  Quantum field theory is the application of quantum physics to fields. It
provides a theoretical framework widely used in particle physics and condensed
matter physics. One of the most distinct features of quantum physics with
respect to classical physics is entanglement or the existence of strong
correlations between subsystems that can even be spacelike separated. In
quantum fields, observables restricted to a region of space define a subsystem.
While there are proofs on the existence of local observables that would allow a
violation of Bell's inequalities in the vacuum states of quantum fields as well
as some explicit but technically demanding schemes requiring an extreme
fine-tuning of the interaction between the fields and detectors, an
experimentally accessible entanglement witness for quantum fields is still
missing. Here we introduce smeared field operators which allow reducing the
vacuum to a system of two effective bosonic modes. The introduction of such
collective observables is motivated by the fact that no physical probe has
access to fields in single spatial (mathematical) points but rather smeared
over finite volumes. We first give explicit collective observables whose
correlations reveal vacuum entanglement in the Klein-Gordon field. We then show
that the critical distance between the two regions of space above which two
effective bosonic modes become separable is of the order of the Compton
wavelength of the particle corresponding to the massive Klein-Gordon field.
