Quantum theory of measurements as quantum decision theory

TL;DR

This paper develops a unified quantum decision theory framework that extends measurement descriptions to uncertain and composite events using positive operator-valued measures, highlighting the role of entanglement in quantum probabilities.

Contribution

It introduces a comprehensive language for quantum decision making and measurements under uncertainty, emphasizing the importance of entanglement and generalized measurement operators.

Findings

Quantum measurements under uncertainty require positive operator-valued measures.

Entangled events lead to quantum probabilities that differ from classical probabilities.

Interference effects in quantum probability arise from entangled prospects and states.

Abstract

Theory of quantum measurements is often classified as decision theory. An event in decision theory corresponds to the measurement of an observable. This analogy looks clear for operationally testable simple events. However, the situation is essentially more complicated in the case of composite events. The most difficult point is the relation between decisions under uncertainty and measurements under uncertainty. We suggest a unified language for describing the processes of quantum decision making and quantum measurements. The notion of quantum measurements under uncertainty is introduced. We show that the correct mathematical foundation for the theory of measurements under uncertainty, as well as for quantum decision theory dealing with uncertain events, requires the use of positive operator-valued measure that is a generalization of projection-valued measure. The latter is appropriate for operationally testable events, while the former is necessary for characterizing operationally uncertain events. In both decision making and quantum measurements, one has to distinguish composite non-entangled events from composite entangled events. Quantum probability can be essentially different from classical probability only for entangled events. The necessary condition for the appearance of an interference term in the quantum probability is the occurrence of entangled prospects and the existence of an entangled strategic state of a decision maker or of an entangled statistical state of a measuring device.