Single-photon observables and preparation uncertainty relations

We propose a procedure to define all single-photon observables in a consistent and unified picture based on operational approach to quantum mechanics. We identify the suppression of zero-helicity states as a projection from an extended Hilbert space onto the physical single-photon Hilbert space. We show that all single-photon observables are in general described by positive-operator valued measures (POVMs), obtained by applying this projection to opportune projection-valued measures (PVMs) defined on the extended Hilbert space. The POVMs associated to momentum and helicity reduce to PVMs, unlike those associated to position and spin. This fact reflects the intrinsic unsharpness of these observables. We apply this formalism to study the preparation uncertainty relations for position and momentum and to compute the probability distribution of spin, for a broad class of Gaussian states. Results show quantitatively the enhancement of the statistical character of the theory.