  Non-Gaussian states and operations are crucial for various
continuous-variable quantum information processing tasks. To quantitatively
understand non-Gaussianity beyond states, we establish a resource theory for
non-Gaussian operations. In our framework, we consider Gaussian operations as
free operations, and non-Gaussian operations as resources. We define
entanglement-assisted non-Gaussianity generating power and show that it is a
monotone that is non-increasing under the set of free super-operations, i.e.,
concatenation and tensoring with Gaussian channels. For conditional unitary
maps, this monotone can be analytically calculated. As examples, we show that
the non-Gaussianity of ideal photon-number subtraction and photon-number
addition equal the non-Gaussianity of the single-photon Fock state. Based on
our non-Gaussianity monotone, we divide non-Gaussian operations into two
classes: (1) the finite non-Gaussianity class, e.g., photon-number subtraction,
photon-number addition and all Gaussian-dilatable non-Gaussian channels; and
(2) the diverging non-Gaussianity class, e.g., the binary phase-shift channel
and the Kerr nonlinearity. This classification also implies that not all
non-Gaussian channels are exactly Gaussian-dilatable. Our resource theory
enables a quantitative characterization and a first classification of
non-Gaussian operations, paving the way towards the full understanding of
non-Gaussianity.
