On relations between probabilities under quantum and classical   measurements
  We show that the so-called quantum probabilistic rule, usually presented in the physical literature as an argument of the essential distinction between the probability relations under quantum and classical measurements, is not, as it is usually considered, in contrast to the rule for the addition of probabilities of incompatible events. The latter is, of course, valid under all measurement situations upon classical and quantum objects. We discuss also quantum measurement situation, which is similar to the classical one, corresponding to classical Bayes' formula for conditional probabilities. We show the compatibility of description of this quantum measurement situation in the frame of purely classical and experimentally justified straight ward frequency arguments [Khrennikov (1999, 2000, 2001)] and in the frame of the quantum stochastic approach to the description of generalized quantum measurements (Loubenets (2000); Barndorff-Nielsen and Loubenets (2001)]. In view of derived results, we argue that even in the classical probability the classical Bayes' formula describes a particular case of the considered measurement situation, which is specific for context-independent measurements. The similarity of the forms of the relation between the transformation of probabilities, which we derive in the frame of quantum stochastic approach and in the frame of straight ward frequency arguments, underlines once more that the projective (von Neumann) measurements correspond only to a very special kind of measurement situations in quantum theory. 
Andrei Khrennikov, Elena Loubenets