Contextuality, Complementarity, Signaling, and Bell Tests

This is a review devoted to the complementarity–contextuality interplay with connection to the Bell inequalities. Starting the discussion with complementarity, I point to contextuality as its seed. Bohr contextuality is the dependence of an observable’s outcome on the experimental context; on the system–apparatus interaction. Probabilistically, complementarity means that the joint probability distribution (JPD) does not exist. Instead of the JPD, one has to operate with contextual probabilities. The Bell inequalities are interpreted as the statistical tests of contextuality, and hence, incompatibility. For context-dependent probabilities, these inequalities may be violated. I stress that contextuality tested by the Bell inequalities is the so-called joint measurement contextuality (JMC), the special case of Bohr’s contextuality. Then, I examine the role of signaling (marginal inconsistency). In QM, signaling can be considered as an experimental artifact. However, often, experimental data have signaling patterns. I discuss possible sources of signaling—for example, dependence of the state preparation on measurement settings. In principle, one can extract the measure of “pure contextuality” from data shadowed by signaling. This theory is known as contextuality by default (CbD). It leads to inequalities with an additional term quantifying signaling: Bell–Dzhafarov–Kujala inequalities.


Introduction
This is a review devoted to the interplay of notions of contextuality and complementarity as the interpretational basis of the violation of the Bell inequalities [1]- [3].We set essential efforts to clarify and logically structure Bohr's views [4] on contextuality and contextuality's crucial role in the derivation of the complementarity principle [5]- [11](see also [12,13]).In fact, in Bohr's writings these two notions are really inseparable.We recommend to the reader the books of Plotnitsky and Jaeger [14]- [17] clarifying Bohr's views on complementarity and contextuality.Bohr did not use the notion of contextuality.He wrote about experimental conditions.But in the modern terminology he appealed to contextuality of quantum measurements.We remark that at the beginning Bell neither used this terminology.This notion was invented in QM by Beltrametti and Cassinelli [18].
In philosophic terms Bohr's contextuality means rejection of "naive realism"; by Bohr the outcomes of quantum measurements cannot be treated as the objective properties of a system under observation.These values cannot be assigned to a system before a measurement, with exception of special system's states -the eigenstates of observables.However, we do not like to operate with the notion of realism including the EPR elements of reality.We leave this field for philosophers who have been working on it during the last two thousands years.Instead we will work with the notion of Bohr's contextuality which is formulated in the heuristically clear physical terms -the interaction between a system and a measurement device.We would neither operate with the notion of local realism.I think that this is an ambiguous notion, but this is just my personal viewpoint.At least one has to split local realism into two components, realism and locality, and then analyze them separately.We will shortly discuss this notion and its components in appendix A.
In this review we do not try to cover all approaches to contextuality; in particular, we do not discuss the Kochen-Specker theorem and the corresponding contextuality (see the recent review of Svozil [19] for the description of the diversity of the views on contextuality).
Starting with mentioning the Bohr principle of complementarity also known as "wave-particle duality" , we analyze the notion of contextuality.The latter is understood very generally, as the irreducible dependence of observable's outcome on the experimental context.Thus, the outcomes of quantum observables are not the objective properties of systems.They are generated in the complex process of interaction between a system and a measurement device.In fact, "Bohr-contextuality" is the seed of complementarity, the existence of incompatible observables [5]- [11].
In the probabilistic terms, incompatibility means that JPD does not exist.Instead of the JPD, one has to operate with a family of probability distributions depending on experimental contexts as in the the Växjö model for contextual probability theory [20]- [31].This model generalizes the notion of conditional probability from classical probability (CP) theory.In some cases the contextual probability update can be represented via the state update of the projection type represented in the complex Hilbert space [21]- [24], [29], [33,34].And, of course, the probability update of quantum theory can be easily realized as update of contextual probability.The update machinery is formalized via introduction of special contexts corresponding to the outcomes of observables [20]- [31].
We continue to analyze the probabilistic structure of QM by considering the Bell inequalities and concentrating on the CHSH-inequality [35] and the Fine theorem [36].This theorem connects Bell inequality with the existing of the JPD for four observables involved in the Bohm-Bell experiment, in fact the group of four separate experiments for the pairwise measurements for some pairs of these observables.We use the Fine theorem as the bridge to the contextual interpretation of the Bell type inequalities.For context dependent probabilities in the absence of JPD unifying them, these inequalities can be violated [29].We point out that contextuality tested by the Bell inequalities is so called joint measurement contextuality (JMC) [2] (and section 2.3) -the very special case of Bohr's contextuality.We stress that consideration of JMC is dominating within the quantum studies of contextuality.On one hand, this simplifies the picture; on the other hand, by reducing Bohr's contextuality to JMC people miss the general contextual perspective as it was established by Bohr at the very beginning of QM.Some authors even define contextuality directly as the violation of some Bell inequality (see, eg., [37] and references herein).We call such type of contextuality Bell contextuality.However, Bell by himself invented contextuality [2] as JMC and then he pointed out that JMC can serve as a source of "Bell contextuality".
We remark that originally Bell explained the violation of his inequality by Einsteinian nonlocality [38], "spooky action at a distance" -Einstein's hype slogan.In article [2] Bell discussed contextuality in the JMC form in connection with nonlocality (see also related papers of Gudder [39]- [41] and Shimony [42,43]).However, JMC per se cannot clarify the origin of Einsteinian nonlocality.In Bell's discussion [2] JMC looks even more mystical than nonlocality.Consideration of JMC as the special case of Bohr contextuality and connecting it with incompatibility, demystifies JMC.And by highlighting the role of incompatibility, the debate on the meaning of the Bell type inequalities turns to the very basics of QM, to Bohr's complementarity principle and the existence of incompatible observables.The Bell inequalities are interpreted as the special tests of contextuality and, hence, incompatibility [6,7].Coupling contextuality-incompatibility is basic in our treatment of the Bell inequalities.This review continues the line of articles -"getting rid off nonlocality from quantum physics" [6]- [9] (see also [44]- [61]).
We also examine signaling which may be better to call marginal inconsistency by following the line of research presented in articles of Adenier and Khrennikov [62]- [67].Typically its role in discussions on the Bell inequalities is not highlighted.In contrast to the majority of authors, we take very seriously complications related to the presence signaling patterns in experimental statistical data [63].It must be noted that the terminology "signaling" is quite ambiguous, since in fact "signaling"is defined not in terms of signals propagating in physical space-time, but in purely probabilistic framework, as non-coincidence of marginal probability distributions corresponding to join measurements of an observables a with other observables which are compatible with it.
In QM, signaling can be considered as an experimental artifacttheoretically there should be no signaling.However, often experimental data has signaling patterns which are statistically non-negligible [63], [68]- [72].We discuss possible sources of signaling, both in the theoretical and experimental frameworks.In particular, we point out to dependence of the state preparation procedure on settings of measurement devices as a signaling source (cf.[68,69,73]): the standard source state generation is supplemented with additional state modification which is setting dependent.We emphasize that in the studies on interrelation between classical and quantum physics, signaling cannot be ignored.The presence of signaling in the experimental statistical data per se means that such data cannot be modeled within QM.So, in such a case there is no need to check whether some Bell inequality is violated or not.In the presence of signaling approaching the high level of the violation of e.g. the CHSH-inequality is totally meaningless.Even tremendous efforts to close all possible loopholes meaningless if data suffers of signaling.
We remark that, as was recently found by Dzhafarov et al. [74]- [78], one can extract the measure of pure contextuality even from statistical data shadowed by signaling.This theory known as Contextuality by Default (CbD) is based on coupling technique of CP.CbD with mathematical technique from CP leads to the Bell inequalities with the additional term quantifying the level of signaling, we call such inequalities the Bell-Dzhafarov-Kujala inequalities (BDK).In this review, we are concentrated on the CHSH-BDK inequality.Generally, CbD can be considered as a part of the project on the CP-treatment of the Bell inequalities and contextuality.Another part of this project was presented in [79]- [82], where quantum probabilities were treated as classical conditional probabilities with conditioning w.r.t. the selection of experimental settings (cf.with Koopman [83], Ballentine [46], [84]- [87]).This is the good place to mention the CP-based tomographic approach to QM which was developed by Vladimir Man'ko and coauthors [88]- [91].We also point out to articles [78] and [92] for a debate on the perspectives of the CP-use in contextual modeling (without direct connection with QM).
I also would like to inform physicists that nowadays quantum theory, its methodology and mathematical formalism, are widely applied outside of physics, to cognition, psychology, decision making, social and political sciences, economics and finances (see, e.g., monographs [93]- [99] and references in them).I called this kind scientific research quantum-like modeling and this terminology was widely spread.In particular, contextuality based on the quantum studies attracted a lot of attention, especially in cognitive psychology and decision making, including the Bell tests [97], [100]- [105].One of the specialties of such studies is the presence of signaling patterns in statistical data collected in all experiments which were done up to now [102].Here the BDK-inequalities are especially useful [104,105].
In this review we discuss mainly the CHSH inequality.This is motivated by two reasons, experimental and theoretical ones.The basic of experiments were done for this inequality [68,69,71,106,107] (with some very important exceptions [70,108], see also [72]).The mathematical structure of this inequality makes it possible to establish the straightforward coupling with incompatibility expressed mathematically in the form of commutators [6] (section 6).From my viewpoint, the original Bell inequality derived under the assumption on the prefect correlations deserves more attention, both theoretically and experimentally; some steps in this direction were done in works [109]- [111].
In this review we are concentrated only on the Bohr contextuality and its "derivatives", JMC and Bell contextuality.We neither discuss hidden variables theory.The latter may be surprising, since from the beginning the Bell inequalities were derived in hidden variables framework.However, we treat these inequalities as statistical tests of incompatibility.In the presence of incompatible observables, it is meaningless to discuss theories with hidden variables, at least theories in which hidden variables are straightforwardly connected with the outcomes of observables as was done by Bell and his followers.Already De Broglie pointed out that such theories have no physical meaning.
In principle, one can consider subquantum models, but variables of such models are only indirectly coupled to outcomes of quantum observables.The latter viewpoint was advertized by Schrödinger [113] who in turn followed the works fo Hertz [114] and Boltzmann [115,116] (see also [117,118]).One of such subquantum theories was developed in the series of author's works on emergence of QM from classical random field theory [119].

