QCQMB Virtual May 17

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Marian Kupczynski

University of Quebec in Outaiais (UQO), Canada

Contextuality-by-Default description of Bell Tests

In a system of $n$ dichotomous jointly distributed random variables ${X1,..,Xn }$ taking $±1$ values some marginal expectations obey several non-contextuality inequalities (NCI). Of particular interest are those satisfied by pairwise cyclic expectations ${⟨X1X2 ⟩,⟨X2X3 ⟩,...,⟨XnX1 ⟩}$ such as Boole, Bell-LHCH and KCBS inequalities. If only pairs of random variables may be measured jointly NCI may be violated by the data due statistical fluctuations in finite samples, even if a studied statistical population may be described by a joint probability distribution of all variables. Quantum mechanics and behavioral studies taught us that there exist contextual random variables ‘measuring’ the same content (an answer to the same Yes or No question), which may vary, if they are ‘measured’ jointly with other random variables. Therefore, according to Contextuality-by-Default (CbD) in a real experiment we do not have in a cyclic system, obeying NCI, but a system ${⟨X ′X2⟩,⟨X′X3⟩ ,⟨X ′X1⟩} 1 2 n$ in which random variables $Xi$ and $X′ i$ are statistically unrelated. In order to test the degree of contextuality of such system one derives, in CbD, modified NCI. It is often believed that, because of Einsteinian no-signaling, we don’t have this problem in Bell Tests because Alice’s and Bob’s measurement may not depend on distant settings. In fact, if Alice and Bob use two different PBS settings and two detectors for each setting, then in synchronized time windows of the width W, they may describe their raw data using 4 random variables $Ai$ and $Bj$ taking the values $±1$ or $0$ . The no-signaling is confirmed and $⟨Ai⟩$ and $⟨Bj⟩$ do not depend on what was ‘measured’ on other side. However, to perform a Bell Test we have to define a W-dependent coupling between Alice’s and Bob’s non vanishing outcomes and new coupled post-selected samples, for different settings are described now by 4 pairs of 8 context dependent random variables ${(A11,B11),(A12,B21),(A21,B12),(A22,B22)}$ . The data show that the resulting random variables are not consistently connected $⟨A11⟩ ⁄= ⟨A12⟩,⟨B21⟩ ⁄= ⟨B22⟩$ , thus to study the contextuality of this system, more in detail, one should use CbD approach instead of CHSH inequalities. It is obvious that the violation of these inequalities has nothing to do with nonlocal physical influences between distant experimental settings, similarly as it is a case in KCBS experiments and in cognitive psychology. This is why one should not talk about signaling and nonlocality, if a significant violation of NCI is observed. Since the correlations observed in Bell Tess may be explained in a locally causal way such terminology is not only imprecise but misleading.

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[2] J. V. Kujala, E. N. Dzhafarov, and J.-Å. Larsson, Necessary and sufficient conditions for extended noncontextuality in a broad class of quantum mechanical systems, Phys. Rev. Lett. 115, 150401 (2015).

[3] Kupczynski M., Can we close the Bohr-Einstein quantum debate?, Phil.Trans.R.Soc.A., 2017, 20160392., DOI: 10.1098/rsta.2016,0392

[4] Kupczynski M. Is Einsteinian no-signalling violated in Bell tests? Open Physics, 2017, 15 , 739-753, DOI: https://doi.org/10.1515/phys-2017-0087,2017

[5] Kupczynski M, Is the Moon there if nobody looks: Bell inequalities and Physical Reality, Front. Phys., 23. https://doi.org/10.3389/fphy.2020.00273