Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> Forty years after the advent of quantum mechanics the problem of hidden variables, that is, the possibility of imbedding quantum theory into a classical theory, remains a controversial and obscure subject. Whereas to most physicists the possibility of a classical reinterpretation of quantum mechanics remains remote and perhaps irrelevant to current problems, a minority have kept the issue alive throughout this period. (See Freistadt [5] for a review of the problem and a comprehensive bibliography up to 1957.) As far as results are concerned there are on the one hand purported proofs of the non-existence of hidden variables, most notably von Neumann’s proof, and on the other, various attempts to introduce hidden variables such as de Broglie [4] and Bohm [1] and [2]. One of the difficulties in evaluating these contradictory results is that no exact mathematical criterion is given to enable one to judge the degree of success of these proposals. <s> BIB001 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> By probabilistic means, the concept of contextuality is extended so that it can be used in non-ideal situations. An inequality is presented, which at least in principle enables a test to discard non-contextual hidden-variable models at low error rates, in the spirit of the Kochen-Specker theorem. Assuming that the errors are independent, an explicit error bound of 1.42% is derived, below which a Kochen-Specker contradiction occurs. <s> BIB002 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> It has recently been questioned whether the Kochen-Specker theorem is relevant to real experiments, which by necessity only have finite precision. We give an affirmative answer to this question by showing how to derive hidden-variable theorems that apply to real experiments, so that non-contextual hidden variables can indeed be experimentally disproved. The essential point is that for the derivation of hidden-variable theorems one does not have to know which observables are really measured by the apparatus. Predictions can be derived for observables that are defined in an entirely operational way. <s> BIB003 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> We presented a contextual statistical model of the probabilistic description of physical reality. Here contexts (complexes of physical conditions) are considered as basic elements of reality. There is discussed the relation with QM. We propose a realistic analogue of Bohr’s principle of complementarity. In the opposite to the Bohr’s principle, our principle has no direct relation with mutual exclusivity for observables. To distinguish our principle from the Bohr’s principle and to give better characterization, we change the terminology and speak about supplementarity, instead of complementarity. Supplementarity is based on the interference of probabilities. It has quantitative expression trough a coefficient which can be easily calculated from experimental statistical data. We need not appeal to the Hilbert space formalism and noncommutativity of operators representing observables. Moreover, in our model there exists pairs of supplementary observables which cannot be represented in the complex Hilbert space. There are discussed applications of the principle of supplementarity outside quantum physics. <s> BIB004 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> The main aim of this report is to inform the quantum information community about investigations on the problem of probabilistic compatibility of a family of random variables: a possibility to realize such a family on the basis of a single probability measure (to construct a single Kolmogorov probability space). These investigations were started hundred of years ago by J. Boole (who invented Boolean algebras). The complete solution of the problem was obtained by Soviet mathematician Vorobjev in 60th. Surprisingly probabilists and statisticians obtained inequalities for probabilities and correlations among which one can find the famous Bell’s inequality and its generalizations. Such inequalities appeared simply as constraints for probabilistic compatibility. In this framework one can not see a priori any link to such problems as nonlocality and “death of reality” which are typically linked to Bell’s type inequalities in physical literature. We analyze the difference between positions of mathematicians and quantum physicists. In particular, we found that one of the most reasonable explanations of probabilistic incompatibility is mixing in Bell’s type inequalities statistical data from a number of experiments performed under different experimental contexts. <s> BIB005 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> The aim of this book is to show that the probabilistic formalisms of classical statistical mechanics and quantum mechanics can be unified on the basis of a general contextual probabilistic model. By taking into account the dependence of (classical) probabilities on contexts (i.e. complexes of physical conditions), one can reproduce all distinct features of quantum probabilities such as the interference of probabilities and the violation of Bells inequality. Moreover, by starting with a formula for the interference of probabilities (which generalizes the well known classical formula of total probability), one can construct the representation of contextual probabilities by complex probability amplitudes or, in the abstract formalism, by normalized vectors of the complex Hilbert space or its hyperbolic generalization. Thus the Hilbert space representation of probabilities can be naturally derived from classical probabilistic assumptions. An important chapter of the book critically reviews known no-go theorems: the impossibility to establish a finer description of micro-phenomena than provided