A measure of the non-Gaussian character of a quantum state

We address the issue of quantifying the non-Gaussian character of a bosonic quantum state and introduce a non-Gaussianity measure based on the Hilbert- Schmidt distance between the state under examination and a reference Gaussian state. We analyze in details the properties of the proposed measure and exploit it to evaluate the non-Gaussianity of some relevant single- and multi- mode quantum states. The evolution of non-Gaussianity is also analyzed for quantum states undergoing the processes of Gaussification by loss and de-Gaussification by photon-subtraction. The suggested measure is easily computable for any state of a bosonic system and allows to define a corresponding measure for the non-Gaussian character of a quantum operation. Also other kind of measures based on entropic quantities like quantum relative entropy are investigated.