Polarization distribution and degree of polarization for three-dimensional quantum light fields

We introduce a probability distribution for polarization of three-dimensional quantum light fields as a marginal of the quadrature Q function for a three-mode field by removing the variables irrelevant for polarization (total intensity and global phase). The probability distribution turns out to be determined by projection on SU(3) coherent states. We introduce a degree of polarization as the distance between the polarization distribution and the uniform distribution associated with completely unpolarized light. We study the relation between two- and three-dimensional polarization by considering field states with a component in the vacuum state. We apply this formalism to some relevant field states