Polarization of Optical Fields beyond the Classical Description

In many measurements, and in communication protocols, the polarization degree of freedom of an optical field is used to pick up or encode the information, both in the classical domain and in quantum optics and quantum information. This is natural since polarization is a relatively robust and an easily and cheaply controllable degree of freedom. The classical description of polarization involves the Stokes parameters and Mller matrices. However, this description characterizes only the first order moments of the Stokes parameters. In contrast, quantum protocols and measurements typically involve coincidence measurements, and such measurement outcomes cannot be described in terms of first order (intensity) moments of the Stokes operators, higher order moments are needed. E.g., classically, in order for a field to be unpolarized (have vanishing Stokes parameters) the field must be in a statistical mixture of polarized states. This is not necessary for quantum states, where there exists large classes of pure states that are (classically) unpolarized. This phenomenon has been called hidden polarization. I therefore suggest that the polarization of optical quantum states be characterized by a hierarchy of moment tensors and propose a method to completely characterize the polarization state using a minimal set of measurements.