State change, quantum probability, and information in operational phase-space measurement

State change, quantum probability, and information gain in the operational phase space measurement are formulated by means of positive operator-valued measure (POVM) and operation. The properties of the operational POVM and its marginal POVM which yield the quantum probability distributions of the measurement outcomes obtained by the operational phase space measurement are investigated. The Naimark extension of the operational POVM can be expressed in terms of the relative-position states and the relative-momentum states in the extended Hilbert space. An observable quantity measured in the operational phase-space measurement becomes a fuzzy or unsharp observable. The state change of a physical system caused by the operational phase-space measurement is described by the operation which is obtained explicitly for the position and momentum measurements and for the simultaneous measurement of position and momentum. Using the results, the entropy change of the measured physical system and the information gain in the operational phase-space measurement are investigated. It is found that the average value of the entropy change is equal to the Shannon mutual information extracted from the outcomes exhibited by the measurement apparatus.