  Setting off from the classic input-output formalism, we develop a theoretical
framework to characterise the Gaussian quantum channels relating the initial
correlations of an open bosonic system to those of properly identified output
modes. We then proceed to apply our formalism to the case of quantum harmonic
oscillators, such as the motional degrees of freedom of trapped ions or
nanomechanical oscillators, interacting with travelling electromagnetic modes
through cavity fields and subject to external white noise. Thus, we determine
the degree of squeezing that can be transferred from an intra-cavity oscillator
to light, and also show that the intra-cavity squeezing can be transformed into
distributed optical entanglement if one can access both output fields of a
two-sided cavity.
