Quantization and noiseless measurements

Abstract

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 $f:\R^2\to \R$ 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 well-known 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 notice that the noiseless measurements are the ones 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 $\Gamma(f^k)$, $k\in \N$, with $\Gamma$ a fixed map from classical variables to Hilbert space operators. In particular, we consider the quantization of classical \emph{questions}, that is, functions $f:\R^2\to\R$ taking only values 0 and 1. We compare two concrete realizations of the map $\Gamma$ in view of their ability to produce noiseless measurements: one being the Weyl map, and the other defined by using phase space probability distributions.