Quantum theory of incompatible observations

Quantum theory describes events on the most fundamental level currently available. The synthesis of information from mutually incompatible quantum measurements plays the key role in testing the structure of the theory. The purpose of this Letter is to show a unique relationship between quantum theory and the mathematical statistics used to obtain optimal information from incompatible observations: Quantum theory prefers the relative entropy (maximum likelihood principle) as the proper measure for evaluation of the distance between measured data and probabilities defined by quantum theory. In the standard textbooks [1], a quantum measurement is represented by a hermitian operator Â, whose spectrum determines the possible results of the measurement