We characterize the class of all multimode quantum Gaussian states that can be reduced to its normal form using only a passive unitary operator. In the language of optics, we find the complete class of Gaussian states of light that can be reduced to normal form using a linear optical device called a Mach-Zehnder interferometer (MZI). We also discuss the complete classification of classical gaussian states and its connection with passive unitary normalizable gaussian states and circularly symmetric gaussian states. This is a joint work with Saikat Guha and Hemant Kumar Mishra.