Quantum-like cognition and decision making in the light of quantum measurement theory

We characterize the class of quantum measurements that matches the applications of quantum theory to cognition (and decision making) - quantum-like modeling. Projective measurements describe the canonical measurements of the basic observables of quantum physics. However, the combinations of the basic cognitive effects, such as the question order and response replicability effects, cannot be described by projective measurements. We motivate the use of the special class of quantum measurements, namely {\it sharp repeatable non-projective measurements} - ${\cal SR\bar{P}}. $ This class is practically unused in quantum physics. Thus, physics and cognition explore different parts of quantum measurement theory. Quantum-like modeling isn't automatic borrowing of the quantum formalism. Exploring the class ${\cal SR\bar{P}}$ highlights the role of {\it noncommutativity of the state update maps generated by measurement back action.} Thus, ``non-classicality'' in quantum physics as well as quantum-like modeling for cognition is based on two different types of noncommutativity, of operators (observables) and instruments (state update maps): {\it observable-noncommutativity} vs. {\it state update-noncommutativity}. We speculate that distinguishing quantum-like properties of the cognitive effects are the expressions of the latter, or possibly both.

Coupling quantum-like cognition with the neuronal networks within generalized probability theory

The recent years are characterized by intensive applications of the methodology and mathematical apparatus of quantum theory, quantum-like modeling, in cognition, psychology, and decision making. In spite of the successful applications of this approach to a variety of psychological effects, e.g., the order, conjunction, disjunction, and response replicability effects, one may (but need not) feel dissatisfaction due to the absence of clear coupling to the neurophysiological processes in the brain. For the moment, this is just a phenomenological approach. In this paper we construct the quantum-like representation of the networks of communicating neurons. It is based not on standard quantum theory, but on generalized probability theory (GPT) with the emphasis of the operational measurement approach. We employ GPT's version which is based on ordered linear state space (instead of complex Hilbert space). A network of communicating neurons is described as a weighted ordered graph that in turn is encoded by its weight matrix. The state space of weight matrices is embedded in GPT with effect-observables and state updates within measurement instruments theory. The latter plays the crucial role. This GPT based model shows the basic quantum-like effects, as e.g. the order, non-repeatability, and disjunction effects; the latter is also known as interference of decisions. This GPT coupling also supports quantum-like modeling in medical diagnostic for neurological diseases, as depression and epilepsy. Although the paper is concentrated on cognition and neuronal networks, the formalism and methodology can be straightforwardly applied to a variety of biological and social networks.