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Andrei Khrennikov - One of the best experts on this subject based on the ideXlab platform.
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Classical versus quantum Probability: Comments on the paper “On universality of Classical Probability with contextually labeled random variables” by E. Dzhafarov and M. Kon
Journal of Mathematical Psychology, 2019Co-Authors: Andrei KhrennikovAbstract:Abstract Recently Dzhafarov and Kon published the paper advertising the possibility to use the coupling technique of Classical Probability theory to model incompatible observables in quantum physics and quantum-like models of psychology. Here I present comments on this paper by stressing advantages and disadvantages.
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bohm bell type experiments Classical Probability approach to no signaling and applications to quantum physics and psychology
arXiv: Quantum Physics, 2018Co-Authors: Andrei Khrennikov, A P AlodjantsAbstract:We consider the problem of representation of quantum states and observables in the framework of Classical Probability theory (Kolmogorov's measure-theoretic axiomatics, 1933). Our aim is to show that, in spite of the common opinion, correlations of observables $A_1, A_2$ and $B_1,B_2$ involved in the experiments of the Bohm-Bell type can be expressed as correlations of Classical random variables $a_1, a_2$ and $b_1, b_2.$ The crucial point is that correlations $\langle A_i, B_j \rangle$ should be treated as conditional on the selection of the pairs $(i, j).$ The setting selection procedure is based on two random generators $R_A$ and $R_B.$ They are also considered as observables, supplementary to the "basic observables" $A_1, A_2$ and $B_1, B_2.$ These observables are absent in the standard description, e.g., in the scheme for derivation of the CHSH-inequality. We represent them by Classical random variables $r_a$ and $r_b.$ Following the recent works of Dzhafarov and collaborators, we apply our conditional correlation approach to characterize (no-)signaling in the Classical probabilistic framework. Consideration the Bohm-Bell experimental scheme in the presence of signaling is important for applications outside quantum mechanics, e.g., in psychology and social science.
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Two-slit experiment: quantum and Classical probabilities
Physica Scripta, 2015Co-Authors: Andrei KhrennikovAbstract:Inter-relation between quantum and Classical Probability models is one of the most fundamental problems of quantum foundations. Nowadays this problem also plays an important role in quantum technologies, in quantum cryptography and the theory of quantum random generators. In this letter, we compare the viewpoint of Richard Feynman that the behavior of quantum particles cannot be described by Classical Probability theory with the viewpoint that quantum–Classical inter-relation is more complicated (cf, in particular, with the tomographic model of quantum mechanics developed in detail by Vladimir Man'ko). As a basic example, we consider the two-slit experiment, which played a crucial role in quantum foundational debates at the beginning of quantum mechanics (QM). In particular, its analysis led Niels Bohr to the formulation of the principle of complementarity. First, we demonstrate that in complete accordance with Feynman's viewpoint, the probabilities for the two-slit experiment have the non-Kolmogorovian structure, since they violate one of basic laws of Classical Probability theory, the law of total Probability (the heart of the Bayesian analysis). However, then we show that these probabilities can be embedded in a natural way into the Classical (Kolmogorov, 1933) Probability model. To do this, one has to take into account the randomness of selection of different experimental contexts, the joint consideration of which led Feynman to a conclusion about the non-Classicality of quantum Probability. We compare this embedding of non-Kolmogorovian quantum probabilities into the Kolmogorov model with well-known embeddings of non-Euclidean geometries into Euclidean space (e.g., the Poincare disk model for the Lobachvesky plane).
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Fundamentals of Classical Probability and Quantum Probability Theory
Quantum Adaptivity in Biology: From Genetics to Cognition, 2015Co-Authors: Masanari Asano, Andrei Khrennikov, Masanori Ohya, Yoshiharu Tanaka, Ichiro YamatoAbstract:In this chapter we present briefly the basic notions of Classical and quantum theories of Probability and information. This chapter is especially important for biologists, psychologists, experts in cognition, and sociologists who were not trained in quantum theory (but even Classical theory is presented in a simple manner). We start with the presentation of the standard measure-theoretic formulation of the modern Classical Probability theory (Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin [1]). Then we turn to fundamentals of quantum formalism, including theory of open quantum systems and its generalizations.
