Quantifying non-Gaussianity for quantum information

We address the quantification of non-Gaussianity (nG) of states and operations in continuous-variable systems
and its use in quantum information. We start by illustrating in detail the properties and the relationships of two recently proposed measures of nG based on the Hilbert-Schmidt distance and the quantum relative entropy (QRE) between the state under examination and a reference Gaussian state. We then evaluate the non-Gaussianities of several families of non-Gaussian quantum states and show that the two measures have the same basic properties and also share the same qualitative behavior in most of the examples taken into account. However, we also show
that they introduce a different relation of order; that is, they are not strictly monotone to each other. We exploit the nG measures for states in order to introduce a measure of nG for quantum operations, to assess Gaussification and de-Gaussification protocols, and to investigate in detail the role played by nG in entanglement distillation protocols. Besides, we exploit the QRE-based nG measure to provide different insight on the extremality of
Gaussian states for some entropic quantities such as conditional entropy, mutual information, and the Holevo bound. We also deal with parameter estimation and present a theorem connecting the QRE nG to the quantum
Fisher information. Finally, since evaluation of the QRE nG measure requires the knowledge of the full density matrix, we derive some experimentally friendly lower bounds to nG for some classes of states and by considering the possibility of performing on the states only certain efficient or inefficient measurements.