In the conventional approach to quantum mechanics, indeterminism is an axiom and nonlocality is a theorem. We consider inverting the logical order, making nonlocality an axiom and indeterminism a theorem. Nonlocal “superquantum” correlations, preserving relativistic causality, can violate the CHSH inequality more strongly than any quantum correlations.

Let A 1 , A 2 , A 3 A 4 be four observables, the compatible observables among them being (A 1 , A 3 ), (A 1 , A 4 ), (A 2 , A 3 ), (A 2 , A 4 ). In order that the empirical data be reproducible by a quantum or a classical theory, the two-point correlation functions $$\{ C_{ij} = \left\langle {A_i A_j } \right\rangle :i,j a compatible pair\} $$ must necessarily satisfy $$|X_{13} X_{14} (...) - X_{23} X_{24} | \leqslant \left( {1 - X_{13} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left( {1 - X_{14} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \left( {1 - X_{23} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left( {1 - X_{24} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} (*)$$ where Xij=CijC ii −1/2 C jj −1/2 . In the case ofGaussian data, this inequality is alsosufficient; If (*) holds, there is a Gaussian joint distribution for A 1 , A 2 , A 3 , A 4 which reproduces the Gaussian data for compatible pairs. It follows that Bell's inequality is satisfied by all true-false propositions about the Gaussian data. A further consequence of the analysis is thatquantum Gaussian fields satisfy Bell's inequality for all true-false propositions aboutfield measurements.The maximum violation of (*) corresponds to Rastall's example in the case of two-valued observables. (shrink)

We present a formal theory of contextuality for a set of random variables grouped into different subsets 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. 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. (shrink)

The Bell/CHSH inequalities of quantum physics are identical with the inequalities derived in mathematical psychology for the problem of selective influences in cases involving two binary experimental factors and two binary random variables recorded in response to them. The following points are made regarding cognitive science applications: (1) compliance of data with these inequalities is informative only if the data satisfy the requirement known as marginal selectivity; (2) both violations of marginal selectivity and violations of the Bell/CHSH inequalities are interpretable (...) as indicating that at least one of the two responses is influenced by both experimental factors. (shrink)