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Classical Probability model for bell inequality
arXiv: Quantum Physics, 2014Co-Authors: Andrei KhrennikovAbstract:We show that by taking into account randomness of realization of experimental contexts it is possible to construct common Kolmogorov space for data collected for these contexts, although they can be incompatible. We call such a construction "Kolmogorovization" of contextuality. This construction of common Probability space is applied to Bell's inequality. It is well known that its violation is a consequence of collecting statistical data in a few incompatible experiments. In experiments performed in quantum optics contexts are determined by selections of pairs of angles $(\theta_i, \theta^\prime_j)$ fixing orientations of polarization beam splitters. Opposite to the common opinion, we show that statistical data corresponding to measurements of polarizations of photons in the singlet state, e.g., in the form of correlations, can be described in the Classical probabilistic framework. The crucial point is that in constructing the common Probability space one has to take into account not only randomness of the source (as Bell did), but also randomness of context-realizations (in particular, realizations of pairs of angles $(\theta_i, \theta^\prime_j)$). One may (but need not) say that randomness of "free will" has to be accounted.
Jerome R. Busemeyer - One of the best experts on this subject based on the ideXlab platform.
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quantum cognition a new theoretical approach to psychology
Trends in Cognitive Sciences, 2015Co-Authors: Peter Bruza, Zheng Wang, Jerome R. BusemeyerAbstract:What type of Probability theory best describes the way humans make judgments under uncertainty and decisions under conflict? Although rational models of cognition have become prominent and have achieved much success, they adhere to the laws of Classical Probability theory despite the fact that human reasoning does not always conform to these laws. For this reason we have seen the recent emergence of models based on an alternative probabilistic framework drawn from quantum theory. These quantum models show promise in addressing cognitive phenomena that have proven recalcitrant to modeling by means of Classical Probability theory. This review compares and contrasts probabilistic models based on Bayesian or Classical versus quantum principles, and highlights the advantages and disadvantages of each approach.
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Insights from quantum cognitive models for organizational decision making
Journal of Applied Research in Memory and Cognition, 2015Co-Authors: Lee C. White, Emmanuel M. Pothos, Jerome R. BusemeyerAbstract:Organizational decision making is often explored with theories from the heuristics and biases research program, which have demonstrated great value as descriptions of how people in organizations make decisions. Nevertheless, rational analysis and Classical Probability theory are still seen by many as the best accounts of how decisions should be made and Classical Probability theory is the preferred framework for cognitive modelling for many researchers. The focus of this work is quantum Probability theory, an alternative probabilistic framework. Results in decision making, which appear paradoxical from a perspective of Classical Probability theory, may make perfect sense if one adopts quantum Probability theory. We review some cognitive models of decision making based on quantum Probability theory. Each of these models is based on a challenge to prescription from Classical Probability theory. The transition from labeling a particular behavior as irrational, by Classical Probability standards, to (potentially) rational (or, at any rate, not fallacious), raises interesting possibilities, including that of characterizing certain heuristics in formal, probabilistic terms.
Elena R. Loubenets - One of the best experts on this subject based on the ideXlab platform.
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All joint von Neumann measurements on a quantum state admit a quasi-Classical Probability model
arXiv: Quantum Physics, 2012Co-Authors: Elena R. LoubenetsAbstract:We prove that the Hilbert space description of all joint von Neumann measurements on a quantum state can be reproduced in terms of a single measure space ({\Omega}, F, {\mu}) with a normalized real-valued measure {\mu}, that is, in terms of a new general Probability model, the quasi-Classical Probability model, developed in [Loubenets: J. Math. Phys. 53 (2012), 022201; J. Phys. A: Math. Theor. 45 (2012), 185306]. In a quasi-Classical Probability model for all von Neumann measurements, a random variable models the corresponding quantum observable in all joint measurements and depends only on this quantum observable. This mathematical result sheds a new light on some important issues of quantum randomness discussed in the literature since the seminal article (1935) of Einstein, Podolsky and Rosen.
