The questions studied in this thesis are centered around the moment operators of a quantum observable, the latter being represented by a normalized positive operator measure. The moment operators of an observable are physically relevant, in the sense that these operators give, as averages, the moments of the outcome statistics for the measurement of the observable. The main questions under consideration in this work arise from the fact that, unlike a projection valued observable of the von Neumann formulation, a general positive operator measure cannot be characterized by its first moment operator. The possibility of characterizing certain observables by also involving higher moment operators is investigated and utilized in three different cases: a characterization of projection valued measures among all the observables is given, a quantization scheme for unbounded classical variables using translation covariant phase space operator measures is presented, and, finally, a mathematically rigorous description is obtained for the measurements of rotated quadratures and phase space observables via the high amplitude limit in the balanced homodyne and eight-port homodyne detectors, respectively. In addition, the structure of the covariant phase space operator measures, which is essential for the above quantization, is analyzed in detail in the context of a (not necessarily unimodular) locally compact group as the phase space.
This paper focuses on the problem of localization in quantum mechanics. It is well known that it is not possible to define a localization observable for the photon by means of projection-valued measures, or conversely, that it is possible by using positive operator-valued measures. On the other hand, projection-valued measures imply a kind of localization which is stronger than that implied by positive operator-valued measures. It has been claimed that the norm-1 property would in some sense reduce the gap between the two kinds of localizations. We give a necessary condition for the norm-1 property and show that it is not satisfied by several important localization observables.
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.
We represent quantum phase observables as phase shift covariant normalized positive operator measures. The phase operators are the first moment operators of the phase observables. A phase operator determines the associated phase observable uniquely. We show that the Cahill–Glauber s -ordered phase operators are determined by phase shift covariant generalized operator measures, which are ordinary operator measures whenever Re s < 0 and phase observables when s ≤ −1. The Wigner–Weyl quantized phase operator is not determined by any phase observable. We investigate the classical limit of covariant (generalized) operator measures in coherent states.
A function of positive type can be defined as a positive functional on a convolution algebra of a locally compact group. In the case where the group is abelian, by Bochner's theorem a function of positive type is, up to normalization, the Fourier transform of a probability measure. Therefore, considering the group of translations on phase space, a suitably normalized phase-space function of positive type can be regarded as a realization of a classical state. Thus, it may be called a function of classical positive type . Replacing the ordinary convolution on phase space with the twisted convolution, one obtains a noncommutative algebra of functions whose positive functionals we may call functions of quantum positive type . In fact, by a quantum version of Bochner's theorem, a continuous function of quantum positive type is, up to normalization, the (symplectic) Fourier transform of a Wigner quasi-probability distribution; hence, it can be regarded as a phase-space realization of a quantum state. Playing with functions of positive type—classical and quantum—one is led in a natural way to consider a class of semigroups of operators, the classical-quantum semigroups . The physical meaning of these mathematical objects is unveiled via quantization, so obtaining a class of quantum dynamical semigroups that, borrowing terminology from quantum information science, may be called classical-noise semigroups .