  We generalize Gisin's theorem on the relation between the entanglement of
pure states and Bell non-classicality to the case of mode entanglement of
separated groups of modes of quantum fields extending the theorem to cover also
states with undefined particle number. We show that any pure state of the field
which contains entanglement between two groups of separated modes violates some
Clauser-Horne inequality. In order to construct the observables leading to a
violation in the first step, we show an isomorphism between the Fock space
built from a single-particle space involving two separated groups of modes and
a tensor product of two abstract separable Hilbert spaces spanned by formal
monomials of creation operators. In the second step, we perform a Schmidt
decomposition of a given entangled state mapped to this tensor product space
and then we map back the obtained Schmidt decomposition to the original Fock
space of the system under consideration. Such obtained Schmidt decomposition in
Fock space allows for construction of observables leading to a violation of the
Clauser-Horne inequality. We also show that our generalization of Gisin's
theorem holds for the case of states on non-separable Hilbert spaces, which
physically represent states with actually infinite number of particles. Such
states emerge, for example, in the discussion of quantum phase transitions.
Finally, we discuss the experimental feasibility of constructed Bell test and
provide a necessary condition for realizability of this test within the realm
of passive linear optics.
