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We address the issue of quantifying the non-Gaussian character of a bosonicquantum state and introduce a non-Gaussianity measure based on theHilbert-Schmidt distance between the state under examination and a referenceGaussian state. We analyze in details the properties of the proposed measureand exploit it to evaluate the non-Gaussianity of some relevant single- andmulti-mode quantum states. The evolution of non-Gaussianity is also analyzedfor quantum states undergoing the processes of Gaussification by loss andde-Gaussification by photon-subtraction. The suggested measure is easilycomputable for any state of a bosonic system and allows to define acorresponding measure for the non-Gaussian character of a quantum operation.

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