Measurements of Nondegenerate Discrete Observables

Abstract

Every measurement on a quantum system causes a state change from the system state just before the measurement to the system state just after the measurement conditional upon the outcome of measurement. This paper determines all the possible conditional state changes caused by measurements of nondegenerate discrete observables. For this purpose, the following conditions are shown to be equivalent for measurements of nondegenerate discrete observables: (i) The joint probability distribution of the outcomes of successive measurements depends affinely on the initial state. (ii) The apparatus has an indirect measurement model. (iii) The state change is described by a positive superoperator valued measure. (iv) The state change is described by a completely positive superoperator valued measure. (v) The output state is independent of the input state and the family of output states can be arbitrarily chosen by the choice of the apparatus. The implications to the measurement problem are discussed briefly.