Europhysics Letters (EPL)

Bell's Inequalities as Triangle Inequalities for Cross-Sections

Nils A. Törnqvist

Published Apr 15, 1986• 5 citations

Abstract

It is observed that Bell's inequalities can be written as triangle inequalities for cross-sections, while quantum mechanics implies triangle inequalities for the square roots of the same cross-sections. A useful diagram displays instructively the domains separated by Bell's inequalities, on the one hand, and by quantum mechanics, on the other hand, and shows how the former lacks quantum-mechanical interference terms.

DOI: 10.1209/0295-5075/1/8/002

Related papers

Mathematical and physical meaning of the Bell inequalities

Emilio Santos

Abstract It is shown that the Bell inequalities are closely related to the triangle inequalities involving distance functions amongst pairs of random variables with values { 0 , 1 } . A hidden variables model may be defined as a mapping between a set of quantum projection operators and a set of random variables. The model is noncontextual if there is a joint probability distribution. The Bell inequalities are necessary conditions for its existence. The inequalities are most relevant when measurements are performed at space-like separation, thus showing a conflict between quantum mechanics and local realism (Bell's theorem). The relations of the Bell inequalities with contextuality, the Kochen–Specker theorem, and quantum entanglement are briefly discussed.

arXiv:1410.4935European Journal of PhysicsJun 29, 20168 citations

And If Bell’s Inequality Were Not Violated

Olivier Serret

It briefly recalls the theory of Bell’s inequality and some experimental measures. Then measurements are processed on one hand according to a property of the wave function, on the other hand according to the sum definition. The results of such processed measures are apparently not the same, so Bell’s inequality would not be violated. It is a use of the wave function which implies the violation of the inequality, as it can be seen on the last flowcharts.

Journal of Modern PhysicsJan 1, 20141 citation

Separation theorems and Bell inequalities in algebraic quantum mechanics

Guido Bacciagaluppi

The paper discusses the concept of separation of quantum mechanical systems in the algebraic approach. We review known theorems, then establish a link between the C*-algebraic and the corresponding W*-algebraic concepts. A characterization of separation in terms of Bell inequalities, due to Raggio (1988), is given a C*-algebraic formulation. Finally, we comment on the implications for the understanding of the Bell inequalities.

arXiv:2306.01909Jun 2, 20237 citations

Testing quantum mechanics against a full set of Bell inequalities

Paul Busch, Peter Kienzler, PEKKA J. LAHTI +1 more

The validity domain, within quantum mechanics, of a full set of eight Bell inequalities for four observables is determined and represented geometrically. It turns out that the incorporation of all inequalities increases considerably the domain of invalidity. For special spin constellations, the validity regions become arbitrarily small. The results are compared with experiment and used to elucidate the status of the existing experimental evidence in favor of quantum theory and against hidden variables.

Physical Review AJun 1, 19936 citations

A simple proof of Bell's inequality

Lorenzo Maccone

Bell’s theorem is a fundamental result in quantum mechanics: it discriminates between quantum mechanics and all theories where probabilities in measurement results arise from the ignorance of pre-existing local properties. We give an extremely simple proof of Bell's inequality; a single figure suffices. This simplicity may be useful in the unending debate over what exactly the Bell inequality means, because the hypotheses underlying the proof become transparent. It is also a useful didactic tool, as the Bell inequality can be explained in a single intuitive lecture.

arXiv:1212.5214American Journal of PhysicsOct 23, 201345 citations

A Reexamination on Bell's Theorem

Tan Tian-rong

It is proved that Bell's inequality can be traced back to a classical probabilistic formula that is invalid clearly, while the spin correlation formula in quantum mechanics to a formula that has confirmed both by facts and quantum mechanics. Two misunderstandings about Bell's theorem are pointed out: Firstly, the hidden variable theory Bell's used is a special one, but it is regarded as a general form of such theories. Secondly, there are two outcomes obtainable from Bell's hypotheses: one is compatible with quantum mechanics and the other leads to Bell's inequality. Unfortunately, the former is regarded as a property of local hidden variable theory, while the latter as self-evident. G. Lochak has revealed the first misunderstanding, and thereby he pointed out that Bell's inequality has nothing to do with locality, but he did not find the second one, so that he still analyzed this problem starting from hidden variable theory, which is actually irrelative to Bell's inequality. (The Journal of American Science. 2005;1(2):42-50).

Jan 1, 20050 citations