Forgotten contribution of Bohr to contextuality theory
Contextuality is one of the hottest topics of modern quantum physics, both theoretical and experimental.During the recent 20 years, it was discussed in numerous papers published in top physical journals.Unfortunate of these discussions is that from the very beginning contextuality (JMC, section 2.3) was coupled to the issue of nonlocality.It was Bell's intention in his analysis of the possible seeds of the violation of the Bell type inequalities [2].
Surprisingly, Bell had never mentioned general contextuality which we call "Bohr contextuality".The latter has no straightforward coupling to the Bell inequalities; it is closely related to the notion of incompatibility of observables -the Bohr principle of complementarity.What is even more surprising that Shimony who was one of authorities in quantum foundations by commenting [42,43] Bell's article [2] had neither mentioned the Bohr principle of complementarity and its contextual dimension.
One of the explanations for this astonishing situation in quantum foundations is that Bohr presented his ideas in a vague way; moreover, he often changed his vague formulations a few times at different occasions.In this section we briefly present Bohr's ideas about contextuality of quantum measurements and its role in his formulation of the complementarity principle (see [5]- [11] for detailed presentations).Then, we move to the Bell inequalities.This pathway towards these inequalities (i.e., via Bohr's contextuality-complementarity) highlights the role of incompatibility of quantum observables in the Bell framework and gives the possibility to operate with the Bell inequalities without mentioning the ambiguous notion of quantum nonlocality (spooky action at a distance).

What does contextuality mean?
In this situation when so many researchers write and speak about quantum contextuality, one should be sure that this notion is well defined and its physical interpretation is clear and well known.In fact, before started to think about the meaning of contextuality, I was completely sure in this.Strangely enough, I was not able to create a consistent picture.And I was really shocked when by visiting the institute of Atom Physics in Vienna and having conversation with Rauch and Hasegawa, I found that they are also disappointed.They asked me about the contextuality meaning.And they performed the brilliant experiments [120,121] to test contextuality in the framework of neuron interferometry.They had a vague picture of what was tested and what is the physical meaning of their experimental results!Then, in Stockholm by being in the PhD defense jury of one student who was supervised by prof.Bengtsson (let call her Alice), I asked Alice about the physical meaning of contextuality.(Her thesis was about it.)Alice answered that she has no idea about the physical interpretation of advanced mathematical results obtained in her thesis.Generally I like discussions.To stimulate a debate, I told that Rauch and Hasegawa had the strange idea that contextuality is just noncommutativity, a sort of the order effect in the sequential measurements (this was the final output of our discussions in Vienna).Unfortunately, in Stockholm the discussion quickly finished with the conclusion that the question is interesting, but not for the PhD-defense.

