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Ehtibar N. Dzhafarov

Purdue University, USA, ehtibar@purdue.edu

free choice, causality, contextuality, and signed measures

free choice vs local causality, contextuality vs causality, negative probabilities

A hidden variable model (HVM) satisfies the assumption of Free Choice (and is denoted HVM-FC) if the settings chosen by experimenters are independent of the values of the hidden variable. An HVM satisfies the assumption of Context-Independence (HVM-CI) if the outcomes of measurements are independent of settings for other measurements. If the measurements are spacelike separated, CI assumption is known as Local Causality. An HVM satisfies Bell-type criteria of contextuality/nonlocality (HVM-B) if and only if it is both HVM-FC and HVM-CI. We show, in complete generality, for any system of random variables with or without disturbance, that any HVM-CI can be reformulated as an HVM-FC, and vice versa. Moreover, an HVM unconstrained by either of these assumptions (HVM-Gen) can always be reformulated as an HVM-CI or HVM-FC (so, by itself, neither of the two assumptions is constraining). It follows that measures of the degree of FC and the degree of CI in an HVM can always be made the same. Contextuality-by-Default (CbD) approach allows one to separate two components of context-dependence in any HVM-FC formulation: (A) direct context-dependence (overt, causal) and (B) contextuality proper (hidden, non-causal). These two components can be separately measured, and in CbD we have developed three distinct measures of contextuality. One of them, CNT3, is based on the use of signed mea- sures, i.e. quasi-probabilities allowed to be negative. Any HVM has a quasi-coupling, in which the HVM is represented by a set of variables with joint signed distribution. CNT3 is the minimal total variation of this distribution.

[1] Dzhafarov, E.N. (2022). Context-independent mapping and free choice are equivalent: A general proof. J. Phys. A: Math. Theor. 55:305304. + Corrigendum Note in J. Phys. A: Math. Theor. (2022) 55:4095020. (arXiv:2110.15910)

[2] Dzhafarov, E.N., Kujala, J.V. (2020). Systems of random variables and the Free Will Theo- rem. Phys. Rev. Res 2:043288. (arXiv:2007.13229)

[3] Dzhafarov, E.N., Cervantes, V.H., Kujala, J.V. (2017). Contextuality in canonical systems of random variables. Phil. Trans. Roy. Soc. A 375:20160389. (arXiv:1703.01252)