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Open Systems & Information Dynamics

On Quantum-Like Probabilistic Structure of Mental Information

Andrei Yu. Khrennikov

Published Sep 1, 2004• 93 citations
Publisher

Abstract

We describe methodology of cognitive experiments (based on interference of probabilities for mental observables) which could verify quantum-like structure of mental information, namely, interference of probabilities for incompatible observables. In principle, such experiments can be performed in psychology, cognitive, and social sciences. In fact, the general contextual probability theory predicts not only quantum-like trigonometric (cos θ) interference of probabilities, but also hyperbolic (cosh θ) interference of probabilities (as well as hyper-trigonometric). In principle, statistical data obtained in experiments with cognitive systems can produce hyperbolic (cosh θ) interference of probabilities. We introduce a wave function of (e.g., human) population. In general, we should not reject the possibility that cognitive functioning is neither quantum nor classical. We discuss the structure of state spaces for cognitive systems.

DOI
10.1023/b:opsy.0000047570.68941.9d

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We describe methodology of cognitive experiments (based on interference of probabilities for mental observables) which could verify quantum-like structure of mental information, namely, interference of probabilities for incompatible observables. In principle, such experiments can be performed in psychology, cognitive, and social sciences. In fact, the general contextual probability theory predicts not only quantum-like trigonometric (cos θ) interference of probabilities, but also hyperbolic (cosh θ) interference of probabilities (as well as hyper-trigonometric). In principle, statistical data obtained in experiments with cognitive systems can produce hyperbolic (cosh θ) interference of probabilities. We introduce a wave function of (e.g., human) population. In general, we should not reject the possibility that cognitive functioning is neither quantum nor classical. We discuss the structure of state spaces for cognitive systems.
We present a contextualist statistical realistic model for quantum-like representations in physics, cognitive science and psychology. We apply this model to describe cognitive experiments to check quantum-like structures of mental processes. The crucial role is played by interference of probabilities for mental observables. Recently one of such experiments based on recognition of images was performed. This experiment confirmed our prediction on quantum-like behaviour of mind. In our approach ``quantumness of mind'' has no direct relation to the fact that the brain (as any physical body) is composed of quantum particles. We invented a new terminology ``quantum-like (QL) mind.'' Cognitive QL-behaviour is characterized by nonzero coefficient of interference λ.λ.λ. This coefficient can be found on the basis of statistical data. There is predicted not only cos⁡θ\cos θcosθ-interference of probabilities, but also hyperbolic cosh⁡θ\cosh θcoshθ-interference. This interference was never observed for physical systems, but we could not exclude this possibility for cognitive systems. We propose a model of brain functioning as QL-computer (there is discussed difference between quantum and QL computers).
In this paper we present quantum-like (QL) representation of the Shafir-Tversky statistical effect. We apply so called contextual approach. The Shafir-Tversky effect is considered as a consequence of combination of a number of incompatible contexts which are involved e.g. in Prisoner's Dilemma or in more general games inducing the disjunction effect. As a consequence, the law of total probability is violated for experimental data obtained by Shafir and Tversky (1992) as well as Tversky and Shafir (1992). Moreover, we can find a numerical measure of contextual incompatibility (so called coefficient of interference) as well as represent contexts which are involved in Prisoner's Dilemma (PD) by probability amplitudes -- normalized vectors (``mental wave functions''). We remark that statistical data from Shafir and Tversky (1992) and Tversky and Shafir (1992) experiments differ crucially from the point of view of mental interference. The second one exhibits the conventional trigonometric (cos⁡\coscos-type) interference, but the first one exhibits so called hyperbolic (cosh⁡\coshcosh-type) interference. We discuss QL processing of information by cognitive systems, in particular, QL decision making as well as classical and QL rationality.
We continue the development of a so-called contextual statistical model (here context has the meaning of a complex of physical conditions). It is shown that, besides contexts producing the conventional trigonometric cos-interference, there exist contexts producing the hyperbolic cos-interference. Starting with the corresponding interference formula of total probability we represent such contexts by hyperbolic probabilistic amplitudes or in the abstract formalism by normalized vectors of a hyperbolic analogue of the Hilbert space. There is obtained a hyperbolic Born’s rule. Incompatible observables are represented by noncommutative operators. This paper can be considered as the first step towards hyperbolic quantum probability. We also discuss possibilities of experimental verification of hyperbolic quantum mechanics: in physics of elementary particles, string theory as well as in experiments with nonphysical systems, e.g., in psychology, cognitive sciences, and economy.
Introduction In the previous chapter, we presented experimental data from cognitive psychology which can be considered as a strong argument in favor of the complex (or more general) Hilbert space representation of information by cognitive systems and the use of the quantum information scheme in information processing. We have seen that the classical Law of Total Probability (LTP) is violated. The interference of probabilities induces a possibility to “reconstruct a mental wave function.” An important lesson we can learn from this is that the standard Dirac-von Neumann approach based on self-adjoint operators is too restrictive to describe known data from cognitive psychology. More general quantum models should be used. Please see also Appendix 2, this chapter. In this chapter, we proceed under the assumption (which we consider as sufficiently confirmed experimentally) that in order to make decisions, cognitive systems operate with “mental wave functions.” We describe the scheme of such decision making based on the lifting formalism, see Appendix 1, this chapter. Since this book is oriented towards non-experts in quantum information theory, we try to avoid the use of the lifting concept as long as is tolerable. In principle, the reader can proceed without using the lifting concept. However, the formulation in terms of lifting is mathematically elegant and it provides a possibility for further generalizations of the model.