Jump from contextuality to Bell inequalities
Typically by writing a paper about contextuality in QM one starts by referring to this notion as joint measurement contextuality (JMC): dependence of the outcomes of some observable a on its joint measurement with another observable b.We note that this definition is countefactual and cannot be used in the experimental framework.
Nevertheless, the "universal contextuality writer" is not disappointed by this situation and he immediately jumps to the Bell inequalities which are treated as noncontextual inequalities (see,e.g., [37]).Moreover, contextuality is often identified with the violation of the Bell inequalities -Bell contextuality in our terminology.This identification shadows the problem of the physical meaning of contextuality.One jumps from the problem of understanding to calculation of a numerical quantity, the degree of the violation of some Bell inequality.Such inequalities are numerous.And they can be tested in different experimental situations and generate the permanent flow of highly recognized papers.
I suggested the following critical illustration to this strategy (contextuality = violation of the Bell inequalities) [122].Consider the notion of a random sequence.Theory of randomness is the result of the intensive research (Mises, Church, Kolmogorov, Solomovov, Chatin, Martin-Löf ; see,e.g., the first part of my book [123]).This theoretical basis led to elaboration of the variety of randomness tests which are used to check whether some sequence of outputs of physical or digital random generator is random.But, in fact, it is possible to check only pseudo-randomness.The universal test of randomness, although exist, but the proof of its existence is nonconstructive and this test cannot be applied to the concrete sequence of outcomes.
In applications the NIST test (a batch of tests for randomness) is the most widely used.So, in theory of randomness we also use tests, but beyond them there is the well developed theory of randomness.In particular, this leads to understanding that even if a sequence x passed the NIST test, this does not imply that it is random.In principle, there can be found another test such that x would not pass it.The latter would not be a surprise.
In contrast to the above illustration, in QM contextuality is per definition the violation of some noncontextual (Bell) inequality (at least for some authors).Hence, the theoretical notion is identified with the Bell test; in fact, the batch of the tests corresponding to different Bell inequalities.(The Bell test for classicality plays the role of the NIST test for randomness).This is really bad!Not only from the theoretical viewpoint, but even from the practical one.As was mentioned, by working with randomness people understand well that even passing the NIST test does not guarantee randomness.In QM, passing the Bell test is per definition is equivalent to contextuality.This is wrong strategy which led to skews in handling quantum contextuality.

Signaling and other anomalies in data
The first signs that addiction to one concrete test of contextuality (Bell inequalities) may lead to the wrong conclusions were observed by Adenier and Khrennikov [62]- [67].Adenier was working on the translation of the PhD thesis of Alain Aspect (due to the joint agreement with prof.Aspect and Springer) and he pointed out to me that he found some strange anomalies in Aspect's data [68].One of them was signaling.i.e., dependence of detection probability on one side (Bob's lab) on the selection of an experimental setting on another side (Alice's lab).
Then, we found signaling in the data from the famous Weihs experiment closing the nonlocality loophole [69].Our publications [62]- [67] attracted attention to the problem of signaling in data collected in quantum experiments.Slowly people started to understand that experimenter cannot be happy by just getting higher degree of the violation of say the CHSH-inequality, with higher confidence.Often this implied the increase of the degree of signaling.Experimenters started to check the hypothesis of signaling in data [108,106].Unfortunately, our message was ignored by some experimenters, e.g., the data from the "the first loophole free experiment" [71] demonstrated statistically significant signaling.
Any Bell test should be combined with the test of experimental statistical data on signaling.
We pointed out that signaling was not the only problem in Aspect's data.As he noted in his thesis [68], the data contains "anomalies" of the following type.Although the CHSH-combination of correlations violates the CHSH-inequality, the correlation for the concrete pair of angles θ 1 , φ, as the function of these angles, does not match the theoretical prediction of QM, the graph of the experimental data differs essentially from the theoretical cos-graph.Our attempts to discuss this problem with other experimenters generated only replies that "we do not have such anomalies in our data". 1

Växjö model: Contextuality-complementarity and probability
In the probabilistic terms complementarity, incompatibility of observables, means that their joint probability distribution (JPD) does not exist.Instead of the JPD, one has to operate with context-dependent family of probability spaces -the Växjö probability model [20]- [31]: where Z is a family of contexts and, for each C ∈ Z, Here Ω C is a sample space, F C is a σ-algebra of subsets of Ω C (events), and P C is a probability measure on F C .All these structures depend on context C. To develop a fruitful theory, Z must satisfy to some conditions on interrelation between contexts.THese conditions give the possibility to create an analog of the CP calculus of conditional probabilities.
In CP the points of Ω C represent elementary events, the most simple events which can happen within context C.Although these events are elementary, their structure can be complex and include the events corresponding to appearance of some parameters ("hidden variables") for a system under observation and measurement devices, times of detection and so on.
Observables are given by random variables on contextually-labeled probability spaces, measurable functions, a C : Ω C → R. The same semantically defined observable a is represented by a family of random variables (a C , C ∈ Z a ), where Z a is the family of contexts for which the a-observable can be measured.In M Z averages and correlations are also labeled by contexts, where P a|C is the probability distribution of a c and P a,b|C is the JPD of the pair of random variables (a C , b C ).In (1) C ∈ Z a and in (2) both observables a and b are represented by random variables, namely, by a C and b C , it is natural to assume that in this context both observables can be measured and the measure-theoretic JPD P a,b|C represents mathematically the JPD for joint measurements of the pair of observables (a, b).
In further sections, we analyze the probabilistic structure of QM by considering the Bell inequalities and concentrating on the CHSHinequality [35] and the Fine theorem [36].

Summary of preliminary discussion
We can conclude the discussion with a few statements: • The theoretical definition of contextuality as JMC suffers of appealing to conterfactuals.
• Identification of contextuality with the violation of the Bell inequalities is not justified, neither physically nor mathematically (in the last case such an approach does not match the mathematical tradition).
• The Bell tests have to accompanied with test on signaling.
• Probabilistically contextuality-complementarity is described by contextual probability (as by the Växjö model).

Rethinking Bohr's ideas
This section is devoted to rethinking of Bohr's foundational works in terms of contextuality.I spent a few years for reading Bohr and rethinking his often fuzzy formulations.

Bohr Contextuality
The crucial question is about the physical meaning of contextuality; without answering to it, JMC (even by ignoring counterfactuality) is mystical, especially for spatially separated systems.Even spooky action at a distance is welcome -to resolve this mystery.
In series of my papers [5]- [11] the physical meaning of contextuality was clarified through referring to the Bohr's complementarity principle.Typically this principle is reduced to wave-particle duality.(In fact, Bohr had never used the latter terminology.)However, Bohr's formulation of the complementarity principle is essentially deeper.Complementarity is not postulated; for Bohr, it is the natural consequence of the irreducible dependence of observable's outcome on the experimental context.Thus, the outcomes of quantum observables are generated in the complex process of the interaction of a system and a measurement device [4] (see also [10], [32]).This dependence on the complex of experimental conditions is nothing else than a form of contextuality, Bohr-contextuality (section 3.2).We remark that JMC is its special case.But, in contrast to JMC, the physical interpretation of Bohr-contextuality is transparent -dependence of results of measurements on experimental contexts.And it does not involve the use of conterfactuals.
Such contextuality is the seed of complementarity, the existence of incompatible observables.(We recall that observables are incompatible if they cannot be measured jointly.)Moreover, contextuality without incompatibility loses its value.
If all observables were compatible, then they might be jointly measured in a single experimental context and multicontextual consideration would be meaningless.

One can go in deeper foundations of QM and ask:
Why is dependence on experimental context (system-apparatus interaction) is irreducible?
Bohr's answer is that irreducibility is due to the existence of indivisible quantum of action given by the Planck constant (see my article [8,9] for discussion and references).

