  Quantum Gaussian states can be considered as the majority of the practical
quantum states used in quantum communications and more generally in quantum
information. Here we consider their properties in relation with the
geometrically uniform symmetry, a property of quantum states that greatly
simplifies the derivation of the optimal decision by means of the square root
measurements. In a general framework of the $N$-mode Gaussian states we show
the general properties of this symmetry and the application of the optimal
quantum measurements. An application example is presented, to quantum
communication systems employing pulse position modulation. We prove that the
geometrically uniform symmetry can be applied to the general class of multimode
Gaussian states.
