Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> Pseudo-quasi-distance tests <s> We provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no‐hidden‐variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem. <s> BIB001 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> Pseudo-quasi-distance tests <s> A general definition and a criterion (a necessary and sufficient condition) are formulated for an arbitrary set of external factors to selectively influence a corresponding set of random entities (generalized random variables, with values in arbitrary observation spaces), jointly distributed at every treatment (a set of factor values containing precisely one value of each factor). The random entities are selectively influenced by the corresponding factors if and only if the following condition, called the joint distribution criterion, is satisfied : there is a jointly distributed set of random entities, one entity for every value of every factor, such that every subset of this set that corresponds to a treatment is distributed as the original variables at this treatment. The distance tests (necessary conditions) for selective influence previously formulated for two random variables in a two-by-two factorial design (Kujala & Dzhafarov, 2008, J. Math. Psychol., 52, 128–144) are extended to arbitrary sets of factors and random variables. The generalization turns out to be the simplest possible one: the distance tests should be applied to all two-by-two designs extractable from a given set of factors. <s> BIB002 </s> Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview <s> Pseudo-quasi-distance tests <s> We construct a class of real-valued nonnegative binary functions on a set of jointly distributed random variables, which satisfy the triangle inequality and vanish at identical arguments (pseudo-quasi-metrics). These functions are useful in dealing with the problem of selective probabilistic causality encountered in behavioral sciences and in quantum physics. The problem reduces to that of ascertaining the existence of a joint distribution for a set of variables with known distributions of certain subsets of this set. Any violation of the triangle inequality or its consequences by one of our functions when applied to such a set rules out the existence of this joint distribution. We focus on an especially versatile and widely applicable pseudo-quasi-metric called an order-distance and its special case called a classification distance. <s> BIB003
The conventional pseudometrics (also called semimetrics) obtain by adding the property d (H 1 , H 2 ) = d (H 2 , H 1 ); the conventional quasimetrics are obtained by adding the property Pr [ A conventional metric is both a pseudometric and a quasimetric. A sequence of input points where w i ∈ α λi for i = 1, . . . , l ≥ 3, is called treatment-realizable if there are treatments φ 1 , . . . , φ l ∈ Φ (not necessarily pairwise distinct), such that If a JDC-set H exists, then for any p.q.-metric d on H we should have and This chain inequality constitutes a p.q.-metric test for selective influences. If this inequality is found not to hold for at least one treatment-realizable sequence of input points, selectivity (7) is ruled out BIB003 . It turns out that one needs to check the chain inequality only for irreducible treatment-realizable sequences x 1 , . . . , x l , i.e., those with x 1 = x l and with the property that the only subsequences {x i1 , . . . , x i k } with k > 1 that are subsets of treatments are pairs {x 1 , x l } and {x i−1 , x i }, for i = 2, . . . , l. Inequality (20) is satisfied for all treatment-realizable sequences if and only if it holds for all irreducible sequences BIB003 . The situation is even simpler if Φ = λ∈Λ W λ (all logically possible treatments are allowable). Then (20) is satisfied for all treatment-realizable sequences if and only if this inequality holds for all tetradic sequences of the form x, y, s, t, with x, s ∈ {λ 1 } × α λ1 , y, t ∈ {λ 2 } × α λ2 , x = s, y = t, λ 1 = λ 2 BIB002 . Order-distances constitute a special class of p.q.-metrics, defined as follows. and let us write a b for (a, b) ∈ R. Let R be a total order (transitive, reflexive, and connected). We assume that for any , a b belongs to the product sigmaalgebra over Σ λ1 and Σ λ2 . Then the function where ≺ is the strict order induced by , is well-defined, and it is a p.q.-metric on H, called order-distance BIB003 . As a simple example, consider the results of a CHSH type experiment with two spin axes per each of two entangled 1 /2-spin particles. Enumerate the spin axes 1, 2 for either particle, enumerate the two outcomes (up and down) of each measurement 1, 2 for particle 1 and 1 ′ , 2 ′ for particle 2, and denote where i, j, k, l ∈ {1, 2}. Define the order-distance D 1 by putting 1 ≃ 1 ′ ≺ 2 ≃ 2 ′ , where ≃ is equivalence induced by . We have then the chain inequality Consider next a similar inequality for the order-distance By simple algebra, denoting the conjunction of (24) and (25) can be shown to be equivalent to One derives analogously The four double-inequalities (27)- (28) can be referred to as the Bell-CHSHFine inequalities BIB001  , necessary and sufficient conditions for the CHSH type experiment to have a "classical" explanation.