Bohr's Principle of Contextuality-complementarity
The Bohr principle of complementarity [4] is typically presented as wave-particle duality, incompatibility of the position and momentum observables.The latter means the impossibility of their joint measurement.We remark that Bohr started with the problem of incompatibility of these observables by discussing the two slit experiment.In this experiment position represented by "which slit?" observable and momentum is determined the detection dot on the registration screen.(This screen is covered by photo-emulsion and placed on some distance beyond the screen with two slits.)Later Bohr extended the wave-particle duality to arbitrary observables which cannot be jointly measured and formulated the principle of complementarity.He justified this principle by emphasizing contextuality of quantum measurements.The Bohr's viewpoint on contextuality was wider than in the modern discussion on quantum contextuality related to the Bell inequality.The later is contextuality of joint measurement with a compatible observable (section 2.3).
In 1949, Bohr [4] presented the essence of complementarity in the following widely citing statement: "This crucial point ... implies the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear.In fact, the individuality of the typical quantum effects finds its proper expression in the circumstance that any attempt of subdividing the phenomena will demand a change in the experimental arrangement introducing new possibilities of interaction between objects and measuring instruments which in principle cannot be controlled.Consequently, evidence obtained under different experimental conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects." In short, Bohr's way to the complementarity principle, the claim on the existence of incompatible quantum observables, can be presented as the following chain of reasoning [5]- [11]: • CONT1 An outcome of any observable is composed of the contributions of a system and a measurement device.2 • CONT2 The whole experimental context has to be taken into account.
• INCOMP1 There is no reason to expect that all experimental contexts can be combined with each other.
• INCOMP2 Therefore one cannot expect that all observables can be measured jointly.Therefore it is more natural to speak about two Bohr's principles: • Contextuality Principle.
• Complementarity Principle.And the second principle is a consequence of the first one.So, contextuality (understood in the Bohr's sense) is the seed of complementarity.We can unify these two principles and speak about the Contextuality-Complementarity Principle.Unfortunately, the contextual dimension of Bohr's complementarity is typically missing in the discussions on quantum foundations.By speaking about the wave-particle duality one typically miss that the wave and particle properties of a system cannot be merged in a single experimental framework, because these properties are contextual; their are determined within two different experimental contexts.
We state once again than the essence of QM is not in complementarity, but in contextuality.The real surprise is not that say position and momentum observables are incompatible, but in contextuality (in Bohr's sense) of each of them.The surprise (for classical physicist) is that neither position nor momentum "exist" before measurements, i.e., they cannot be considered as the objective properties of the quantum systems..
In the light of Bohr-contextuality, the following natural question arises: How can one prove that the concrete observables a and b cannot be jointly measured (i.e., that they are incompatible)?
From the viewpoint of experimental verification, the notion of incompatibility is difficult.How can one show that the joint measurement of a and b is impossible?One can refer to the mathematical formalism of quantum theory and say that the observables a and b cannot be jointly measurable if the corresponding Hermitian operators A and B do not commute.But, another debater can say that may be this is just the artifact of the quantum formalism: yes, the operators do not commute, but observables still can be jointly measured.

Probabilistic Viewpoint on Contextuality-Complementarity
The basic analysis on the (in)compatibility problem is done in the probabilistic terms.Suppose that observables a, b, c, ... can be in principle jointly measured, but we are not able to design the corresponding measurement procedure.Nevertheless, the assumption of joint measurability, even hypothetical, implies the existence of JPD.

What are consequences of JPD's existence?
We shall comeback to this question in section 4.1.Now we remark that the principle of contextuality-complementarity can be reformulated in probabilistic terms.In short, we can say that the measurement part of QM is a (special) calculus of context-dependent probabilities.This viewpoint was presented in a series of works summarized in monograph [29] devoted to the calculus of context dependent probability measures (P C ), C ∈ Z, where Z is a family of contexts constrained by some consistency conditions.
We emphasize that QP is a special contextual probabilistic calculus.Its specialty consists in the possibility to use a quantum state (the wave function) |ψ to unify generally incompatible contexts.This is the important feature of QP playing the crucial role in quantum foundations.
In classical statistical physics the contextuality of observations is not emphasized.Here it is assumed that it is possible to proceed in the CP-framework: to introduce a single context-independent probability measure P and reproduce the probability distributions of all physical observables on the basis of P.This is really possible.However, the careful analysis of interplay of probability measures appearing in classical physics shows that even here contexuality cannot be ignored.In articles [125,126], there are considered models, e.g., in theory of complex disordered systems (spin glasses), such that it is impossible to operate with just one fixed probability measure P. A variety of context dependent probabilities have to be explored.We especially emphasize the paper on classical probabilistic entanglement [127].

Existence vs. Non-existence of Joint Probability Distribution
Let P = (Ω, F, P ) be a Kolmogorov probability space [128].Each random variable a : Ω → R determines the probability distribution P a .The crucial point is that all these distributions are encoded in the same probability measure P : P a (α) = P (ω ∈ Ω : a(ω) = α).(We consider only discrete random variables.) In CP, the probability distributions of all observables (represented by random variables) can be consistently unified on the basis of P.
For any pair of random variables a, b, their JPD P a,b is defined and the following condition of marginal consistency holds: This condition means that observation of a jointly with b does not change the probability distribution of a. Equality (3) implies that, for any two observables b and c, In fact, condition (4) is equivalent to (3): by selecting the random variable c such that c(ω) = 1 almost everywhere, we see that (4) implies (3).These considerations are easily generalized to a system of k random variables a 1 , ..., a k .Their JPD is well defined, And marginal consistency conditions holds for all subsets of random variables (a i 1 , ..., a im ), m < k).Consider now some system of experimental observables a 1 , ..., a k .If the experimental design for their joint measurement exists, then it is possible to define their JPD P a 1 ,...,a k (α 1 , ..., α k ) (as the relative frequency of their joint outcomes).This probability measure P ≡ P a 1 ,...,a k can be used to define the Kolmogorov probability space, i.e., the case of joint measurement can be described by CP.Now consider the general situation: only some groups of observables can be jointly measured.For example, there are three observables a, b, c and only the pairs (a, b) and (a, c) can be measurable, i.e., only JPDs P a,b and P a,c can be defined and associated with the experimental data.There is no reason to assume the existence of JPD P a,b,c .In this situation equality (4) may be violated.In the terminology of QM, this violation is called signaling.
Typically one considers two labs, Alice's and Bob's labs.Alice measures the a-observable and Bob can choose whether to measure the b-or c-observable.
one says that the a-measurement procedure is (in some typically unknown way) is disturbed by the selection of a measurement procedure by Bob, some signal from Bob's lab approaches Alice's lab and changes the probability distribution.This terminology, signaling vs. no-signaling, is adapted to measurements on spatially separated systems and related to the issue of nonlocality.In quantum-like models, one typically works with spatially localized systems and interested in contextuality (what ever it means).Therefore we called condition (4) marginal consistency (consistency of marginal probabilities) and ( 5) is marginal inconsistency.In the further presentation we shall use changeably both terminologies, marginal consistency vs. inconsistency and no-signaling vs. signaling.
In future we shall be mainly interested in the CHSH inequality.In this framework, we shall work with four observables a 1 , a 2 and b 1 , b 2 ; experimenters are able to design measurement procedures only for some pairs of them, say (a i , b j ), i, j = 1, 2. In this situation, there is no reason to expect that one can define (even mathematically) the JPD P a 1 ,a 2 ,b 1 ,b 2 (α 1 , α 2 , β 1 , β 2 ).This situation is typical for QM.This is a complex interplay of theory and experiment.Only probability distributions P a i ,b j can be experimentally verified.However, in theoretical speculation, we can consider JPD P a 1 ,a 2 ,b 1 ,b 2 as mathematical quantity.If it were existed, we might expect that there would be some experimental design for joint measurement of the quadruple of observables (a 1 , a 2 , b 1 , b 2 ).On the other hand, if it does not exist, then it is meaningless even to try to design an experiment for their joint measurement.
Now we turn back to marginal consistency; in general (if P a 1 ,a 2 ,b 1 ,b 2 does not exist), it may be violated.However, in QM it is not violated: there is no signaling.This is the miracle feature of QM.Often it is coupled to spatial separation of systems: a 1 or a 2 are measured on S 1 and b 1 or b 2 on S 2 .And these systems are so far from each other that the light signal emitted from Bob's lab cannot approach Alice's lab during the time of the measurement and manipulation with the selection of experimental settings.However, as we shall see nosignaling is the general feature of the quantum formalism which has nothing to do with spatial separability nor even with consideration of the compound systems.

