Entropy and information gain in quantum continual measurements

The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument, positive operator valued (POV) measure, etc. [1, 2], by using quantum stochastic differential equations [3, 4] and by using classical stochastic differential equations (SDE’s) for vectors in Hilbert spaces or for trace-class operators [5, 6, 7, 8]. In the same times Ozawa made developments in the theory of instruments [9, 10] and introduced the related notions of a posteriori states [11] and of information gain [12]. In Section 2 we introduce a simple class of SDE’s relevant to the theory of continual measurements and we recall how they are related to instruments and a posteriori states and, so, to the general formulation of quantum mechanics [13]. In Section 3 we shall introduce and use the notion of information gain and the other results of paper [12] inside the theory of continual measurements.