Why fixed probe noise is not negligible

Prepare a reference probe in an independent copy ${\sigma }_{\mathrm{ref}}$ of the bulk Unruh state. This is a centered regular quasifree Hadamard state of the same massless probe equation. For its real s-wave tests,

The probe reference could be replaced by another explicitly controlled covariance. It must not be replaced by a formal “zero-noise” state.

The leading residual test in (eq:residualprobe) is $g/ϵ$. Since $[g]\ne 0$, $V_{\mathrm{ref}}(g,g)>0$, and

for fixed $R,T$. Thus a fixed reference preparation gives $c_{ϵ}\to 0$, not one. Its induced effect tends toward $I/2$.

Dividing the measured Weyl observable by $c_{ϵ}$ would in general destroy the effect bound. The following construction suppresses noise through the preparation instead.