Polarization monotones of two-dimensional and three-dimensional random electromagnetic fields

We propose a formal resource-theoretic approach to quantify the degree of polarization of two- and three-dimensional random electromagnetic fields. This endows the space of spectral polarization matrices with the orders induced by majorization or convex mixing that naturally recover the best-known polarization measures.

Figure 1

Polarization space representation. (a) For $d=2$ (solid line), $\mathcal{P}$ is embedded in the 1-simplex defined by ${\rho }^{1\mathrm{P}}$ and ${\rho }^{2\mathrm{P}}$. (b) For $d=3$ (gray triangle), $\mathcal{P}$ is embedded in the 2-simplex formed by the convex hull of ${\rho }^{1\mathrm{P}},{\rho }^{2\mathrm{P}}$, and ${\rho }^{3\mathrm{P}}$.

Figure 2

Given the state $\rho =\mathrm{diag}(0.5,0.4,0.1)$ (black point), we depict the sets ${\rho }^{\prec }$ (gray region), ${\rho }^{\succ }$ (meshed region), and ${\rho }^{\mathrm{inc}}$ (white region) given by (a) majorization partial order and (b) convex preorder.

Figure 3

Geometric representation of the isopolarization curves for (a) $P^{\mathrm{SSKF}}$, (b) $P^{\mathrm{vN}}$, (c) $P^{\mathrm{lin}}$, and (d) $P^{\mathrm{EDPW}}$. All of them monotonically increase from the center towards the vertex.