Gaussian dynamics in the double Siegel disk

Abstract: We show that multimode deterministic (CPTP) Gaussian channels admit a symmetric-space description: by passing from the (n)-mode Siegel disk (Δ{n}) to the (2n)-mode \emph{double Siegel disk} (Δ{2n}), general Gaussian dynamics becomes a linear-fractional (Möbius) action on a single matrix representative. Concretely, (Δ{2n}) naturally parametrizes Gaussian kernels in the Fock--Bargmann representation, and we identify an explicit physical subset (\sspace\subsetΔ{2n}) corresponding to valid mixed Gaussian states. We then construct, from the standard ((X,Y)) parametrization of a deterministic Gaussian channel, a normalized oscillator-semigroup element (\bar E) whose fractional action implements the channel update (\amat\mapstoφ_{\bar E}(\amat)) on (\sspace); Gaussian unitaries arise as the symplectic (isometric) subcase. The resulting calculus bridges covariance-matrix channel theory with the adjacency-matrix/symmetric-space picture, retains a simple composition law (matrix multiplication of the acting blocks), and suggests a direct route to graphical update rules beyond the pure-state setting.