by quantum mechanics; and, in particular, the commonly accepted consequences of Bells theorem (including quantum non-locality). Also, possible applications of the contextual probabilistic model and its quantum-like representation in complex Hilbert spaces in other fields (e.g. in cognitive science and psychology) are discussed. <s> BIB006 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <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> BIB007 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> From behavioral sciences to biology to quantum mechanics, one encounters situations where (i) a system outputs several random variables in response to several inputs, (ii) for each of these responses only some of the inputs may “directly” influence them, but (iii) other inputs provide a “context” for this response by influencing its probabilistic relations to other responses. These contextual influences are very different, say, in classical kinetic theory and in the entanglement paradigm of quantum mechanics, which are traditionally interpreted as representing different forms of physical determinism. One can mathematically construct systems with other types of contextuality, whether or not empirically realizable: those that form special cases of the classical type, those that fall between the classical and quantum ones, and those that violate the quantum type. We show how one can quantify and classify all logically possible contextual influences by studying various sets of probabilistic couplings, i.e., sets of joint distributions imposed on random outputs recorded at different (mutually incompatible) values of inputs. <s> BIB008 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> The amount of contextuality is quantified in terms of the probability of the necessary violations of noncontextual assignments to counterfactual elements of physical reality. <s> BIB009 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> Correlations of spins in a system of entangled particles are inconsistent with Kolmogorov’s probability theory (KPT), provided the system is assumed to be non-contextual. In the Alice–Bob EPR paradigm, non-contextuality means that the identity of Alice’s spin (i.e., the probability space on which it is defined as a random variable) is determined only by the axis \(\alpha _{i}\) chosen by Alice, irrespective of Bob’s axis \(\beta _{j}\) (and vice versa). Here, we study contextual KPT models, with two properties: (1) Alice’s and Bob’s spins are identified as \(A_{ij}\) and \(B_{ij}\), even though their distributions are determined by, respectively, \(\alpha _{i}\) alone and \(\beta _{j}\) alone, in accordance with the no-signaling requirement; and (2) the joint distributions of the spins \(A_{ij},B_{ij}\) across all values of \(\alpha _{i},\beta _{j}\) are constrained by fixing distributions of some subsets thereof. Of special interest among these subsets is the set of probabilistic connections, defined as the pairs \(\left( A_{ij},A_{ij'}\right) \) and \(\left( B_{ij},B_{i'j}\right) \) with \(\alpha _{i}\not =\alpha _{i'}\) and \(\beta _{j}\not =\beta _{j'}\) (the non-contextuality assumption is obtained as a special case of connections, with zero probabilities of \(A_{ij}\not =A_{ij'}\) and \(B_{ij}\not =B_{i'j}\)). Thus, one can achieve a complete KPT characterization of the Bell-type inequalities, or Tsirelson’s inequalities, by specifying the distributions of probabilistic connections compatible with those and only those spin pairs \(\left( A_{ij},B_{ij}\right) \) that are subject to these inequalities. We show, however, that quantum-mechanical (QM) constraints are special. No-forcing theorem says that if a set of probabilistic connections is not compatible with correlations violating QM, then it is compatible only with the classical–mechanical correlations. No-matching theorem says that there are no subsets of the spin variables \(A_{ij},B_{ij}\) whose distributions can be fixed to be compatible with and only with QM-compliant correlations. <s> BIB010 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> We compare two approaches to embedding joint distributions of random variables recorded under different conditions (such as spins of entangled particles for different settings) into the framework of classical, Kolmogorovian probability theory. In the contextualization approach each random variable is"automatically"labeled by all conditions under which it is recorded, and the random variables across a set of mutually exclusive conditions are probabilistically coupled (imposed a joint distribution upon). Analysis of all possible probabilistic couplings for a given set of random variables allows one to characterize various relations between their separate distributions (such as Bell-type inequalities or quantum-mechanical constraints). In the conditionalization approach one considers the conditions under which the random variables are recorded as if they were values of another random variable, so that the observed distributions are interpreted as conditional ones. This approach is uninformative with respect to relations between the distributions observed under different conditions, because any set of such distributions is compatible with any distribution assigned to the conditions. <s> BIB011 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> We describe a mathematical language for determining all possible patterns of contextuality in the dependence of stochastic outputs of a system on its deterministic inputs. The central notion is that of all possible couplings for stochastically unrelated outputs indexed by mutually incompatible values of inputs. A system is characterized by a pattern of which outputs can be"directly influenced"by which inputs (a primitive relation, hypothetical or normative), and by certain constraints imposed on the