-type inequalities

Bell-type inequalities involve coincidence counting rates and are tested in coincidence experiments. Time is ontologically intertwined with the definition of coincidence. An example is proposed in which Bell's inequality is violated by a local model, simply because the interaction between a photon and a photomultiplier is not instantaneous, but requires a finite elapse of time. Models are only illustrative examples and there is no reason why they should reflect reality. We only claim that maybe Bell-type inequalities are not as general as they are supposed to be in ruling out local hidden variable theories.     

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.     

Bell-type inequalities for cold heteronuclear molecules

We introduce Bell-type inequalities allowing for nonlocality and entanglement tests with two cold heteronuclear molecules. The proposed inequalities are based on correlations between each molecule spatial orientation, an observable which can be experimentally measured with present day technology. Orientation measurements are performed on each subsystem at different times. These times play the role of the polarizer angles in Bell tests realized with photons. We discuss the experimental implementations of the proposed tests, which could also be adapted to other high dimensional quantum angular momenta systems.     

An algorithm for finding Bell-type inequalities

This paper deals with the calculation of all the necessary and sufficient conditions for the classical characterization of the probabilities of some events. It shows how a scarcely known method devised by Boole is particularly useful to the solution of this problem. Based on this method, a general and simple algorithm is found that enables the deduction of the inequalities that, like Bell inequalities, express the conditions for classical characterization. The results of the application of the algorithm to some open problems are reported.This work follows the graduation thesis work of the author under the supervision of Professor Enrico G. BNeltrametti at the University of Genoa, Italy.     

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.     

Molecular orientation entanglement and temporal Bell-type inequalities

We make a detailed study of Bell-type inequalities based on correlations between measurements of continuous observables performed on trapped molecular systems. We show that, in general, when an observable has a continuous spectrum which is bounded, one is able to construct non-locality tests sharing common properties with those for two-level systems. The specific observable studied here is molecular spatial orientation, and it can be experimentally measured for single molecules, as required in our protocol. We also provide some useful general properties of the derived inequalities and study their robustness to noise. Finally, we detail possible experimental scenarios and analyse the role played by different experimental parameters.     

Bell-type inequalities to detect true n-body nonseparability

We analyze the structure of correlations among more than two quantum systems. We introduce a classification of correlations based on the concept of nonseparability, which is different a priori from the concept of entanglement. Generalizing a result of Svetlichny [Phys. Rev. D 35, 3066 (1987)] on three-particle correlations, we find an inequality for n-particle correlations that holds under the most general separability condition and that is violated by some quantum-mechanical 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.     

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.     

Bell-type inequalities and upper bounds for multiqudit states

Multiqudit systems are studied in the tomographic-probability representation of quantum states. Results of calculations for the Bell-type numbers within the framework of classical probability theory and in quantum tomography are compared. Violations of the Bell-type inequalities are shown explicitly using the method of averaging in the tomographic picture of quantum states.     

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.     

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.     

Characterizations of Boolean algebras of idempotents

The notions of the left (right) Jordan groupoids are introduced. IfR is an associative^* ring with the identity and ifU(R) [resp.P(R)] denotes the set of all idempotents (resp. projections) of the^* ringR, then the operationsp q =p – 2pq – qp + 4qpq andp q =q – 2pq – 2qp + 4pqp ifp, q U(R) [resp.p, q P(R)] are the nonassociative linear operations inU(R) [resp. inP(R)]. The present paper shows that the operations and are associative iffpq=qp forp, q U(R) [resp.p, q P(R)]. As a corollary it follows from this that the orthomodular poset (U(R), , 0, 1,) is a Boolean algebra [which is commutative, i.e.,pq= qp, p, q U(R)] iff (U(R), , 0, 1,) or (U(R), , 0, 1,) are Jordan associative groupoids. Similar results hold for (P(R), ,0, 1, ).     

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.     

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.     

Bell-type inequalities in the nonideal case: Proof of a conjecture of bell

Recently Bell has conjectured that, with epsilonics, one should be able to argue, à la EPR, from almost ideal correlations (in parallel Bohm-Bell pair experiments) to almost determinism, and that this should suffice to derive an approximate Bell-type inequality. Here we prove that this is indeed the case. Such an inequality—in principle testable—is derived employing only weak locality conditions, imperfect correlation, and a propensity interpretation of certain conditional probabilities. Outcome-independence (Jarrett's completeness condition), hence factorability of joint probabilities, is not assumed, but rather an approximate form of this is derived. An alternative proof to the original one of Bell [1971] constraining stochastic, contextual hidden-variables theories is thus provided.     

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.     

Semi-direct sums of Lie algebras and continuous integrable couplings

A relation between semi-direct sums of Lie algebras and integrable couplings of continuous soliton equations is presented, and correspondingly, a feasible way to construct integrable couplings is furnished. A direct application to the AKNS spectral problem leads to a novel hierarchy of integrable couplings of the AKNS hierarchy of soliton equations. It is also indicated that the study of integrable couplings using semi-direct sums of Lie algebras is an important step towards complete classification of integrable systems.