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Contextuality in Three Types of Quantum-Mechanical Systems

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Abstract

We present a formal theory of contextuality for a set of random variables grouped into different subsets (contexts) corresponding to different, mutually incompatible conditions. Within each context the random variables are jointly distributed, but across different contexts they are stochastically unrelated. The theory of contextuality is based on the analysis of the extent to which some of these random variables can be viewed as preserving their identity across different contexts when one considers all possible joint distributions imposed on the entire set of the random variables. We illustrate the theory on three systems of traditional interest in quantum physics (and also in non-physical, e.g., behavioral studies). These are systems of the Klyachko–Can–Binicioglu–Shumovsky-type, Einstein–Podolsky–Rosen–Bell-type, and Suppes–Zanotti–Leggett–Garg-type. Listed in this order, each of them is formally a special case of the previous one. For each of them we derive necessary and sufficient conditions for contextuality while allowing for experimental errors and contextual biases or signaling. Based on the same principles that underly these derivations we also propose a measure for the degree of contextuality and compute it for the three systems in question.

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Notes

  1. A sequence is an indexed set, and “generalized” means that the indexing is not necessarily finite or countable. We try to keep the notation simple, omitting technicalities. A sequence of random variables that are jointly distributed is a random variable, if the latter term is understood broadly, as anything with a well-defined probability distribution, to include random vectors, random sets, random processes, etc.

  2. This means that the directions in the 3D real Hilbert space are chosen strictly in accordance with [23], with no experimental errors, signaling, or contextual biases involved. In this case, the values \(\left( +1,+1\right) \) for paired measurements \(\left( V_{i},W_{i\oplus _{5}1}\right) \) are excluded by quantum theory.

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Acknowledgments

This work is supported by NSF Grant SES-1155956 and AFOSR Grant FA9550-14-1-0318. We have benefited from collaboration with J. Acacio de Barros, and Gary Oas, as well as from discussions with Samson Abramsky, Guido Bacciagaluppi, and Andrei Khrennikov. An abridged version of this paper was presented at the Purdue Winer Memorial Lectures in November 2014.

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Authors and Affiliations

  1. Purdue University, West Lafayette, USA

    Ehtibar N. Dzhafarov

  2. University of Jyväskylä, Jyväskylä, Finland

    Janne V. Kujala

  3. Linköping University, Linköping, Sweden

    Jan-Åke Larsson

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  1. Ehtibar N. Dzhafarov
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  2. Janne V. Kujala
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  3. Jan-Åke Larsson
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Correspondence to Ehtibar N. Dzhafarov.

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Dzhafarov, E.N., Kujala, J.V. & Larsson, JÅ. Contextuality in Three Types of Quantum-Mechanical Systems. Found Phys 45, 762–782 (2015). https://doi.org/10.1007/s10701-015-9882-9

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  • DOI: https://doi.org/10.1007/s10701-015-9882-9

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