Developing improved measures of non-Gaussianity and Gaussianity for quantum states based on normalized Hilbert-Schmidt distance

Non-Gaussianity of quantum states is a very important source for quantum information technology and can be quantified by using the known squared Hilbert-Schmidt distance recently introduced by Genoni et al. (Phys. Rev. A 78 042327 (2007)). It is, however, shown that such a measure has many imperfects such as the lack of the swapping symmetry and the ineffectiveness evaluation of even Schrodinger-cat-like states with small amplitudes. To deal with these difficulties, we propose an improved measure of non-Gaussianity for quantum states and discuss its properties in detail. We then exploit this improved measure to evaluate the non-Gaussianities of some relevant single-mode non-Gaussian states and multi-mode non-Gaussian entangled states. These results show that our measure is reliable. We also introduce a modified measure for Gaussianity following Mandilara and Cerf (Phys. Rev. A 86 030102(R) (2012)) and establish a conservation relation of non-Gaussianity and Gaussianity of a quantum state.