  We compare two sets of multimode quantum channels acting on a finite
collection of harmonic oscillators: (a) the set of linear bosonic channels,
whose action is described as a linear transformation at the phase space level;
and (b) Gaussian dilatable channels, that admit a Stinespring dilation
involving a Gaussian unitary. Our main result is that the set (a) coincides
with the closure of (b) with respect to the strong operator topology. We also
present an example of a channel in (a) which is not in (b), implying that
taking the closure is in general necessary. This provides a complete resolution
to the conjecture posed in Ref. [K.K. Sabapathy and A. Winter, Phys. Rev. A
95:062309, 2017]. Our proof technique is constructive, and yields an explicit
procedure to approximate a given linear bosonic channel by means of Gaussian
dilations. It turns out that all linear bosonic channels can be approximated by
a Gaussian dilation using an ancilla with the same number of modes as the
system. We also provide an alternative dilation where the unitary is fixed in
the approximating procedure. Our results apply to a wide range of physically
relevant channels, including all Gaussian channels such as amplifiers,
attenuators, phase conjugators, and also non-Gaussian channels such as additive
noise channels and photon-added Gaussian channels. The method also provides a
clear demarcation of the role of Gaussian and non-Gaussian resources in the
context of linear bosonic channels. Finally, we also obtain independent proofs
of classical results such as the quantum Bochner theorem, and develop some
tools to deal with convergence of sequences of quantum channels on continuous
variable systems that may be of independent interest.
