Quantization and noiseless measurements

In accordance with the fact that quantum measurements are described in terms of positive operator measures (POMs), we consider certain aspects of a quantization scheme in which a classical variable f : ℝ² → ℝ is associated with a unique positive operator measure (POM) E^f , which is not necessarily projection valued. The motivation for such a scheme comes from the wellknown fact that due to the noise in a quantum measurement, the resulting outcome distribution is given by a POM and cannot, in general, be described in terms of a traditional observable, a selfadjoint operator. Accordingly, we note that the noiseless measurements are those which are determined by a selfadjoint operator. The POM E^f in our quantization is defined through its moment operators, which are required to be of the form ⌈(f^k), k ε ℕ, with ⌈ being a fixed map from classical variables to Hilbert space operators. In particular, we consider the quantization of classical questions, that is, functions f : ℝ² → ℝ taking only values 0 and 1. We compare two concrete realizations of the map ⌈ in view of their ability to produce noiseless measurements: one being the Weyl map, and the other defined by using phase space probability distributions.