Theorem: Normal two-mode Gaussian restriction

Under Assumption ass:radiation, the two-packet state is the unique normal density on ${\mathcal{H}}_{12}={\mathcal{F}}_{\mathrm{s}}({\mathbb{C}}^2)$ with characteristic function

If $A=W\mathrm{diag}({\lambda }_1,{\lambda }_2)W^{†}$, then

A zero eigenvalue means vacuum in that diagonal mode. No density operator for the entire stationary continuum is required.

Proof. Restriction and complex-linear scattering preserve the inherited centered quasifree prescription. Its zero anomalous covariance and number matrix (eq:A) give (eq:characteristic).

For completeness, let ${\beta }_1,{\beta }_2$ be independent circular complex Gaussian random variables with variances ${\lambda }_1,{\lambda }_2$, with a zero-variance variable identically zero, and put $\alpha =W\beta$. The positive trace-class mixture

has trace one and characteristic function

For a single variance $\lambda >0$, angular and radial integration give the number probabilities

Consequently the mixture is (eq:jointdensity). Equality of Weyl characteristic functions identifies the abstract restricted state and uniquely determines its normal density. This proves normality by construction.