Clauser, Horne, Shimony, and Holt (CHSH) Inequality
We restrict further considerations to the CHSH-framework, i.e., we shall not consider other types of Bell inequalities.
How can one get to know whether JPD exists?The answer to this question is given by a theorem of Fine [36] concerning the CHSH inequality.
Consider dichotomous observables a i and b j (i, j = 1, 2) taking values ±1.In each pair (a i , b j ) observables are compatible, i.e., they can be jointly measurable and pairwise JPDs P a i ,b j are well defined.Consider correlation By Fine's theorem JPD P a 1 ,a 2 ,b 1 ,b 2 exists if and only if the CHSHinequality for these correlations is satisfied: and the three other inequalities corresponding to all possible permutations of indexes i, j = 1, 2.

Derivation of CHSH Inequality within Kolmogorov Theory
The crucial assumption for derivation of the CHSH-inequality is that all correlations are w.r.t. the same Kolmogorov probability space P = (Ω, F, P ) and that all observables a i , b j , i, j = 1, 2, can be mathematically represented as random variables on this space.Under the assumption of the JPD existence, one can select the sample space Ω = {−1, +1} 4 and the probability measure P = P a 1 ,a 2 ,b 1 ,b 2 .Thus, the CHSH inequality has the form, (7) The variable ω can include hidden variables of a system, measurement devices, detection times, and so on.It is only important the possibility to use the same probability space to model all correlations.The latter is equivalent to the existence of JPD P A 1 ,a 2 ,b 2 ,b 2 .This is the trivial part of Fine's theorem, JPD implies the CHSH inequality.Another way around is more difficult [36].
This inequality can be proven by integration of the inequality which is the consequence of the inequality which holds for any quadrupole of real numbers belonging [−1, +1].

Role of No-signaling in Fine Theorem
The above presentation of Fine's result is common for physics' folklore.However, Fine did not consider explicitly the CHSH inequalities presented above, see (6).He introduced four inequalities that are necessary and sufficient for the JPD to exist, but these inequalities are expressed differently to the CHSH inequalities.The CHSH inequalities are derivable from Fine's four inequalities stated in Theorem 3 of his paper.We remark that the existence of the quadruple JPD implies marginal consistency (no-signaling), And the Fine theorem presupposed that marginal consistency.This is the good place to make the following remark.In quantum physics this very clear and simple meaning of violation of the CHSH-inequality (non-existence of JPD) is obscured by the issue of nonlocality.However, in this book we are not aimed to criticize the nonlocal interpretation of QM.If some physicists have fun by referring to spooky action at a distance and other mysteries of QM, it is not disturbing for us, since we only use the quantum formalism, not its special interpretation.In any event, non-locality may be relevant only to space separated systems.However, except parapsychology, cognitive psychology does not handle space separated systems.Finally, we point out that the Bell type inequalities were considered already by Boole (1862) [130,131] as necessary conditions for existence of a JPD.

Violation of CHSH inequality for Växjö model
If it is impossible to proceed with the same probability space for all correlations, one has to use the Växjö model (section 2.5), and there is no reason to expect that the following inequality (and the corresponding permutations) would hold, ' where C ij is the context for the joint measurement of the observables a i and b j .Here a i -observable is represented by random variables In the Växjö model the condition of no-signaling may be violated; for discrete variables, signaling means that

CHSH-inequality for quantum observables: representation via commutators
In this section we present the purely quantum treatment of the CHSH inequality and highlight the role of incompatibility in its violation (we follow article [6]).Although in QM the CHSH inequality is typically studied for compound systems with the emphasis to the tensor product structure of the state space, in this section we shall not emphasize the latter and proceed for an arbitrary state space and operators.Consequences and simplifications for the tensor product case will be presented in section 6.1.
Observables a i , b j are described by (Hermitian) operators We remark that generally i.e., the observables in the pairs a 1 , a 2 and b 1 , b 2 do not need to be compatible.
Observables under consideration are dichotomous with values ±1.Hence, the corresponding operators are such that A 2 i = B 2 j = I.The latter plays the crucial role in derivation of the Landau equality (13).
Consider the CHSH correlation represented in the quantum formalism and normalized by 1/2, This correlation is expressed via the Bell-operator: as Simple calculations lead to the Landau identity [132,133]: If at least one commutator equals to zero, i.e., then, for quantum observables, we obtain the inequality Derivation of ( 16) was based solely on quantum theory.This inequality is the consequence of compatibility for at least one pair of observables, A 1 , A 2 or B 1 , B 2 .Symbolically equation ( 16) is the usual CHSH-inequality, but its meaning is different.Equation ( 16) can be called the quantum CHSH inequality.Now suppose that A i -observables as well as B j -observables are incompatible, i.e., corresponding operators do not commute: i.e., M A = 0 and M B = 0, (18) where the commutator observables are defined The Landau identity can be written as where Weremark that if M AB = 0, then, in spite the incompatibility condition (17), the quantum QCHSH-inequality cannot be violated.So, we continue under condition This condition is not so restrictive.In my interpretation, the quantum CHSH-inequality is simply one of possible statistical tests of incompatibility.It provides the possibility to estimate the degree of incompatibility in a pair of observables, e.g., in the A-pair.The Bpair is the axillary; it can be selected.
The condition in equation ( 20) is guaranteed via selection of the B-operators in such a way that the operator M B is invertible.We point out that the case of compound systems (see section 6.1) incompatibility of the A-observables and the B-observables implies the non-degeneration condition (20).
Under condition (20), there exists common eigenvector ψ AB of commuting commutator-operators, Consider the case when µ A > 0 and µ B > 0. Such ψ AB is an eigenvector of operator B 2 with eigenvalue ( Operator B is Hermitian and this implies that Finally, we obtain the following estimate: We demonstrated that, for some pure states, the quantum CHSHinequality f is violated.
Consider now the case µ A > 0, but µ B < 0. The sign of µ B can be changed via interchange the B-observables.

