  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.
