Classical distinguishability as an operational measure of polarization

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 of its polarization-transformed counterparts. 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.

Figure 1

Mach-Zehnder setup to interfere the state $\rho$ (lower arm) with its SU($n$) transformed partners (upper arm). The distinguishability is related to the visibility of the interference pattern at the detector $D$.

Figure 2

A 3D plot locating the overlap $\mathrm{Tr}\left(\rho {\rho }_g\right)$ as a function of the parameters $n_3$ and $n_8$. At the bottom, we show a contour plot of the surface.

Figure 3

Isocontour lines (in the same color scale as in Fig. 2) of the different degrees of polarization as a function of the parameters $n_3$ and $n_8$. In the top panel, ${\mathbb{P}}_{\mathrm{HS}}^{\left(3\right)}$ (left) and ${\mathbb{P}}_{\mathrm{PP}}^{\left(3\right)}$ (right); in the bottom panel, ${\mathbb{P}}_{\mathrm{PU}}^{\left(3\right)}$ (left) and ${\mathbb{P}}_{\mathrm{U}}^{\left(3\right)}$ (right).