We conclude:
Conjunction of incompatibilities of the A-observables and the Bobservables constrained by equation ( 20) is sufficient for violation of the quantum CHSH-inequality (for some quantum state).
The degree of violation can serve as an incompatibility measure in two pairs of quantum observables, A 1 , A 2 and B 1 , B 2 .Testing the degree of incompatibility is testing the degree of noncommutativity, or in other words, the "magnitudes" of observables corresponding to commutators, The incompatibility-magnitude can be expressed via the maximal value of averages of commutator-operators, i.e., by their norms, for example, sup By interpreting quantity ψ|M A |ψ as the theoretical counterpart of experimental average M A ψ of observable M A , we can measure experimentally the incompatibility-magnitude, i.e., norm M A from measurements of commutator-observable M A .(The main foundational problem is that measurement of such commutator-observables is challenging.Recently some progress was demonstrated on the basis of weak measurements, but generally we are not able to measure commutatorquantities.) We remark that (from the quantum mechanical viewpoint) the CHSH-test estimates the product of incompatibility-magnitudes for the A-observables and B-observables, i.e., the quantity M A M B .By considering the B-observables as axillary and selecting them in a proper way (for example, such that the B-commutator is a simple operator), we can use the CHSH-test to obtain the experimental value for the incompatibility-magnitude given by M A .

Compound Systems: Incompatibility as Necessary and Sufficient Condition of Violation of Quantum CHSH-Inequality
Here, H = H A ⊗ H B and A j = A j ⊗ I, B j = I ⊗ B j , where Hermitian operators A j and B j act in H A and H B , respectively.
Here, the joint incompatibility-condition in Equation ( 17) is equivalent to incompatibility of observables on subsystems: We have As mentioned above, constraint M AB = 0 is equivalent to (23).Thus, conjunction of local incompatibilities is the sufficient condition for violation of the quantum CHSH-inequality.And we obtain: Conjunction of local incompatibilities is the necessary and sufficient condition for violation of the quantum CHSH-inequality.

Tsirelson bound
By using Landau identity (13) we can derive the Tsirelson bound 2 √ 2 for the CHSH correlation of quantum observables, i.e., observables which are represented by Hermitian operators A i , B j , i, j = 1, 2, with spectrum ±1, so A 2 i = B 2 j = I.For such operators, for any state |ψ , we have: On the other hand, if observables are not described by QM, then this bound can be exceeded.For the Växjö contextual probability model, the CHSH correlation may approach the value 4.

Signaling in Physical and Psychological Experiments
By using the quantum calculus of probabilities, it is easy to check whether the no-signaling condition holds for quantum observables, which are represented mathematically by Hermitian operators.Therefore Fine's theorem is applicable to quantum observables.This theoretical fact played an unfortunate role in hiding from view signaling in experimental research on the violation of the CHSH-inequality.Experimenters were focused on observing as high violation of (6) as possible and they ignored the no-signaling condition.However, if the latter is violated, then a JPD automatically does not exist, and there is no reason to expect that (6) would be satisfied.The first paper in which the signaling issue in quantum experimental research was highlighted was Adenier and Khrennikov (2006) [62].There it was shown that statistical data collected in the basic experiments (for that time) performed by Aspect [68] and Weihs [69] violates the no-signaling condition.
After this publication experimenters became aware of the signaling issue and started to check it [108,106].However, analysis presented in Adenier and Khrennikov [67] demonstrated that even statistical data generated in the first loophole-free experiment to violate the CHSHinequality [71] exhibits very strong signaling.Nowadays no signaling condition is widely discussed in quantum information theory, but without referring to the pioneer works of Adenier and Khrennikov [62]- [67].
The experiments to check CHSH and other Bell-type inequalities were also performed for mental observables in the form of questions asked to people [97], [100]- [105].The first such experiment was done in 2008 [100] and was based on the theoretical paper of Khrennikov [129].As was found by Dzhafarov et al. [102], all known experiments of this type suffer of signaling.Moreover, in contrast to physics, in psychology there are no theoretical reasons to expect no-signaling.In this situation Fine's theorem is not applicable.And Dzhafarov and his coauthors were the first who understood the need of adapting the Belltype inequalities to experimental data exhibiting signaling.Obviously, the interplay of whether or not a JPD exists for quadruple can't be considered for signaling data.

Coupling Method (Contextuality-by-Default)
Dzhafarov and his coauthors [74]- [77] proposed considering, instead of quadruple S, some octuple S generated by doubling each observable and associating S with four contexts of measurements of pairs, so, e.g., observable a 1 measured jointly with observable b j is denoted a 1j .
It is assumed that this system of observables can be realized by random variables on the same Kolmogorov probability space P S = (Ω, F, P).We shall use bold symbols for sample spaces and probabilities realizing the octuple representation of observables by random variables.For example, A ij = A ij (ω), ω ∈ Ω, is a random variable representing observable a i measured jointly with the observable b j .
By moving from quadruple S to octuple S, one confronts the problem of identity of an observable which is now represented by two different random variables, e.g., the observable a i is represented by the random variables A ij (ω), j = 1, 2. In the presence of signaling one cannot expect the equality of two such random variables almost everywhere.Dzhafarov et al. came up with a novel treatment of the observable-identity problem.
It is assumed that averages and covariation are fixed.These are measurable quantities.They can be statistically verified by experiment.Set and This is the experimentally verifiable measure of signaling.We remark that in the coupling representation the joint satisfaction of the CHSH inequalities, i.e., (6) and other inequalities obtained from it via permutations, can be written in the form: In the signaling-free situation, e.g., in quantum physics, the difference between the left-hand and right-hand sides is considered as the measure of contextuality.Denote (1/2 times) this quantity by ∆ CHSH .It is also experimentally verifiable.
Then Dzhafarov and coauthors introduced quantity where Here ∆ a i (P) characterizes mismatching of representations of observable a i by random variables A i1 and A i2 with respect to probability measure P; ∆ b j (P) is interpreted in the same way.The problem of the identity of observables is formulated as the mismatching minimization or identity maximization problem ∆(P) → min (36) with respect to all octuple probability distributions P satisfying constraints ( 29), (30).And it turns out, that It is natural to consider the solutions of the identity maximization problem (36) as CP-representations for contextual system S.The corresponding random variables have the highest possible, in the presence of signaling, degree of identity.
The quantity ∆ min − ∆ 0 is considered as the measure of "genuine contextuality".This approach is very useful to study contextuality in the presence of signaling.The key point is the coupling of this measure of contextuality with the problem of the identity of observables measured in different contexts.As was pointed out in article [76] : "...contextuality means that random variables recorded under mutually incompatible conditions cannot be join together into a single system of jointly distributed random variables, provided one assumes that their identity across different conditions changes as little as possibly allowed by direct cross-influences (equivalently, by observed deviations from marginal selectivity)." This approach to contextuality due to Dzhafarov-Kujala can be reformulated in the CHSH-manner by using what we can call CHSH-BDK inequality: It was proven that octuple-system S exhibits no genuine contextuality, i.e., ∆ min = ∆ 0 , if and only if the CHSH-BDK inequality is satisfied.
8 Sources of Signaling Compatible with Quantum Formalism and, hence, both marginal probability distributions coincide with the probability of measurement of the a-observable alone.We remark that this proof of no-signaling can be easily extended to generalized quantum observables given by POVMs.So, in quantum measurement theory there is no place for signaling.We also recall that signaling (marginal inconsistency) is absent in classical (Kolmogorov) probability theory.On the other hand, it is natural for contextual probability (as in the Växjö model).