outputs (such as Bell-type inequalities or their quantum analogues). The set of couplings compatible with these constraints represents a form of contextuality in the dependence of outputs on inputs with respect to the declared pattern of direct influences. <s> BIB012 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> We present general principles underlying analysis of the dependence of random variables (outputs) on deterministic conditions (inputs). Random outputs recorded under mutually exclusive input values are labeled by these values and considered stochastically unrelated, possessing no joint distribution. An input that does not directly influence an output creates a context for the latter. Any constraint imposed on the dependence of random outputs on inputs can be characterized by considering all possible couplings (joint distributions) imposed on stochastically unrelated outputs. The target application of these principles is a quantum mechanical system of entangled particles, with directions of spin measurements chosen for each particle being inputs and the spins recorded outputs. The sphere of applicability, however, spans systems across physical, biological, and behavioral sciences. <s> BIB013 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> This is a chapter for the forthcoming New Handbook of Mathematical Psychology, to be published by Cambridge University Press. A systematic theory of random variables and joint distributions under varying conditions is presented. This is a Kolmogorovian theory most generally understood, in which random variables recorded under mutually exclusive conditions do not possess joint distributions but can be coupled in multiple ways. <s> BIB014 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> Contextuality is a central property in comparative analysis of classical, quantum, and supercorrelated systems. We examine and compare two well-motivated approaches to contextuality. One approach (“contextuality-by-default”) is based on the idea that one and the same physical property measured under different conditions (contexts) is represented by different random variables. The other approach is based on the idea that while a physical property is represented by a single random variable irrespective of its context, the joint distributions of the random variables describing the system can involve negative (quasi-)probabilities. We show that in the Leggett-Garg and EPR-Bell systems, the two measures essentially coincide. <s> BIB015 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> Contextual situations are those in which seemingly 'the same' random variable changes its identity depending on the conditions under which it is recorded. Such a change of identity is observed whenever the assumption that the variable is one and the same under different conditions leads to contradictions when one considers its joint distribution with other random variables (this is the essence of all Bell-type theorems). In our Contextuality-by-Default approach, instead of asking why or how the conditions force 'one and the same' random variable to change 'its' identity, any two random variables recorded under different conditions are considered different 'automatically.' They are never the same, nor are they jointly distributed, but one can always impose on them a joint distribution (probabilistic coupling). The special situations when there is a coupling in which these random variables are equal with probability 1 are considered noncontextual. Contextuality means that such couplings do not exist. We argue that the determination of the identity of random variables by conditions under which they are recorded is not a causal relationship and cannot violate laws of physics. <s> BIB016 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> Contextuality means non-existence of a joint distribution for random variables recorded under mutually incompatible conditions, subject to certain constraints imposed on how the identity of these variables may change across these conditions. In simple quantum systems contextuality is indicated by violations of Bell-type or Leggett-Garg-type inequalities. These inequalities, however, are predicated on the assumption of no-signaling, defined as invariance of the distributions of measurement results with respect to other (e.g., earlier in time) measurements' settings. Signaling makes the inequalities inapplicable: a non-signaling system with any degree of contextuality, however high, loses any relation to this concept as soon as it exhibits any degree of signaling, however small. This is unsatisfactory. We describe a principled way of defining and measuring contextuality in arbitrary systems with random outputs, whether signaling is absent or present. <s> BIB017 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> The possibility of experimentally testing the Bell–Kochen–Specker theorem is investigated critically, following the demonstrations by Meyer, Kent, and Clifton–Kent that the predictions of quantum mechanics are indistinguishable (up to arbitrary precision) from those of a non-contextual model, and the subsequent debate about the extent to which these models are actually classical or non-contextual. The present analysis starts from a careful consideration of these ‘finite-precision’ approximations. A stronger condition for non-contextual models, dubbed ontological faithfulness, is exhibited. It is shown that this allows us to approximately formulate the constraints in Bell–Kochen–Specker theorems, such as to render the usual proofs robust. Consequently, one can experimentally test to finite precision ontologically faithful non-contextuality, and thus experimentally refute explanations from this smaller class. We include a discussion of the relation of ontological faithfulness to other proposals to overcome the finite precision objection.