  The coherence resource theory needs to study the operational value and
efficiency which can be broadly formulated as the question: when can one
coherent state be converted into another under incoherent operations. We answer
this question completely for one-mode continuous-variable systems by
characterizing conversion of coherent Gaussian states under incoherent Gaussian
operations in terms of their first and second moments. The no-go theorem of
purification of coherent Gaussian states is also built. The structure of
incoherent Gaussian operations of two-mode continuous-variable systems is
discussed further and is applied to coherent conversion for pure Gaussian
states with standard second moments. The standard second moments are images of
all second moments under local linear unitary Bogoliubov operations. As
concrete applications, we obtain some peculiarities of a Gaussian system: (1)
There does not exist a maximally coherent Gaussian state which can generate all
coherent Gaussian states; (2) The conversion between pure Gaussian states is
reversible; (3) The coherence of input pure state and the coherence of output
pure state are equal.
