Contextuality in Three Types of Quantum-Mechanical Systems
- 181 Downloads
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.
KeywordsCHSH inequalities Contextuality Klyachko inequalities Leggett–Garg inequalities Probabilistic couplings Signaling
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.
- 1.Bacciagaluppi, G.: Leggett–Garg inequalities, pilot waves and contextuality. Int. J. Quant. Found. 1, 1–17 (2015)Google Scholar
- 2.Bell, J.: On the Einstein–Podolsky–Rosen paradox. Physics 1, 195–200 (1964)Google Scholar
- 7.de Barros, J.A., Dzhafarov, E.N., Kujala, J.V., Oas, G.: Measuring observable quantum contextuality (2014). arXiv:1406.3088
- 9.Dzhafarov, E.N., Kujala, J.V.: Probability, random variables, and selectivity (2013). arXiv:1312.2239
- 12.Dzhafarov, E.N., Kujala, J.V.: Generalizing Bell-type and Leggett–Garg-type inequalities to systems with signaling (2014). arXiv:1407.2886
- 13.Dzhafarov, E.N., Kujala, J.V.: Probabilistic contextuality in EPR/Bohm-type systems with signaling allowed (2014). arXiv:1406.0243
- 16.Dzhafarov, E.N., Kujala, J.V.: A Qualified Kolmogorovian account of probabilistic contextuality. Lect. Not. Comp. Sci. Springer, Berlin (2014)Google Scholar
- 18.Dzhafarov, E.N., Kujala, J.V.: Random variables recorded under mutually exclusive conditions: contextuality-by-default. In: Liljenström, H. (ed.) Advances in Cognitive Neurodynamics IV, pp. 405–410 (2015)Google Scholar
- 24.Kujala, J.V., Dzhafarov, E.N., Larsson, J.-Å.: Necessary and sufficient conditions for maximal noncontextuality in a broad class of quantum mechanical systems (2014). arXiv:1412.4724
- 29.Specker, E.: Die Logik Nicht Gleichzeitig Entscheidbarer Aussagen. Dialectica 14, 239–246 (1960). (English translation by Seevinck, M.P., available as arXiv:1103.4537.)