Classical coherence matrices and measurement scope — Examples

Given: The two-point probability space with probability $1\mathrm{/}2$ at each point, and $a\in \mathbb{C}\setminus \{0\}$ in V m${}^{-1}$.

Classical two-component coherency matrix: $\mathrm{\Gamma }=\mathbb{E}[Z{Z}^{†}]$ and the Stokes coordinates are the trace, diagonal difference, twice the real off-diagonal part, and minus twice its imaginary part.

Stokes cone and deterministic Jones transformations: a positive-intensity state with $S_1=S_2=S_3=0$ has degree of polarization zero; the theorem's algebra is conditional on this classical second-moment model.

Linearity, monotonicity, and the modulus bound for expectation: finite expectations distribute over the two outcomes with their stated probabilities.

The ensemble consists exactly of the two stated realizations with probability $1\mathrm{/}2$ each; no physical preparation or observation is claimed.

Given: The one-point probability space $\mathrm{\Xi }=\{{\sigma }_0\}$ with $\mathbb{P}(\{{\sigma }_0\})=1$, and the square-integrable two-component complex amplitude $Z({\sigma }_0)=0$.

Classical two-component coherency matrix: $\mathrm{\Gamma }=\mathbb{E}[Z{Z}^{†}]$, $S_0={\mathrm{\Gamma }}_{11}+{\mathrm{\Gamma }}_{22}$, and the normalized degree is defined only when $S_0>0$.

$Z({\sigma }_0)=0$ on the declared one-point probability space.