The notion of ideal detectors, as discussed here, should not be confused with that of ideal quantum measurements. In this context, ideal detectors are defined as those that remain inactive in the vacuum state but produce clicks in certain excited states. In contrast, ideal quantum measurements refer to projective measurements onto the eigenbasis of the corresponding observable.

This follows from microcausality, which ensures that $\widehat{U}$ and ${\widehat{E}}_{\mathrm{click}}$ commute [25], and from the unitarity of $\widehat{U}$, so that $⟨\mathrm{\Omega }|{\widehat{U}}^{†}{\widehat{E}}_{\mathrm{click}}\widehat{U}|\mathrm{\Omega }⟩=⟨\mathrm{\Omega }|{\widehat{U}}^{†}\widehat{U}{\widehat{E}}_{\mathrm{click}}|\mathrm{\Omega }⟩=⟨\mathrm{\Omega }|{\widehat{E}}_{\mathrm{click}}|\mathrm{\Omega }⟩$ [27].

This can be shown by contradiction: suppose the family ${\widehat{A}}_{\zeta }$ converges to a bounded operator $\widehat{A}$. Then, $(\widehat{U}-\widehat{A})|\mathrm{\Omega }⟩=0$, and by the separating property of the vacuum [4], it would follow that $\widehat{U}=\widehat{A}$. This contradicts the fact that the two operators are localized in disjoint regions. Hence, ${\widehat{A}}_{\zeta }$ must become unbounded as ${\mathcal{E}}_{\zeta }\to 0$.