The aim of this chapter is to attract attention of experimenters to the originalBell (OB) inequality which was shadowed by the common considerationof the CHSH inequality. Since this chapter is directed to experimenters, herewe present the standard viewpoint on the violation of the Bell inequality andthe EPR argument. There are two reasonsto test the OB inequality and not the CHSH inequality. First, theOB inequality is a straightforward consequence of the EPR argumentation.And only this inequality is related to the EPR–Bohr debate.The second distinguishing feature of the OB inequality was emphasizedby Pitowsky. He pointed out that the OB inequality provides a higherdegree of violations of classicality than the CHSH inequality. Thus, by violating the OBinequality it is possible to approach a higher degree of deviation from classicality.The main problem is that the OB inequality is derived under theassumption of perfect (anti-)correlations. However, the last few years have been characterizedby the amazing development of quantum technologies. Nowadays,there exist sources producing with very high probability the pairs of photonsin the singlet state. Moreover, the efficiency of photon detectors wasimproved tremendously. In any event one can start by proceeding with thefair sampling assumption.

The aim of this chapter is to highlight the possibility of applying the mathematicalformalism and methodology of quantum theory to model behaviourof complex biosystems, from genomes and proteins to animals, humans, ecologicaland social systems. Such models are known as quantum-like and theyshould be distinguished from genuine quantum physical modeling of biologicalphenomena. One of the distinguishing features of quantum-like models istheir applicability to macroscopic biosystems, or to be more precise, to informationprocessing in them. Quantum-like modeling has the base in quantuminformation theory and it can be considered as one of the fruits of the quantuminformation revolution. Since any isolated biosystem is dead, modelingof biological as well as mental processes should be based on theory of opensystems in its most general form – theory of open quantum systems. In thischapter we advertise its applications to biology and cognition, especiallytheory of quantum instruments and quantum master equation. We mentionthe possible interpretations of the basic entities of quantum-like models withspecial interest to QBism as maybe the most useful interpretation.

In this chapter we develop the contextual approach to quantum mechanics.This approach matches with the views ofBohr who emphasized that the quantum description represents complexes ofexperimental physical conditions, in the modern terminology – experimentalcontexts. In this chapter we formalize the contextual approach on the basisof contextual probability theory which is closely connected with generalizedprobability theory (but interpretationally not identical with it). The contextualprobability theory serves as the basis of the contextual measurementmodel (CMM). The latter covers measurements in classical, quantum, andquasi-classical physics.

This chapter is a step towards understanding why quantum nonlocalityis a misleading concept. Metaphorically speaking, quantum nonlocality isJanus faced. One face is an apparent nonlocality of the state update basedon the Luders projection postulate. It can be referred as intrinsic quantumnonlocality. And the other face is subquantumnonlocality: by introducing a special model with hidden variables onederives the Bell inequality and claims that its violation implies the existenceof mysterious instantaneous influences between distant physical systems(Bell nonlocality). According to the Luders projection postulate, aquantum measurement performed on one of the two distant entangled physicalsystems, say on S1, modifies instantaneously the state of S2. Therefore, ifthe quantum state is considered to be an attribute of the individual physicalsystem (Copenhagen interpretation) and if one assumes thatexperimental outcomes are produced in a random way one arrives at the contradiction. It is a primary source of speculation about aspooky action at the distance. But Einstein had already pointed out that the quantum paradoxes disappear, ifone adopts the statistical interpretation.

The hidden variable project realized by Bell contradicts the uncertaintyand complementarity principles. The inequalities derived with Bell’s hidden variablesare violated for quantum observables. Thus, Bell’s hidden variables shouldbe rejected and the validity of quantum theory is confirmed. (This foundationalachievement deserved the Nobel Prize in 2022.) This scientific loop,ignorance of the uncertainty and complementarity principles – hidden variablesmodel – Bell’s inequalities – their violation – reestablishing the validityof the uncertainty and complementarity principles, was stimulating for quantumfoundations. However, Bohr and Heisenberg might say that such resultscan be expected from the very beginning. For them, the uncertainty andcomplementarity principles form the basis of quantum physics. And theycan’t be rejected, since they are the consequences of the so-called quantum postulate– the existence of an indivisible quantum of action given by Planck’sconstant h. The quantum postulate is the ontological basis of quantum theory.I formulated its epistemic counterpart in the form of the principle ofquantum action invariance.

We analyze interrelation of quantum and classical entanglement. The latternotion is widely used in classical optic simulation of the quantum-likefeatures of light. We criticize the common interpretation that quantum nonlocalityis the basic factor differentiating these two sorts of entanglement. Instead,we point to the Grangier experiment on photon existence, the experimenton the coincidence detection. Classical entanglement sources produce lightbeams with the coefficient of second-order coherence g(2)(0) ≥ 1. This featureof classical entanglement is obscured by using intensities of signals indifferent channels, instead of counting clicks of photodetectors. Interplaybetween intensity and clicks counting is not just a technicality. We emphasizethe foundational dimension of this issue and its coupling with theBohr’s statement on individuality of quantum phenomena.

