ABSTRACT
Gaussian operations, as one usually refers to CP-maps that send Gaussian states into Gaussian states, are apt to model a wide range of situations of practical interest and may often be controlled and applied on demand in the laboratory, in a variety of set-ups. This chapter starts from start from Gaussian unitary operations, corresponding to the symplectic group, and moves on to clarify how other basic manipulations such as tensoring and partial tracing carry over to the Gaussian picture. Then, it includes all deterministic (trace-preserving) maps resulting from Gaussian unitary dilations, which are often known as the set of "bosonic Gaussian channels" and encompasses the open dynamics subject to noise and 'decoherence'. Next, the chapter considers POVMs corresponding to the well-known homodyne and heterodyne detection schemes. Finally, it introduces the Gaussian version of the Choi-Jamiolkowski isomorphism, and attains a unified description of all the Gaussian maps.
