The characterization of quantum polarization of light requires knowledge of all the moments of the Stokes variables, which are appropriately encoded in the multipole expansion of the density matrix. We look into the cumulative distribution of those m
ultipoles and work out the corresponding extremal pure states. We find that SU(2) coherent states are maximal to any order whereas the converse case of minimal states (which can be seen as the most quantum ones) is investigated for a diverse range of the number of photons. Taking advantage of the Majorana representation, we recast the problem as that of distributing a number of points uniformly over the surface of the Poincare sphere.
We capitalize on a multipolar expansion of the polarisation density matrix, in which multipoles appear as successive moments of the Stokes variables. When all the multipoles up to a given order $K$ vanish, we can properly say that the state is $K$th-
order unpolarized, as it lacks of polarization information to that order. First-order unpolarized states coincide with the corresponding classical ones, whereas unpolarized to any order tally with the quantum notion of fully invariant states. In between these two extreme cases, there is a rich variety of situations that are explored here. The existence of textit{hidden} polarisation emerges in a natural way in this context.
We put forward an operational degree of polarization that can be extended in a natural way to fields whose wave fronts are not necessarily planar. This measure appears as a distance from a state to the set of all its polarization-transformed counterp
arts. By using the Hilbert-Schmidt metric, the resulting degree is a sum of two terms: one is the purity of the state and the other can be interpreted as a classical distinguishability, which can be experimentally determined in an interferometric setup. For transverse fields, this reduces to the standard approach, whereas it allows one to get a straight expression for nonparaxial fields.
We advocate a simple multipole expansion of the polarisation density matrix. The resulting multipoles appear as successive moments of the Stokes variables and can be obtained from feasible measurements. In terms of these multipoles, we construct a wh
ole hierarchy of measures that accurately assess higher-order polarization fluctuations.
We present a moment expansion method for the systematic characterization of the polarization properties of quantum states of light. Specifically, we link the method to the measurements of the Stokes operator in different directions on the Poincar{e}
sphere, and provide a method of polarization tomography without resorting to full state tomography. We apply these ideas to the experimental first- and second-order polarization characterization of some two-photon quantum states. In addition, we show that there are classes of states whose polarization characteristics are dominated not by their first-order moments (i.e., the Stokes vector) but by higher-order polarization moments.
We discuss different proposals for the degree of polarization of quantum fields. The simplest approach, namely making a direct analogy with the classical description via the Stokes operators, is known to produce unsatisfactory results. Still, we argu
e that these operators and their properties should be basic for any measure of polarization. We compare alternative quantum degrees and put forth that they order various states differently. This is to be expected, since, despite being rooted in the Stokes operators, each of these measures only captures certain characteristics. Therefore, it is likely that several quantum degrees of polarization will coexist, each one having its specific domain of usefulness.
We propose an operational degree of polarization in terms of the variance of the projected Stokes vector minimized over all the directions of the Poincare sphere. We examine the properties of this degree and show that some problems associated with th
e standard definition are avoided. The new degree of polarization is experimentally determined using two examples: a bright squeezed state and a quadrature squeezed vacuum.
