Foundations of Physics

, Volume 45, Issue 10, pp 1179–1189 | Cite as

Unconditional Quantum Correlations do not Violate Bell’s Inequality

  • Andrei KhrennikovEmail author


In this paper I demonstrate that the quantum correlations of polarization (or spin) observables used in Bell’s argument against local realism have to be interpreted as conditional quantum correlations. By taking into account additional sources of randomness in Bell’s type experiments, i.e., supplementary to source randomness, I calculate (in the standard quantum formalism) the complete quantum correlations. The main message of the quantum theory of measurement (due to von Neumann) is that complete correlations can be essentially smaller than the conditional ones. Additional sources of randomness diminish correlations. One can say another way around: transition from unconditional correlations to conditional can increase them essentially. This is true for both classical and quantum probability. The final remark is that classical conditional correlations do not satisfy Bell’s inequality. Thus we met the following conditional probability dilemma: either to use the conditional quantum probabilities, as was done by Bell and others, or complete quantum correlations. However, in the first case the corresponding classical conditional correlations need not satisfy Bell’s inequality and in the second case the complete quantum correlations satisfy Bell’s inequality. Thus in neither case we have a problem of mismatching of classical and quantum correlations. The whole structure of Bell’s argument was based on identification of conditional quantum correlations with unconditional classical correlations.


Conditional and unconditional correlations Quantum and classical correlations Bell’s inequality Quantum measurement theory Random generators 


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science, Linnaeus UniversityVäxjöSweden

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