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Positive operator-valued measures in quantum decision theory

We show that the correct mathematical foundation of quantum decision theory, dealing with uncertain events, requires the use of positive operator-valued measure that is a generalization of the projection-valued measure. The latter is appropriate for operationally testable events, while the former is necessary for characterizing operationally uncertain events. In decision making, one has to distinguish composite non-entangled events from composite entangled events. The mathematical definition of entangled prospects is based on the theory of Hilbert-Schmidt spaces and is analogous to the definition of entangled statistical operators in quantum information theory. We demonstrate that 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. The origin of uncertainties in standard lotteries is explained.