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)
Ef, 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
Ef in our quantization is defined through its moment operators, which are required to be of the form
Γ(fk),
k∈N, with
Γ a fixed map from classical variables to Hilbert space operators. In particular, we consider the quantization of classical \emph{questions}, that is, functions
f:R2→R taking only values 0 and 1. We compare two concrete realizations of the map
Γ in view of their ability to produce noiseless measurements: one being the Weyl map, and the other defined by using phase space probability distributions.
In these notes we review the foundations of Banach–Poisson geometry and explain how in this framework one obtains a unified approach to the Hamiltonian and the quantum mechanical description of the physical systems. Our considerations will be based on the notion of Banach Lie–Poisson space (see Odzijewicz and Ratiu [25]) and the notion of the coherent state map (see Odzijewicz [22]), which appear to be the crucial instrument for the clarifying what is the quantization of the classical physical (Hamiltonian) system.
We consider questions related to a quantization scheme in which a classical variable f:Ω\to R on a phase space Ωis associated with a semispectral measure E^f, such that the moment operators of E^f are required to be of the form Γ(f^k), with Γa suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in some Hilbert space. In particular, we investigate the situation where the map Γis implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ωis first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R^2, where we determine explicitly the relevant operators Γ(f^k) for certain variables f, in the case where the quantization map Γis implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.
We outline a new approach to the characterization as well as to the classification of positive maps. This approach is based on the facial structures of the set of states and of the cone of positive maps. In particular, the equivalence between Schroedinger's and Heisenberg's pictures is reviewed in this more general setting. Furthermore, we discuss in detail the structure of positive maps for two and three dimensional systems. In particular, the explicit form of decomposition of a positive map and the uniqueness of this decomposition for extremal positive maps for 2 dimensional case are described. The difference of the structure of positive maps between 2 dimensional and 3 dimensional cases is clarified. The resulting characterization of positive maps is applied to the study of quantum correlations and entanglement