A-entropy for spectrally absolutely continuous observables

In this paper the notion of entropy of a density operator with respect to spectrally absolutely continuous observable is investigated. The concept of such an entropy is introduced and various possibilities of defining it are discussed. These entropies are examined with regard to their usual properties. We show that this kind of entropy increases after measurement of an observable with a continuous spectrum in the sense of von Neumann and assumes its maximum on a Gaussian state.