Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0705.3898

Quantum-like Representation of Macroscopic Configurations

Andrei Yu. Khrennikov

Published May 26, 2007• 1 citation
arXivHTMLPDF

Abstract

The aim of this paper is to apply a contextual probabilistic model (in the spirit of Mackey, Gudder, Ballentine) to represent and to generalize some results of quantum logic about possible macroscopic quantum-like (QL) behaviour. The crucial point is that our model provides QL-representation of macroscopic configurations in terms of complex probability amplitudes -- wave functions of such configurations. Thus, instead of the language of propositions which is common in quatum logic, we use the language of wave functions which is common in the conventional presentation of QM. We propose a quantum-like representation algorithm, QLRA, which maps probabilistic data of any origin in complex (or even hyperbolic) Hilbert space. On the one hand, this paper clarifyes some questions in foundations of QM, since some rather mystical quantum features are illustrated on the basis of behavior of macroscopic systems. On the other hand, the approach developed in this paper may be used e.g. in biology, sociology, or psychology. Our example of QL-representation of hidden macroscopic configurations can find natural applications in those domains of science.

DOI
10.48550/arxiv.0705.3898

Related papers

A model of quantum-like decision-making with applications to psychology and cognitive science

Andrei Yu. Khrennikov

We consider the following model of decision-making by cognitive systems. We present an algorithm -- quantum-like representation algorithm (QLRA) -- which provides a possibility to represent probabilistic data of any origin by complex probability amplitudes. Our conjecture is that cognitive systems developed the ability to use QLRA. They operate with complex probability amplitudes, mental wave functions. Since the mathematical formalism of QM describes as well (under some generalization) processing of such quantum-like (QL) mental states, the conventional quantum decision-making scheme can be used by the brain. We consider a modification of this scheme to describe decision-making in the presence of two ``incompatible'' mental variables. Such a QL decision-making can be used in situations like Prisoners Dilemma (PD) as well as others corresponding to so called disjunction effect in psychology and cognitive science.

arXiv:0711.1366•Nov 8, 2007•18 citations

Quantum-like Probabilistic Models Outside Physics

Andrei Yu. Khrennikov

We present a quantum-like (QL) model in that contexts (complexes of e.g. mental, social, biological, economic or even political conditions) are represented by complex probability amplitudes. This approach gives the possibility to apply the mathematical quantum formalism to probabilities induced in any domain of science. In our model quantum randomness appears not as irreducible randomness (as it is commonly accepted in conventional quantum mechanics, e.g., by von Neumann and Dirac), but as a consequence of obtaining incomplete information about a system. We pay main attention to the QL description of processing of incomplete information. Our QL model can be useful in cognitive, social and political sciences as well as economics and artificial intelligence. In this paper we consider in a more detail one special application -- QL modeling of brain's functioning. The brain is modeled as a QL-computer.

arXiv:physics/0702250•WORLD SCIENTIFIC eBooks•Jun 1, 2008•8 citations

Quantumlike representation of extensive form games: Probabilistic aspects

Andrei Yu. Khrennikov

We consider an application of the mathematical formalism of quantum mechanics outside physics, namely, to game theory. We present a simple game between macroscopic players, say, Alice and Bob (or in a more complex form—Alice, Bob, and Cecilia), which can be represented in the quantumlike (QL) way—by using a complex probability amplitude (game’s “wave function”) and noncommutative operators. The crucial point is that games under consideration are so-called extensive form games. Here the order of actions of players is important; such a game can be represented by the tree of actions. The QL probabilistic behavior of players is a consequence of incomplete information (which is available to, e.g., Bob) about the previous action of Alice. In general one could not construct a classical probability space underlying a QL game. This can happen even in a QL game with two players. In a QL game with three players Bell’s inequality can be violated. The most natural probabilistic description is given by the so-called contextual probability theory completed by the frequency definition of probability.

arXiv:0705.1260•Journal of Mathematical Physics•Jul 1, 2007•6 citations

Quantum-like Representation Algorithm: Transformation of Probabilistic Data into vectors on Bloch's Sphere

