Piron's and Bell's Geometrical Lemmas

The famous Gleason's Theorem gives a characterization of measures on lattices of subspaces of Hilbert spaces. The attempts to simplify its proof lead to geometrical lemmas that possess also easy proofs of some consequences of Gleason's Theorem. We contribute to these results by solving two open problems formulated by Chevalier, Dvure?enskij and Svozil. Besides, our use of orthoideals provides a unified approach to finite and infinite measures.     

Coexistence of questions versus Piron's compatibility

The notion of coexistence between questions is introduced in the framework of Piron's approach to quantum physics, using Aerts' notion of performable-together questions. Relationships between coexistence of questions and Piron's compatibility of propositions are investigated. In particular, properties generated by coexistent and primitive questions are compatible.     

A note on misunderstandings of Piron's axioms for quantum mechanics

Piron's axioms for a realistically interpreted quantum mechanics are analyzed in detail within the context of a formal mathematical structure expressed in the conventional set-theoretic idiom of mathematics. As a result, some of the serious misconceptions that have encouraged recent criticisms of Piron's axioms are exposed.     

Supplement to a critique of Piron's system of questions and propositions

A previous critique of the relevance of Piron's questions-propositions system as a generator of quantum mechanics by interpretation is reinforced and brought to a close by further investigation.     

Axiomatic foundations of quantum physics: Critiques and misunderstandings. Piron's question-proposition system

Some of the most frequent misconceptions about axiomatic quantum physics are discussed with the aim of clarifying their true significance, taking Piron's approach as conceptual framework. In particular, we deal with the following topics: the wrong identification of Piron's questions and Mackey's questions, and some curious alleged empirical consequences; the role of propositions as suitable equivalence classes of questions, their partial order structure, and the paradoxical consequences of the erroneous assignment to questions of some lattice properties involving propositions; the logical and the empirical purport of some negative theorems; the standard Hilbert space model of the theory and the consequent metaphysical disasters related to some identifications, which are peculiar of this model. A controversy between Foulis-Piron-Randall and Hadjisavvas-Thieffine-Mugur-Schächter is analyzed on the basis of the proposed Hilbert space model (in which Piron's questions are realized by Hilbertian effects, i.e., linear bounded operatorsF such that which clarify the different point of views. As an example, we treat the unsharp localization operators inL ₂().     

Hidden variables and locality

Bell's problem of the possibility of a local hidden variable theory of quantum phenomena is considered in the context of the general problem of representing the statistical states of a quantum mechanical system by measures on a classical probability space, and Bell's result is presented as a generalization of Maczynski's theorem for maximal magnitudes. The proof of this generalization is shown to depend on the impossibility of recovering the quantum statistics for sequential probabilities in a classical representation without introducing a randomization process for the hidden variables. Hidden variable theories that exclude such a randomization process are termed strict, and it is shown that the class of local hidden variable theories is included in the class of strict theories. A counterargument by Freedman and Wigner is evaluated with reference to Clauser's extension of a hidden variable model proposed by Bell.     

Testing locality at colliders via Bell''s inequality?

We consider a measurement of correlated spins at LEP and show that it does not constitute a general test of local-realistic theories via Bell's inequality. The central point of the argument is that such tests, where the spins of two particles are inferred from a scattering distribution, can be described by a local hidden variable theory. We conclude that with present experimental techniques it is not possible to test locality via Bell's inequality at a collider experiment. Finally we suggest an improved fixed-target it is not possible to test locality via Bell's inequality at a collider experiment. Finally we suggest an improved fixed-target experiment as a viable test of Bell's inequality.     

A remark on Bell's inequality and decomposable normal states

Bell's correlation inequality is considered in the algebraic framework as discussed by Baez. It is shown that all normal states of the tensor product of two W ^*-algebras satisfy Bell's inequality, if and only if every normal state lies in the closure of the convex hull of the normal product states, if and only if one of the algebras is commutative.     

