Proposition: Admissible squeezed preparation

The state ${\sigma }_{R,ϵ}$ is centered, regular, quasifree, and Hadamard for every allowed finite $R,ϵ$. Its residual variance is

Proof. The scaling on a Darboux plane and identity on its symplectic complement preserve the symplectic form. Hence $W(u)\mapsto W(S_{ϵ}u)$ is a Weyl automorphism. Pullback of the reference state proves positivity and quasifreeness. Its covariance remains continuous on compact test spaces, so regularity is preserved.

The difference between the reference and transformed two-point distributions is a finite sum of products of smooth free solutions associated with $g_{ϵ},p_{ϵ}$ and their contractions against the reference covariance. For a Hadamard bisolution, smearing one variable with a smooth compact test gives a smooth solution in the other variable. The covariance difference is therefore smooth. The Hadamard singularity is unchanged.

Finally $S_{ϵ}{g}_{ϵ}={ϵ}^2{g}_{ϵ}$, which proves (eq:squeezedvariance).