Canonical approximate quantum measurements

In order to generalize the Wigner formula for the joint probability distribution of successive canonical measurements of discrete observables to continuous observables, a mathematical model is investigated of a canonical approximate measurement of an arbitrary observable, which generalizes von Neumann’s model of measuring interaction and Davies’s covariant instrument for approximate position measurement. Calculations of two types of root-mean-square errors show that this model clears certain requirements for good approximate measurements. A modified Wigner formula for successive canonical approximate measurements of arbitrary observables is established. It is also shown that the canonical approximate measurement of a discrete observable is reduced to the conventional measurement satisfying the von Neumann–Lüders collapse postulate by a minor modification in the process of measurement.