In this chapter we start with methodological analysis of the notion of scientifictheory and its interrelation with reality. This analysis is based onthe works of Helmholtz, Hertz, Boltzmann, and Schrödinger (and reviewsof D’ Agostino). Following Helmholtz, Hertz established the “Bild concept”for scientific theories. Here “Bild” (“picture”) carries the meaning “model”(mathematical). The main aim of natural sciences is construction of thecausal theoretical models (CTMs) of natural phenomena. Hertz claimed thatCTM cannot be designed solely on the basis of observational data; it typicallycontains hidden quantities. Experimental data can be described by anobservational model (OM), often at the price of acausality. CTM-OM interrelationcan be tricky. Schrödinger used the Bild concept to create CTM forquantum mechanics (QM) and QM was treated as OM. We follow him andsuggest a special CTM for QM, the so-called prequantum classical statisticalfield theory (PCSFT). QM can be considered as a PCSFT-image, but notas straightforward as in Bell’s model with hidden variables. The commoninterpretation of the violation of the Bell inequality is criticized from theperspective of the two-level structuring of scientific theories.

The contextual measurement model (CMM) that was invented in Chapter 10 representsthe wide range of non-Bayesian procedures for probability updatebased on context updates (or state updates). In this chapter we compareBayesian classical probability inference with general contextual probabilityinference. CMM is the basis of the Växjö interpretation of quantum mechanics,one of the contextual probabilistic interpretations. This interpretationhighlights that quantum update of probability (based on the state update)is one of the non-Bayesian updates. Quantum mechanics is interpreted as aprobability update machinery.

We start with discussion on Bohr’s response to the EPR argument andexplain how Bohr was able to sail between Scylla (incompleteness) andCharybdis (nonlocality) towards the consistent interpretation of quantumtheory. We call the latter the Bohr interpretation and distinguish it fromthe commonly used orthodox Copenhagen interpretation. We point to connectionbetween the complementarity principle and the information interpretationof QM and briefly discuss its versions, starting withSchrödinger and continuing to the information quantization interpretation(Zeilinger, Brukner), QBism (Fuchs et al.), reality without realism (RWR,Plotnitsky), the Växjö interpretation (Khrennikov), and derivations of thequantum formalism from the information axioms (e.g., D’Ariano et al.). Oneof the main distinguishing features of the information interpretation is the possibility of structuring thequantum foundations without nonlocality and spooky actionat a distance.

In this chapter the contextual measurement model (CMM) is employed forprobabilistic structuring of classical and quantum physics. We start with CMM framing of classical probability theory (Kolmogorov 1933) servingas the basis of classical statistical physics and thermodynamics. Then weconsider the von Neumann quantum measurement theory with observablesgiven by Hermitian operators and the state update of the projective typeand represent it as CMM. The quantum instrument theory is a generalizationof the von Neumann theory permitting state updates of the non-projectivetype and it also can be represented as CMM. We also show connectionof the generalized probability theory with the space of probability measureswith CMM. Finally, linear space representation for contextual probabilityspace is constructed by using the construction going back to Mackey.

This chapter is aimed to dissociate nonlocality fromquantum theory. We indicate that the tests on violation of the Bell inequalitiescan be interpreted as statistical tests of observables local incompatibility.In fact, these are tests on violation of the Bohr complementarityprinciple. Thus, the attempts to couple experimental violations of the Bell-type inequalities with “quantum nonlocality” are misleading. These violationsare explained by the standard quantum theory as exhibitions of observablesincompatibility even for a single quantum system. Mathematically this chapter is based on the Landau equality. Thequantum CHSH-inequality is considered withoutcoupling to the tensor product, We point out that the notion of local realism isambiguous. The main impact of the Bohm–Bell experiments is on the developmentof quantum technology: creation of efficient sources of entangledsystems and photodetectors.

In this chapter contextual probabilistic entanglement is represented withinthe Hilbert space formalism. The notion of entanglement is clarified anddemystified through decoupling it from the tensor product structure andtreating it as a constraint posed by probabilistic dependence of quantum observablesA and B. In this framework, it is meaningless to speak aboutentanglement without pointing to the fixed observables A and B, so thisis AB-entanglement. Dependence of quantum observables is formalized asnon-coincidence of conditional probabilities. Starting with this probabilisticdefinition, we achieve the Hilbert space characterization of the AB-entangledstates as amplitude non-factorisable states. In the tensor productcase, AB-entanglement implies standard entanglement, but not vice versa.AB-entanglement for dichotomous observables is equivalent to their correlation. Finally, observables entanglement is compared with dependence of random variables in classical probability theory.

This short chapter contains basics of the mathematical formalism for thequantum measurement theory. In this book we proceed mainly withthe von Neumann measurement theory in which observables are given byHermitian operators and the state update by projections. However, we alsomention the measurement formalism based on quantum instruments, sinceit gives the general framework for quantum measurements. This formalismis used only in Chapters 10 and 18. The latter chapter is devoted to quantum-likemodeling – the applications of the mathematical formalism and methodologyof quantum mechanics (QM) to cognition, psychology, and decision making.Surprisingly, in such applications even the simplest effects can’t be described bythe von Neumann theory. One should use quantum instruments (compare withquantum physics where the main body of theory can be presented solelywithin the von Neumann measurement theory).

QBism’s foundational statement that “the outcome of a measurement ofan observable is personal” is in direct contradiction with Ozawa’sIntersubjectivity Theorem (OIT). The latter (proven within the quantummeasurement theory) states that two observers, agents within the QBismterminology, performing joint measurements of the same observable A on asystem S in the state ψ should get the same outcome A = x. In Ozawa’s terminology,this outcome is intersubjective and it can’t be treated as personal.This is the strong objection to QBism which can’t survive without updatingits principles. The essential aspect in understanding of the OIT impact onQBism’s foundations takes the notion of quantum observable. We discussthe difference between the accurate, von Neumann, and inaccurate, noisy,quantum observables which are represented by the projection valued measures(PVMs) and positive operator valued measures (POVMs), respectively.Moreover, we discuss the OIT impact on the Copenhagen interpretation ofquantum mechanics.