An alternative to the Lüders generalization of the von Neumann projection, and its interpretation

An alternative to the Lüders generalization of the von Neumann projection is formulated allowing a generalization to measurements represented by positive operator-valued measures. This generalization is based on the notions of non-orthogonal projections and bi-orthonormal systems in Hilbert-Schmidt space, and is closely related to phase-space representations of quantum mechanics. A number of applications will be discussed. It is demonstrated that the alternative projection has a natural interpretation as a representation of the information on the initial state, to be obtained by the measurement.