Symplectic Dilations, Gaussian States and Gaussian Channels

By elementary matrix algebra we show that every real $2n \times 2n$ matrix admits a dilation to an element of the real symplectic group $Sp (2(n+m))$ for some nonnegative integer $m.$ Our methods do not yield the minimum value of $m,$ for which such a dilation is possible. After listing some of the main properties of Gaussian states in $L^2 (\mathbb{R}^n),$ we analyse the implications of symplectic dilations in the study of quantum Gaussian channels which lead to some interesting open problems, particularly, in the context of the work of Heinosaari, Holevo and Wolf \cite{3}.