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’. <s> BIB018 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> In this paper we first analyse Leggett and Garg's argument to the effect that macroscopic realism contradicts quantum mechanics. After making explicit all the assumptions in Leggett and Garg's reasoning, we argue against the plausibility of their auxiliary assumption of non-invasive measurability, using Bell's construction of stochastic pilot-wave theories as a counterexample. Violations of the Leggett-Garg inequality thus do not provide a good argument against macrorealism per se. We then apply Dzhafarov and Kujala's analysis of contextuality in the presence of signalling to the case of the Leggett-Garg inequalities, with rather surprising results. An analogy with pilot-wave theory again helps to clarify the situation. <s> BIB019 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> The history of the contextuality-by-default approach <s> We present a proof for a conjecture previously formulated by Dzhafarov, Kujala, and Larsson (Foundations of Physics, in press, arXiv:1411.2244). The conjecture specifies a measure for the degree of contextuality and a criterion (necessary and sufficient condition) for contextuality in a broad class of quantum systems. This class includes Leggett-Garg, EPR/Bell, and Klyachko-Can-Binicioglu-Shumovsky type systems as special cases. In a system of this class certain physical properties $q_{1},...,q_{n}$ are measured in pairs $(q_{i},q_{j})$; every property enters in precisely two such pairs; and each measurement outcome is a binary random variable. Denoting the measurement outcomes for a property $q_{i}$ in the two pairs it enters by $V_{i}$ and $W_{i}$, the pair of measurement outcomes for $(q_{i},q_{j})$ is $(V_{i},W_{j})$. Contextuality is defined as follows: one computes the minimal possible value $\Delta_{0}$ for the sum of $\Pr[V_{i}\not=W_{i}]$ (over $i=1,...,n$) that is allowed by the individual distributions of $V_{i}$ and $W_{i}$; one computes the minimal possible value $\Delta_{\min}$ for the sum of $\Pr[V_{i}\not=W_{i}]$ across all possible couplings of (i.e., joint distributions imposed on) the entire set of random variables $V_{1},W_{1},...,V_{n},W_{n}$ in the system; and the system is considered contextual if $\Delta_{\min}>\Delta_{0}$ (otherwise $\Delta_{\min}=\Delta_{0}$). This definition has its justification in the general approach dubbed Contextuality-by-Default, and it allows for measurement errors and signaling among the measured properties. The conjecture proved in this paper specifies the value of $\Delta_{\min}-\Delta_{0}$ in terms of the distributions of the measurement outcomes $(V_{i},W_{j})$. <s> BIB020
A systematic realization of the idea of contextually labeling a system of measurements R, considering all possible couplings S for it, and characterizing it by the marginals S q that form couplings for the connections R q of the system was developing through a series of publications BIB008 BIB007 BIB010 BIB011 BIB012 . The idea of maximally connected couplings as the central concept for contextuality in consistently connected systems was proposed in Refs. BIB015 BIB016 BIB013 and then generalized to inconsistently connected systems  BIB017 . In the latter two references the measure of contextuality BIB001 and the criterion of contextuality (13) were defined and computed for simple QM systems (cyclic systems of rank 3 and 4, as defined below). Later we added to this list cyclic systems of rank 5, and formulated a conjecture for the measure and criterion formulas for cyclic systems of arbitrary rank  . In Refs.   the contextuality-by-default theory is presented in its current form. The conjecture formulated in Ref.  was proved in Ref. BIB020 . A cyclic system (with binary measurements) is defined as one involving n "objects" (n being called the rank of the system) measured two at a time, For i = 1, . . . , n, the pair (q i , q i⊕1 ) forms the context C i (⊕ standing for circular shift by 1). Each object q i enters in precisely two consecutive contexts, C i 1 and C i . Denoting the measurement of q i in context C j by R , R i i . The formula for the measure of contextuality conjectured in Ref.  and proved in Ref. BIB020 is The function s odd is defined for an arbitrary set of argument x 1 , . . . , x k as where the maximum is taken over all assignments of + and − signs with an odd number of −'s. The criterion of contextuality readily derived from (17) is: the system is contextual if ands only if For consistently connected systems, the sum on the right vanishes, and we can derive the traditional formulas for Legget-Garg (n = 3), EPR/Bell (n = 4), and Klyachko-Can-Binicioglu-Shumovsky-type (KCBS) systems (n = 5). But the formula also allows us to deal with the same experimental paradigm when they create inconsistently connected systems, due to signaling or contextual biases in experimental design (see Refs.  BIB017   for details). The contextuality-by-default theory does have precursors in the literature. The idea that random variables in different contexts are stochastically unrelated was prominently considered in Refs. BIB014 BIB004 BIB005 BIB006 . Probabilities of the eq-type with the contextual labeling of random variables, as defined in (8), were introduced in Refs. BIB002 BIB003 BIB009 BIB018 . The distinguishing feature of the contextuality-by-default theory is the notion of a maximally connected coupling, which in turn is based on the idea of comparing maximal couplings for the connections taken in isolation and those extracted as marginals from the couplings of the entire system. Contextuality-by-default is a more systematic and more general theory of contextuality than those proposed previously, also more readily applicable to experimental data BIB019  .