ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Mixed classical polarization ensembles

Physical scope

A finite, ideal classical ensemble of two transverse electric phasors at a fixed paraxial detector point. The example distinguishes a vanishing ensemble polarization vector from nonzero field amplitude in every realization. It is a constructed model example, not a reported preparation, measurement, or observation and not a quantum density-operator interpretation.

Example

At one fixed detector point, let a nonzero complex electric amplitude $a$ , with units V m $^{- 1}$ , occur with equal probability in the two realizations $Z = (a, 0)^{T}$ and $Z = (0, a)^{T}$ . Then $Γ = (∣ a ∣^{2} / 2) I_{2}$ , $S_{0} = ∣ a ∣^{2} > 0$ , and $S_{1} = S_{2} = S_{3} = 0$ , so the ensemble is unpolarized by the two-component definition even though each realization is a definite nonzero classical field amplitude.

Example references

Classical two-component coherency matrix

Stokes cone and deterministic Jones transformations

Facts & Assumptions

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

[F1]

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

[F2]

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.

[F3]

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

[A1]

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

Verification

1.1givenF1F3A1algebra

The two realization matrices are $Z_{1} Z_{1}^{†} = (\begin{array}{cc} ∣ a ∣^{2} & 0 \\ 0 & 0 \end{array})$ and $Z_{2} Z_{2}^{†} = (\begin{array}{cc} 0 & 0 \\ 0 & ∣ a ∣^{2} \end{array})$ . Averaging them with the probabilities in [A1] gives $Γ = (∣ a ∣^{2} / 2) I_{2}$ .

2.1givenstep 1.1F1F2algebra∎

Substituting this matrix into [F1] gives $S_{0} = ∣ a ∣^{2} > 0$ and $S_{1} = S_{2} = S_{3} = 0$ , hence $P = 0$ ; each of the two sample values remains a specified nonzero classical amplitude because $a \neq 0$ .

Remarks

The example is a finite mathematical ensemble, not a simulated or measured dataset. It does not assert that the realizations occur with equal empirical frequency in any laboratory.

Depends on

Dependency tree · 2 levels

Sources

José J. Gil, Polarimetric Characterization of Light and Media, §2.1–2.6, author-revised 2020 version of Eur. Phys. J. Appl. Phys. 40 (2007)