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.

In this chapter we introduce the general class of symmetric two-qubit statesguaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomeswhenever some spin observable is measured at both sites. It is proventhat, for all states from this class, the maximal violation of the original Bellinequality (OB) is upper bounded by 3/2 and specify the two-qubit stateswhere this quantum upper bound is attained. The case of two-qutrit statesis more complicated. Here, for all two-qutrit states, we obtain the same upperbound 3/2 for violation of the original Bell inequality under Alice and Bobspin measurements. But it has not yet been shown that this quantum upperbound is the least one. The experimental consequences of this mathematicalstudy are discussed.