Independent probe squeezing and its cost

Take an independent probe reference state with the same Unruh covariance. A fixed reference would give $V(z_{ϵ},z_{ϵ})\sim {ϵ}^{-2}V(z,z)$; its attenuation would tend to zero. Weak coupling alone does not remove probe noise.

Choose a compact real test $p_0$ with $c_0={\sigma }_{\psi }(z,p_0)\ne 0$. Such a test exists by nondegeneracy of the free scalar test symplectic quotient. Set

For sufficiently small $ϵ$ this is defined and ${\sigma }_{\psi }(z_{ϵ},p_{ϵ})=1$.

On their Darboux plane scale $z_{ϵ}$ by ${ϵ}^2$ and $p_{ϵ}$ by ${ϵ}^{-2}$, fixing its symplectic orthogonal complement. Explicitly,

Prepare

This is a centered regular quasifree state. Its covariance differs from the reference by finite sums of products of smooth solutions: smearing one variable of a Hadamard bisolution with a compact smooth test gives a smooth solution in the other variable. Thus its Hadamard singularity is unchanged.

Choose $X_z$ containing the support of $z_{\mathrm{red}}$, and put

For $ϵ\le 1,{ϵ}^2{\Lambda }^2\le 1/2$,

A sufficient nonzero-denominator condition is

It follows by expanding ${\sigma }_{\psi }(z_{ϵ},p_0)=c_0/ϵ+ϵ{\sigma }_{\psi }(j_{ϵ},p_0)$ and applying the covariance bound to the symplectic form.

The conjugate variance satisfies

This is the uncertainty inequality for a pair of commutator $\mathrm{i}I$. The squeeze magnitude is $2\log (1/ϵ)$; the processing test is $z/ϵ$. No bounded-effort preparation limit is asserted.