A continuous positive operator-valued measurement (POVM) is described for the polarization state of a single photon propagating in a known spatiotemporal mode. This POVM is shown to be the maximum-likelihood quantum measurement for estimating that polarization state. Moreover, the maximum-likelihood polarization estimate derived from this POVM enables uniformly optimal measure-and-prepare state transmission via classical communication---at the
F=2/3 classical limit on average teleportation fidelity---for any pure state of polarization. When two continuous-POVM polarimeters are used to detect the singlet state of a pair photons that are in distinct spatial modes, the conditional probability density for one polarimeter's output given the other's encompasses the quantum interference patterns seen in standard singlet-state polarization analysis. For single-photon illumination, a
1:M optical splitter followed by a collection of
M projective polarization measurements, whose bases are uniformly distributed over the Poincar\'e sphere, yields observation statistics that converge in distribution to those of the continuous-POVM polarimeter as $M\ensuremath{\rightarrow}\ensuremath{\infty}$.
The present dissertation focuses on the analysis of the mathematical structure of the Positive Operator Valued Measures (POVM) and their relevance to quantum mechanics. In particular we analyze: 1. The relationships between POVMs and PVMs (Projection Valued Measures) and prove that each commutative POVM F is the smearing (realized by a Feller Markov Kernel) of a spectral measure. That suggests an interpretation of commutative POVMs as the randomization of real PVMs. Moreover, we characterize the POVMs whose smearing can be realized by strong Feller Markov kernels. 2. The relationships between the characterization of commutative POVMs in item 1) and Naimark's dilation theorem. We prove that the self-adjoint operator A corresponding to the spectral measure E, of which F is the smearing, is the projection of a Naimark operator. 3. Analysis of the informational content of a POVM. We introduce an equivalence relation on the set of observables which we compare with other well known equivalence relations and prove that it is the only one for which E is always equivalent to F. 4. The uniform continuity of a POVM and its relevance to the problem of localization. We take into consideration a non-commutative POVM defined on a locally compact second countable Haussdorf topological space and give necessary and sufficient conditions for it to be uniformly continuous. Moreover, we show the relevance of this result to relativistic quantum mechanics.