Andrei Yu. Khrennikov

In this paper we present a simple algorithm for representation of statistical data of any origin by complex probability amplitudes. Numerical simulation with Mathematica-6 is performed. The Bloch's sphere is used for visualization of results of numerical simulation. On the one hand, creation of such a quantum-like (QL) representation and its numerical approval is an important step in clarification of extremely complicated interrelation between classical and quantum randomness. On the other hand, it opens new possibilities for application the mathematical formalism of QM in other domains of science.

arXiv:0803.1391•Mar 10, 2008•0 citations

Quantum-like modeling of cognition

Andrei Yu. Khrennikov

This paper begins with a historical review of the mutual influence of physics and psychology, from Freud's invention of psychic energy inspired by von Boltzmann' thermodynamics to the enrichment quantum physics gained from the side of psychology by the notion of complementarity (the invention of Niels Bohr who was inspired by William James), besides we consider the resonance of the correspondence between Wolfgang Pauli and Carl Jung in both physics and psychology. Then we turn to the problem of development of mathematical models for laws of thought starting with Boolean logic and progressing towards foundations of classical probability theory. Interestingly, the laws of classical logic and probability are routinely violated not only by quantum statistical phenomena but by cognitive phenomena as well. This is yet another common feature between quantum physics and psychology. In particular, cognitive data can exhibit a kind of the probabilistic interference effect. This similarity with quantum physics convinced a multi-disciplinary group of scientists (physicists, psychologists, economists, sociologists) to apply the mathematical apparatus of quantum mechanics to modeling of cognition. We illustrate this activity by considering a few concrete phenomena: the order and disjunction effects, recognition of ambiguous figures, categorization-decision making. In Appendix 1 we briefly present essentials of theory of contextual probability and a method of representations of contextual probabilities by complex probability amplitudes (solution of the ``inverse Born's problem'') based on a quantum-like representation algorithm (QLRA).

Frontiers in Physics•Sep 22, 2015•91 citations

Algorithm for Quantum-like Representation: Transformation of Probabilistic Data into Vectors on Bloch's Sphere

Andrei Yu. Khrennikov

In this paper, we present a simple algorithm for representation of statistical data of any origin by complex probability amplitudes. Numerical simulation with Mathematica-6 is performed, cf. [1] and [2] on recent numerical simulation for quantum correlations. The Bloch's sphere is used for the visualization of results of numerical simulation. On the one hand, creation of such a quantum-like (QL) representation and its numerical approval is an important step in clarification of extremely complicated interrelation between classical and quantum randomness. On the other hand, it opens new possibilities for applying the mathematical formalism of QM in other domains of science.

