In experiments of games, players frequently make choices which are regarded as irrational in game theory. In papers of Khrennikov (Information Dynamics in Cognitive, Psychological and Anomalous Phenomena. Fundamental Theories of Physics, Kluwer Academic, Norwell, 2004; Fuzzy Sets Syst. 155:4–17, 2005; Biosystems 84:225–241, 2006; Found. Phys. 35(10):1655–1693, 2005; in QP-PQ Quantum Probability and White Noise Analysis, vol. XXIV, pp. 105–117, 2009), it was pointed out that statistics collected in such the experiments have “quantum-like” properties, which can not be explained in (...) classical probability theory. In this paper, we design a simple quantum-like model describing a decision-making process in a two-players game and try to explain a mechanism of the irrational behavior of players. Finally we discuss a mathematical frame of non-Kolmogorovian system in terms of liftings (Accardi and Ohya, in Appl. Math. Optim. 39:33–59, 1999). (shrink)

We discuss foundational issues of quantum information biology —one of the most successful applications of the quantum formalism outside of physics. QIB provides a multi-scale model of information processing in bio-systems: from proteins and cells to cognitive and social systems. This theory has to be sharply distinguished from “traditional quantum biophysics”. The latter is about quantum bio-physical processes, e.g., in cells or brains. QIB models the dynamics of information states of bio-systems. We argue that the information interpretation of quantum mechanics (...) is the most natural interpretation of QIB. Biologically QIB is based on two principles: adaptivity; openness. These principles are mathematically represented in the framework of a novel formalism— quantum adaptive dynamics which, in particular, contains the standard theory of open quantum systems. (shrink)

There exist several phenomena breaking the classical probability laws. The systems related to such phenomena are context-dependent, so that they are adaptive to other systems. In this paper, we present a new mathematical formalism to compute the joint probability distribution for two event-systems by using concepts of the adaptive dynamics and quantum information theory, e.g., quantum channels and liftings. In physics the basic example of the context-dependent phenomena is the famous double-slit experiment. Recently similar examples have been found in biological (...) and psychological sciences. Our approach is an extension of traditional quantum probability theory, and it is general enough to describe aforementioned contextual phenomena outside of quantum physics. (shrink)

A lifting is a map from the state of a system to that of a compound system, which was introduced in Accardi and Ohya (Appl. Math. Optim. 39:33–59, 1999). The lifting can be applied to various physical processes.In this paper, we defined a quantum mutual entropy by the lifting. The usual quantum mutual entropy satisfies the Shannon inequality (Ohya in IEEE Trans. Inf. Theory 29(5):770–774, 1983), but the mutual entropy defined through the lifting does not satisfy this inequality unless some (...) conditions hold. (shrink)