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nonsignaling as the consistency condition for local quasi Classical Probability modeling of a general multipartite correlation scenario
Journal of Physics A, 2012Co-Authors: Elena R. LoubenetsAbstract:We specify for a general correlation scenario a particular type of local quasi hidden variable (LqHV) model (Loubenets 2012 J. Math. Phys. 53 022201)—a deterministic LqHV model, where all joint Probability distributions of a correlation scenario are simulated via a single measure space with a normalized bounded real-valued measure not being necessarily positive and random variables, each depending only on a setting of the corresponding measurement at the corresponding site. We prove that an arbitrary multipartite correlation scenario admits a deterministic LqHV model if and only if all its joint Probability distributions satisfy the consistency condition, constituting the general nonsignaling condition formulated in Loubenets (2008 J. Phys. A: Math. Theor. 41 445303). This mathematical result specifies a new Probability model that has a measure-theoretic structure resembling the structure of the Classical Probability model but incorporates the latter only as a particular case. The local version of this quasi-Classical Probability model covers the probabilistic description of each nonsignaling correlation scenario, in particular, each correlation scenario on a multipartite quantum state.
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nonsignaling as the consistency condition for local quasi Classical Probability modelling of a general multipartite correlation scenario
arXiv: Quantum Physics, 2011Co-Authors: Elena R. LoubenetsAbstract:We specify for a general correlation scenario a particular type of a local quasi hidden variable (LqHV) model [J. Math. Phys. 53 (2012), 022201] -- a deterministic LqHV model, where all joint Probability distributions of a correlation scenario are simulated via a single measure space with a normalized bounded real-valued measure not necessarily positive and random variables, each depending only on a setting of the corresponding measurement at the corresponding site. We prove that an arbitrary multipartite correlation scenario admits a deterministic LqHV model if and only if all its joint Probability distributions satisfy the consistency condition constituting the general nonsignaling condition formulated in [J. Phys. A: Math. Theor. 41 (2008), 445303]. This mathematical result specifies a new Probability model that has the measure-theoretic structure resembling the structure of the Classical Probability model but incorporates the latter only as a particular case. The local version of this quasi Classical Probability model covers the probabilistic description of every nonsignaling correlation scenario, in particular, each correlation scenario on an multipartite quantum state.
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Quantum states satisfying Classical Probability constraints
2006Co-Authors: Elena R. LoubenetsAbstract:For linear combinations of quantum product averages in an arbitrary bipartite state, we derive new quantum Bell-form and CHSH-form inequalities with the right-hand sides expressed in terms of a bipartite state. This allows us to specify in a general setting bipartite state properties sufficient for the validity of a Classical CHSH-form inequality and the perfect correlation form of the original Bell inequality for any bounded quantum observables. We also introduce a new general condition on a bipartite state and quantum observables sufficient for the validity of the original Bell inequality, in its perfect correlation or anticorrelation forms. Under this general sufficient condition, a bipartite quantum state does not necessarily exhibit perfect correlations or anticorrelations.
Tim Rakow - One of the best experts on this subject based on the ideXlab platform.
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If quantum Probability = Classical Probability + bounded cognition; is this good, bad, or unnecessary?
The Behavioral and brain sciences, 2013Co-Authors: Tim RakowAbstract:Quantum Probability models may supersede existing probabilistic models because they account for behaviour inconsistent with Classical Probability theory that are attributable to normal limitations of cognition. This intriguing position, however, may overstate weaknesses in Classical Probability theory by underestimating the role of current knowledge states and may under-employ available knowledge about the limitations of cognitive processes. In addition, flexibility in model specification has risks for the use of quantum Probability.
Jochen Rau - One of the best experts on this subject based on the ideXlab platform.
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On quantum vs. Classical Probability
Annals of Physics, 2009Co-Authors: Jochen RauAbstract:Quantum theory shares with Classical Probability theory many important properties. I show that this common core regards at least the following six areas, and I provide details on each of these: the logic of propositions, symmetry, probabilities, composition of systems, state preparation and reductionism. The essential distinction between Classical and quantum theory, on the other hand, is shown to be joint decidability versus smoothness; for the latter in particular I supply ample explanation and motivation. Finally, I argue that beyond quantum theory there are no other generalisations of Classical Probability theory that are relevant to physics.