No Signaling for Nonlocal Quantum Observables
Now let H = H 1 ⊗ H 2 , where H 1 , H 2 be the state spaces of the subsystems S 1 , S 2 of the compound system S = (S 1 , S 2 ) and let the observables a, b, c are nonlocal, in the sense that their measurements are not localized to subsystems.The corresponding operators have the form , where outcomes of a are labeled by pairs of numbers (x 1 , x 2 ) → x (the map from pairs to the a-outcomes is not one to one).However, the above general scheme based on ( 41) is still valid.The tensor product decomposition of projections does not play any role in summation in (41).

Nonlocality of observables cannot generate signaling.
This is unexpected fact, because typically signaling is associated with nonlocality.But, as we have seen, this is not nonlocality of observables.
Now we turn to the quantum CHSH inequality.As we seen in section 6, for quantum observables its violation is rigidly coupled only to their incompatibility.Even if So, by quantum theory signaling is impossible.But, e.g., in decision making, signaling patterns (expressing marginal inconsistency) were found in all known experiments.This is the contradiction between the quantum-like model for decision making and experiment.This situation questions the whole project on applications of the quantum formalism to modeling behavior of cognitive systems.
However, there are some "loopholes" which can lead to marginal inconsistency.

Signaling on Selection of Experimental Settings
Consider the Bohm-Bell experiment: a source of photons' pairs S = (S 1 , S 2 ) and two polarization beam splitters (PBSs) in Alice's and Bob's labs; their output channels are coupled to the photo-detectors.Denote orientations of PBSs by θ and φ.Suppose now that the quantum observables representing measurements on S 1 and S 2 depend on both orientations, a = a(θ, φ), b = b(θ, φ).
They are represented by operators Thus selection of setting φ for PBS in Bob's lab changes the observable (measurement procedure) in Alice's lab and vice verse.This is a kind of signaling between Bob's lab and Alice's lab, signaling carrying information about selection of experimental settings. 3In such a situation, and hence = TrρE a(θ,φ) = P(a(θ, φ) = x|ρ), or, in the probabilistic terms, We remark that decomposition of S into subsystems S 1 and S 2 and association of observables a and b with these subsystems did not play any role in quantum calculations.Such decomposition and coupling it with spatial locality is important only in the physics as the sufficient condition to prevent signaling on selection of experimental settings.
In the probabilistic terms each pair of settings determines context C = (θ, φ) and the corresponding probability space.Thus, we are in the framework of the Växjö model for contextual probability.Here the possibility of signaling and violation of the Bell type inequalities is not surprising.
In cognitive experiments, observables are typically questions asked to a system S (e.g., a human).As we have seen, dependence of questions a and b on the same set of parameters can generate signaling.This dependence is not surprising.Even if questions a and b are processed by different regions of the brain, the physical signaling between these regions cannot be neglected.If θ and φ are the contents of the a-and b-questions, then after a few milliseconds the area of the brain processing a = a(θ) would get to "know" about the content of the b-question and thus a-processing would depend on both parameter, a = a(θ, φ).We remark that an essential part of information processing in the brain is performed via electromagnetic field; such signals propagate with the light velocity and the brain is very small as a physical body.
On the other hand, some kind of mental localization must be taken into account; mental functions performing different tasks use their own information resources (may be partially overlapping).Without such mental localization, the brain 4 would not be able to discriminate different mental tasks and their outputs.At least for some mental tasks (e.g., questions), dependence of a on the parameter φ (see (43)) can be weak.For such observables, signaling can be minimized.
Are there other sources of signaling compatible with quantum formalism?

State Dependence on Experimental Settings
Let us turn to quantum physics.Here "signaling" often has the form of real physical signaling and it can reflect the real experimental situation.We now discuss the first Bell-experiment in which the detection loophole was closed [70].It was performed in Vienna by Zelinger's group and it was characterized by statistically significant signaling.By being in Vienna directly after this experiment, I spoke with people who did it.They told the following story about the origin of signaling -marginal inconsistency.The photon source was based on laser generating emission of the pairs of entangled photons from the crystal.It happened (and it was recognized only afterwards) that the polarization beam splitters (PBSs) reflected some photons backward and by approaching the laser they changed its functioning and backward flow of photons depended on the orientations of PBSs.In this situation "signaling" was not from b-PBS to a-PBS, but both PBSs sent signals to the source.Selection of the concrete pair of PBSs changed functioning of the source; in the quantum terms this means modification of the state preparation procedure.In this case selection of a pair of orientations leads to generation of a quantum state depending on this pair, ρ ab .This state modification contributed into the signaling pattern in data.
The above physical experimental illustration pointed out to state's dependence on experimental context as a possible source of signaling.It is clear that, for ρ = ρ a,b , generally This dependence also may lead to violation of the Bell inequalities.In the probabilistic terms this is again the area of application of the Växjö model with contexts associated with quantum states, the probability measures depend on the experimental settings.We remark that it seems that the state variability depending on experimental settings was the source of signaling in Weihs' experiment [69] which closed nonlocality loophole.At least in this way we interpreted his reply [73] to our (me and Guillaume Adenier) paper [63].Since Weihs [69] was able to separate two "labs" to a long distance, the signals from one lab could not approach another during the process of measurement.
In quantum physics experimenters were able to block all possible sources of state's dependence on the experimental settings.Thus, it is claimed that one can be sure that ρ does not depend on a and b.By using the orientations of PBSs θ, φ, i.e., ρ = ρ(θ, φ), the latter condition can be written as Stability of state preparation is the delicate issue.As we have seen, the source by itself can be stable and generate approximately the same state ρ, but the presence of measurement devices can modify its functioning.Moreover, even if any feedback to the source from measuring devices is excluded, laser's functioning can be disturbed by fluctuations.Typically violation of state statsbility cannot be observed directly and the appearance of a signaling pattern can be considered as a sign on state's variation.In physics the signaling can be rigidly associated with fluctuations in state preparation.Spatial separation leads to local parameter dependence of observables, i.e., a = a(θ) and b = b(φ).
For cognitive systems, it seems to be impossible to distinguish two sources of signaling: • joint dependence on parameters θ, φ determining contents of questions, • state dependence on θ, φ.

Nonconetxtual inequalities
As before, we consider dichotomous observables taking values ±1.We follow paper [37] (one of the best and clearest representations of noncontextuality inequalities).Consider a set of observables {x 1 , ..., x n }; contexts C ij determined by the pairs of indexes such that observables x i , x j are compatible, i.e., the pair (x i , x j ) is jointly measurable; set Z = {C ij }.For each context C ij , we measure correlations for observables x i and x j as well as averages x i and x j .o The n-cycle contextuality scenario is given by collection of contexts Statistical data associated with this set of contexts is given by the collection of averages and correlations: Theorem 1 from paper [37] describes all tight noncontextuality inequalities.We are not interested in their general form.For n = 4, we have inequality: This inequality can be rewritten in the QM notation which we have used in the previous sections by setting Theorem 2 from article [37] demonstrates that, for n ≥ 4, aforementioned tight noncontexuality inequalities and, in particular, inequality (54), are violated by quantum correlations.