Robustness of entanglement for Bell decomposable states

We propose a simple geometrical approach for finding robustness of entanglement for Bell decomposable states of two-qubit quantum systems. It is shown that for these states robustness is equal to the concurrence. We also present an analytical expression for two separable states that wipe out all entanglement of these states. Random robustness of these states is also obtained. We also obtain robustness of a class of states obtained from Bell decomposable states via some special local operations and classical communications (LOCC).Received: 28 October 2002, Published online: 5 August 2003PACS: 03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)     

Comparison between local Kolmogorovian models and Bell's inequality

The conditions on the relative frequencies of coincidence between the measurements on two physical systems are deduced, in the particular case of four different directions, from Kolmogorovian probability and the Gutkowski and Valdes-Franco computational method. These conditions are compared with those imposed by Bell's inequality. It is proved that Bell's inequality is a necessary but not a sufficient condition for local Kolmogorovian probability. The further assumptions to be added to Bell's inequality, in order to prove the equivalence with local Kolmogorovian probability, are studied. The connection with the results obtained by other authors on the subject is discussed.     

Bell's inequalities and quantum communication complexity

We prove that for every Bell's inequality, including those which are not yet known, there always exists a communication complexity problem, for which a protocol assisted by states which violate the inequality is more efficient than any classical protocol. Violation of Bell's inequalities is the necessary and sufficient condition for quantum protocol to beat the classical ones.     

Locality and Bell's Theorem

It is shown that the violation of Bell's inequality allowed by quantum mechanics and the related Bell's theorem without inequalities is accounted for by local commutations of operators representing single-particle observables. It is argued that the idea of nonlocal influencing of one particle on another when they are in spacelike separated regions clearly has neither empirical nor theoretical support.     

The geometry of violation of Bell''''s inequality

The purpose of this paper is to deduce an analytical expression for the violation of Bell's inequality by quantum theory and plane trigonometry, and expound the violation and maximal violation of the first, second type Bell's inequality in detail. Further, we find out the sufficient conditions for the region in which Bell's inequalities are violated.     

Piron's axioms for quantum mechanics: A reply to Foulis and Randall

In their paper A note on Misunderstandings of Piron's Axioms for Quantum Mechanics, Foulis and Randall undertake a reply to our critique of Piron's question-proposition system (qp-s) which appeared in previous issues of this journal. In the present paper, we want briefly to refute the points of criticism raised by Foulis and Randall (FR). We argue that the misunderstandings are not ours, and we prove it.     

On the real meaning of Bell's theorem

In the last couple of years many important results have been derived showing that Bell's inequalities are nothing else but the indicator whether certain events and their probabilities can be represented or not within a Kolmogorovian probabilistic model. It has become evident that one can derive the Bell's inequalities without mentioning locality, causality hidden variable, etc. Many authors jumped to conclusion that the original content of the Bell's theorem had lost its meaning. I reconsider original problem posed by Bell and I show that the Bell's theorem is still valid.1. On leave from the Institute for Theoretical Physics, Eötvös University, Budapest.     

On the real meaning of Bell's theorem

In the last couple of years many important results have been derived showing that Bell's inequalities are nothing else but the indicator of whether certain events and their probabilities can be represented within a Kolmogorovian probabilistic model. It has become evident that one can derive Bell's inequalities without mentioning locality, causality, hidden variables, etc. Many authors jumped to the conclusion that the original content of Bell's theorem had lost its meaning. I reconsider the original problem posed by Bell and I show that Bell's theorem is still valid.On leave from the Institute for Theoretical Physics, Eötvös University, Budapest, Hungary.     

Bell's theorem does not eliminate fully causal hidden variables

Bell's theorem applies only to a hybrid universe in which hidden variables determine only part of the outcomes of experiments. When applied to a fully causal hidden variable theory, in which detector settings as well as their interaction with particles during observation are determined by the variables, Bell's analysis must be modified. The result is that a fully causal hidden variable model can be produced for which a properly chosen spread of hidden variables gives precisely the same prediction as standard quantum theory.     

On the relation between Bell's inequalities and nonlocal games

We investigate the relation between Bell's inequalities and nonlocal games by presenting a systematic method for their bilateral conversion. In particular, we show that while to any nonlocal game there naturally corresponds a unique Bell's inequality, the converse is not true. As an illustration of the method we present a number of nonlocal games that admits better odds when played using quantum resources.     

Numerical Simulation of Bell Inequality's Violation Using Optical Transverse Modes in Multimode Waveguides

We numerically demonstrate that ＇mode-entangled states＇ based on the transverse modes of classical optical fields in multimode waveguides violate Bell＇s inequality. Numerically simulating the correlation measurement scheme of Bell＇s inequality, we obtain the normalized correlation functions of the intensity fluctuations for the two entangled classical fields. By using the correlation functions, the maximum violations of Bell＇s inequality are obtained. This implies that the two classical fields in the mode-entangled states, although spatially separated, present a nonlocal correlation.