Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Stokes cone and deterministic Jones transformations

Physical scope

Conditional algebra for the classical two-component coherency matrix defined in this library. For transverse electric components its four real coordinates are the stated Stokes parameters; for other classical two-channel amplitudes the same cone is an algebraic second-moment constraint, not a polarization interpretation. No quantum-state identification, apparatus model, empirical observation, or finite-sample claim is made.

Statement

Let $Z = (Z_{1}, Z_{2})^{T}$ be a measurable square-integrable complex two-component vector on a probability space and define $Γ : = E [Z Z^{†}]$ , with $Γ_{i j} : = E [Z_{i} \overline{Z_{j}}]$ . Put $S_{0} : = Γ_{11} + Γ_{22}$ , $S_{1} : = Γ_{11} - Γ_{22}$ , $S_{2} : = 2 Re Γ_{12}$ , and $S_{3} : = - 2 Im Γ_{12}$ ; when $S_{0} > 0$ , put $P : = \sqrt{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}} / S_{0}$ . Then $Γ$ is positive semidefinite, meaning $v^{†} Γ v \geq 0$ for every $v \in C^{2}$ ; its rank is the complex dimension of its image. The Stokes coordinates satisfy $S_{0} \geq 0, S_{0}^{2} \geq S_{1}^{2} + S_{2}^{2} + S_{3}^{2} .$ Equality in the second relation holds exactly when $Γ = 0$ or $Γ$ has rank one. Consequently $P \in [0, 1]$ when $S_{0} > 0$ , and it is undefined when $S_{0} = 0$ . If a fixed deterministic dimensionless complex-linear map $M : C^{2} \to C^{2}$ (a Jones matrix in this model) sends each realization $Z$ to $M Z$ , then $Γ$ transforms to $M Γ M^{†}$ .

Statement references

Classical two-component coherency matrix

Facts & Assumptions

Given: A measurable square-integrable $Z : Ξ \to C^{2}$ and its matrix $Γ = E [Z Z^{†}]$ and Stokes coordinates from the Definition.

[F1]

Classical two-component coherency matrix: $Γ_{i j} = E [Z_{i} \overline{Z_{j}}]$ , $Γ^{†} = Γ$ , $S_{0} = Γ_{11} + Γ_{22}$ , $S_{1} = Γ_{11} - Γ_{22}$ , $S_{2} = 2 Re Γ_{12}$ , $S_{3} = - 2 Im Γ_{12}$ , and $P$ is defined only when $S_{0} > 0$ .

[F2]

Linearity, monotonicity, and the modulus bound for expectation: expectation of an integrable complex variable is linear.

[A1]

$M$ is a fixed deterministic linear map, not a random matrix correlated with $Z$ ; its entries are dimensionless and the output amplitudes use the same component units as the input.

Proof

1.1givenF1algebra

For $v = (v_{1}, v_{2})^{T}$ , the finite sum $\overline{v_{1}} Z_{1} + \overline{v_{2}} Z_{2}$ has an integrable squared modulus, because it is bounded by $2 (∣ v_{1} ∣^{2} ∣ Z_{1} ∣^{2} + ∣ v_{2} ∣^{2} ∣ Z_{2} ∣^{2})$ . Expanding the finite matrix product therefore gives $v^{†} Γ v = E [∣ v^{†} Z ∣^{2}] \geq 0$ . Hence $Γ$ is positive semidefinite.

2.1step 1.1F1algebra

Write $Γ = (\begin{array}{cc} a & c \\ \overline{c} & b \end{array})$ , where $a, b \geq 0$ . If $a > 0$ , positivity applied to $v = (- c / a, 1)^{T}$ gives $b - ∣ c ∣^{2} / a \geq 0$ , hence $a b \geq ∣ c ∣^{2}$ . If $a = 0$ and $c \neq 0$ , positivity at $v = (- t c, 1)^{T}$ gives $b - 2 t ∣ c ∣^{2} \geq 0$ for every $t > 0$ , which fails for sufficiently large $t$ ; thus $c = 0$ and again $a b \geq ∣ c ∣^{2}$ .

3.1step 2.1F1algebra

The definitions give $S_{0}^{2} - S_{1}^{2} = 4 a b$ and $S_{2}^{2} + S_{3}^{2} = 4 ∣ c ∣^{2}$ . Step 2.1 therefore proves $S_{0}^{2} \geq S_{1}^{2} + S_{2}^{2} + S_{3}^{2}$ . Equality is equivalent to $\det Γ = a b - ∣ c ∣^{2} = 0$ , which for a positive-semidefinite $2 \times 2$ matrix means rank below two; it is rank zero exactly when $Γ = 0$ , and otherwise rank one. When $S_{0} > 0$ , taking nonnegative square roots gives $0 \leq P \leq 1$ ; when $S_{0} = 0$ the quotient is undefined by its Definition.

4.1givenF1F2A1algebra∎

Since $M$ is fixed, $(M Z) (M Z)^{†} = M Z Z^{†} M^{†}$ for every realization. Taking the componentwise expectation and using [F2]'s linearity gives $Γ^{'} = E [(M Z) (M Z)^{†}] = M Γ M^{†}$ .

Remarks

This is a conditional identity for an ensemble second-moment model. A deterministic Jones transformation is not a model of a general random, depolarizing, or statistically estimated optical device.

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)