Concluding Remarks
This article is aimed to decouple the Bell tests from the issue of nonlocality via highlighting the contextuality role.We started with discussing the physical meaning of contextuality.The common identification of contextuality with violation of the Bell type inequalities (noncontextual inequalities) cannot be accepted.This situation is illustrated by randomness theory.Here the notion of randomness is based on rigorous mathematical formalization.Statistical tests, as e.g. the NIST test, are useful only to check for randomness the outputs of random or pseudo-random generators.We are also critical to appealing to JMC and not only because it is based on counterfactuals.
Here it is the good place to recall that Svozil [134,135]) and Griffiths [59], [136]- [138] have the different viewpoint and they suggested experimental tests for JMC.Moreover, Griffiths [136] even claimed that QM is noncontextual.So, the diversity of opinions about "quantum contextuality" is really amazing.
Bell considered JMC as an alternative to Einsteinian nonlocality.However, in the framework of the Bohm-Bell experiments, the physical meaning of JMC is even more mysterious than the physical meaning of EPR-nonlocality.JMC gains clear meaning only as the special case of Bohr contextuality.By the latter outcomes of quantum observables are generated in the complex process of the interaction between a system and a measurement apparatus.
Bohr contextuality is the real seed of the complementarity principle leading to the existence of incompatible observables.This principle is also essentially clarified and demystified through connection with contextuality.Our analysis led to the conclusion that contextuality and complementarity are two supplementary counterparts of one principle.It can be called the contextuality-complementarity principle.This is the good place to mention the studies of Grangier, e.g., [139,140], as an attempt to suggest a heuristically natural interpretation for contextuality, which is different from JMC and Bell contextualities.Grangier contextuality is in fact also closely coupled to the Bohr complementarity principle, although this was not pointed out.
In the probabilistic terms, Bohr contextuality is represented via the use of a family of Kolmogorov probability spaces which are labeled by experimental contexts.Such formalism, the Växjö model for contextual probability.
In this review the problem of signaling (marginal inconsistency) is taken very seriously.We (Adenier and Khrennikov) paid attention to this problem for many years ago [62]- [67].These publications attracted attention of experimenters to signaling problem.Nowadays it is claimed that experimental data does not contain signaling patters.However, our analysis of the first loophole free Bell experiment [71] demonstrated that the statistical data suffers of signaling.
In fact, all data sets which we were able to get from experimenters and then analyze contain statistically significant signaling patters.By using induction one may guess that even data which owners claimed no-signaling might suffer of signaling.Unfortunately, I simply do not have resources to lead a new project on data analysis.Moreover, it is still difficult and often not possible at all to receive rough click-by-click data.Creation of the data-base for all basic quantum foundational experiments is very important for quantum foundations -starting with photo-effect and interference experiments and finalizing with the recent Bell type experiments.
Can one work with statistical data shadowed by signaling?The answer to this question is positive as was shon within recently developed CbD-theory.It led a new class of inequalities, the Bell-Dzhafarov-Kujala (BDK) inequalities.These inequalities are especially important in quantum-like studies, applications of the quantum formalism outside of physics.Here up to now, all experimental statistical data contains signaling patterns.
Since incompatibility of quantum observables is mathematically encoded in noncommutativity of corresponding operators, it is natural to try to express Bell contextuality with operator commutators.As was shown in article [6], this is possible at least for the CHSH-inequality.The basic mathematical result beyond such expression is the Landau inequality [132,133].In the light of commutator representation of the degree of violation of the CHSH inequality, we suggest to interpret this inequality as a special test of incompatibility of observables.The commutator representation is valid for any state space, i.e., the tensor product structure does not play any role.In this way we decouple the CHSH inequality from the problem of quantum nonlocality which was so highlighted by Bell.Incompatibility in each pair of local observables and only incompatibility is responsible for the inequality violation.
Finally, we study the possible sources of signaling which are not in the direct contradiction with the quantum formalism.One of such sources is disturbance of the state preparation procedure by the selection of the experimental settings.And we discuss this setting dependent preparations in coupling to the concrete experimental situations.
with Einsteinian (non)locality [141,142].Note that the difference between the notions of Bell locality, EPR locality, and nonsignaling was first specified mathematically in article [144].See also [145]- [147] for Bell locality and nonlocality.Bell locality is formulated via the introduction of hidden variables as the factorzation condition, see, e.g.[146], eq. ( 3).In fact, Bell nonlocality is a form of JMC expressed in term of hidden variables, as Bell pointed out by himself [2].This is the good place to remark that by considering the EPR-Bohm correlations in the space-time within the quantum field formalism, one finds that these correlations should decrease with the distance [148,149].The declared conservation of correlations which is apparently confirmed in the Bell experiments is the consequence of the normalization procedure used in these experiments [149].Now we present some logical considerations: • Local realism = realism and locality • Not(Local realism)= Not(realism and locality)= nonrealism or nonlocality, where "or" is the non-exclusive or operation.The crucial point is that here nonlocality is Bell nonlocality, not Einsteinian one.Hence, nonlocality = JMC (expressed with hidden variables).And it is a consequence of Bohr contextuality; this can also be said about nonrealism.
Thus, the whole Bell consideration can be reduced to showing that by rejecting the Bohr contextuality-complementarity principle one can derive special inequalities for correlations.From my viewpoint, the violation of these inequalities implies only that the Bohr principles hold true.Roughly speaking one can come back to the foundations of QM which were set 1920th.The experimental Bell tests are advanced tests of the Bohr contextuality-complementarity principle; in this sense they are tests of quantumness.
We remark that original Bohr and Heisenberg appealing to the Heisenberg uncertainty relation as the basic test of incompatibility for quantum observables, e.g., [4], [150]- [152]was strongly criticized, e.g., by Margenau [153] and Ballentine [86,87].Since direct measurement of the commutator observable C = i[A, B] is difficult, the Bell tests became the most popular tests of incompatibility and, hence, quantumness.Unfortunately, the issue of incompatibility was shadowed by "quantum nonlocality".

8. 1
Quantum Theory: No-signaling Consider the quantum Hilbert space formalism, a state given by density operator ρ; three observables a, b, c represented by operators A, B, C (acting in H) with spectral families of projectors E a (x), E b (x), E c (x).It is assumed that in each pair (a, b) and (a, c) the observables are compatible, [A, B] = 0, [A, C] = 0. Then P (a = x, b = y|ρ) = TrρE a (x)E b (y), P(a = x, c = y|ρ) = TrρE a (x)E c (y) (40) and hence y P (a = x, b = y|ρ) = TrρE a (x) y E b (y) = TrρE a (x) (41) = TrρE a (x) y E c (y) = y P(a = x, b = y|ρ).and we remark that TrρE a (x) = P(a = x|ρ)