Non-Gaussian preparations that concentrate

Fix the nonzero direction $v$ above and $e=v/\Vert v\Vert$. Use the Fock factorization

Let the residual state be one particle in the $e$ mode and vacuum in $e^{\perp }$. Its mean field is zero, its number expectation is one, and

Therefore the displaced relational observable has exact mean $o$ and variance $3ϵ\Vert v{\Vert }^2$.

Its characteristic function is

To verify it, normal order $e^{\mathrm{i}s(a+a^{†})}$; between one-particle vectors only the constant and the single annihilation–creation term survive, giving $e^{-s^2/2}(1-s^2)$. This is generally not a Gaussian law, but its $W_2$ distance to ${\delta }_o$ is exactly $\sqrt{3ϵ}\Vert v\Vert$.

More generally every fixed Fock-normal state of finite number expectation, including finite mixtures of finite-particle vectors, satisfies (eq:number-condition). No infinite-volume thermal density is assumed to belong to this class.