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.

This is the first ever English translation of Heisenberg’s unpublished response to the EPR paper. In this chapter, Heisenberg uses his famous cut argument to argue against the possibility of hidden variables.

The famously controversial 1935 paper by Einstein, Podolsky, and Rosen (EPR) took aim at the heart of quantum mechanics. The paper provoked responses from leading theoretical physicists of the day, and brought entanglement and nonlocality to the forefront of discussion. This book looks back at when the EPR paper was published and explores those intense. conversations in print and in private correspondence. These offer significant insight into the minds of pioneering quantum physicists, including Bohr, Schrödinger and Einstein himself. Offering the most complete collection of sources to date – many published or translated here for the first time – this text brings a rich new context to this pivotal moment in physics history.

Schrödinger’s reaction to the EPR paper is less widely known than, say, Bohr’s, and yet our analysis shows that it fits rather nicely with contemporary concerns in foundations of quantum mechanics. Taking the lead both from the EPR paper and from Pauli’s remarks in their correspondence, Schrödinger shows that EPR’s locality considerations lead to the assignment of values to all quantum mechanical observables, but that under apparently mild assumptions this then leads to contradictions of the von Neumann type. This dilemma (as he explicitly calls it) is thus similar to more recent debates between nonlocality on the one hand and no-go results on the other (whether through violation of the Bell inequalities, the Kochen–Specker theorem, or what you will). We shall first look at Schrödinger’s fundamental worries in the years leading up to 1935. The chapter then discusses in detail the direct reaction by Schrödinger to EPR. It will, however, not exhaust our discussion of Schrödinger, who is a recurring character in the book, having poked and prodded his peers on EPR during the whole summer and autumn of 1935.

This is a reprinting of Bohr’s response to the EPR paper, wherein Bohr relies on his principle of complementarity to demonstrate an ambiguity in the criterion of reality as described by EPR and to argue that quantum mechanics is in fact a complete description of reality given the bounds of complementarity.

This is a reprinting of Margenau’s response to EPR (and to some extent, his evaluation of previous responses to EPR by Bohr, Kemble and Ruark). Margenau’s contribution to the EPR debate is certainly one of the most original, no doubt at least in part due to the meaty correspondence he had with Einstein while producing it. Margenau’s main strategy in this paper is to argue against the standard collapse postulate of quantum mechanics, suggesting that the EPR argument only applies to quantum mechanics with this postulate added. He also argues against the statistical interpretation of the collapse postulate suggested by Kemble and others.

This is a reprinting from Jammer (1974) of Podolsky’s unpublished response to Kemble’s criticisms of the EPR paper. Podolsky rightly criticises Kemble for missing the point of EPR’s argument and adds a few comments agreeing with Kemble that a statistical interpretation of quantum mechanics is best – yet Podolsky maintains such an interpretation is incomplete.