Open Systems & Information Dynamics•Sep 1, 2008•7 citations
The aim of this paper is to apply a contextual probabilistic model (in the spirit of Mackey, Gudder, Ballentine) to represent and to generalize some results of quantum logic about possible macroscopic quantum-like (QL) behaviour. The crucial point is that our model provides QL-representation of macroscopic configurations in terms of complex probability amplitudes -- wave functions of such configurations. Thus, instead of the language of propositions which is common in quatum logic, we use the language of wave functions which is common in the conventional presentation of QM. We propose a quantum-like representation algorithm, QLRA, which maps probabilistic data of any origin in complex (or even hyperbolic) Hilbert space. On the one hand, this paper clarifyes some questions in foundations of QM, since some rather mystical quantum features are illustrated on the basis of behavior of macroscopic systems. On the other hand, the approach developed in this paper may be used e.g. in biology, sociology, or psychology. Our example of QL-representation of hidden macroscopic configurations can find natural applications in those domains of science.
We consider the following model of decision-making by cognitive systems. We present an algorithm -- quantum-like representation algorithm (QLRA) -- which provides a possibility to represent probabilistic data of any origin by complex probability amplitudes. Our conjecture is that cognitive systems developed the ability to use QLRA. They operate with complex probability amplitudes, mental wave functions. Since the mathematical formalism of QM describes as well (under some generalization) processing of such quantum-like (QL) mental states, the conventional quantum decision-making scheme can be used by the brain. We consider a modification of this scheme to describe decision-making in the presence of two ``incompatible'' mental variables. Such a QL decision-making can be used in situations like Prisoners Dilemma (PD) as well as others corresponding to so called disjunction effect in psychology and cognitive science.
We present a quantum-like (QL) model in that contexts (complexes of e.g. mental, social, biological, economic or even political conditions) are represented by complex probability amplitudes. This approach gives the possibility to apply the mathematical quantum formalism to probabilities induced in any domain of science. In our model quantum randomness appears not as irreducible randomness (as it is commonly accepted in conventional quantum mechanics, e.g., by von Neumann and Dirac), but as a consequence of obtaining incomplete information about a system. We pay main attention to the QL description of processing of incomplete information. Our QL model can be useful in cognitive, social and political sciences as well as economics and artificial intelligence. In this paper we consider in a more detail one special application -- QL modeling of brain's functioning. The brain is modeled as a QL-computer.
We consider an application of the mathematical formalism of quantum mechanics outside physics, namely, to game theory. We present a simple game between macroscopic players, say, Alice and Bob (or in a more complex form—Alice, Bob, and Cecilia), which can be represented in the quantumlike (QL) way—by using a complex probability amplitude (game’s “wave function”) and noncommutative operators. The crucial point is that games under consideration are so-called extensive form games. Here the order of actions of players is important; such a game can be represented by the tree of actions. The QL probabilistic behavior of players is a consequence of incomplete information (which is available to, e.g., Bob) about the previous action of Alice. In general one could not construct a classical probability space underlying a QL game. This can happen even in a QL game with two players. In a QL game with three players Bell’s inequality can be violated. The most natural probabilistic description is given by the so-called contextual probability theory completed by the frequency definition of probability.
In this paper we present a simple algorithm for representation of statistical data of any origin by complex probability amplitudes. Numerical simulation with Mathematica-6 is performed. The Bloch's sphere is used for visualization of results of numerical simulation. On the one hand, creation of such a quantum-like (QL) representation and its numerical approval is an important step in clarification of extremely complicated interrelation between classical and quantum randomness. On the other hand, it opens new possibilities for application the mathematical formalism of QM in other domains of science.
This paper begins with a historical review of the mutual influence of physics and psychology, from Freud's invention of psychic energy inspired by von Boltzmann' thermodynamics to the enrichment quantum physics gained from the side of psychology by the notion of complementarity (the invention of Niels Bohr who was inspired by William James), besides we consider the resonance of the correspondence between Wolfgang Pauli and Carl Jung in both physics and psychology. Then we turn to the problem of development of mathematical models for laws of thought starting with Boolean logic and progressing towards foundations of classical probability theory. Interestingly, the laws of classical logic and probability are routinely violated not only by quantum statistical phenomena but by cognitive phenomena as well. This is yet another common feature between quantum physics and psychology. In particular, cognitive data can exhibit a kind of the probabilistic interference effect. This similarity with quantum physics convinced a multi-disciplinary group of scientists (physicists, psychologists, economists, sociologists) to apply the mathematical apparatus of quantum mechanics to modeling of cognition. We illustrate this activity by considering a few concrete phenomena: the order and disjunction effects, recognition of ambiguous figures, categorization-decision making. In Appendix 1 we briefly present essentials of theory of contextual probability and a method of representations of contextual probabilities by complex probability amplitudes (solution of the ``inverse Born's problem'') based on a quantum-like representation algorithm (QLRA).
In this paper, we present a simple algorithm for representation of statistical data of any origin by complex probability amplitudes. Numerical simulation with Mathematica-6 is performed, cf. [1] and [2] on recent numerical simulation for quantum correlations. The Bloch's sphere is used for the visualization of results of numerical simulation. On the one hand, creation of such a quantum-like (QL) representation and its numerical approval is an important step in clarification of extremely complicated interrelation between classical and quantum randomness. On the other hand, it opens new possibilities for applying the mathematical formalism of QM in other domains of science.