Conversion of Gaussian states under incoherent Gaussian operations

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.