Gaussian dynamics in the double Siegel disk

TL;DR

This paper introduces a symmetric-space framework for deterministic multimode Gaussian channels, enabling a unified geometric description and explicit physical subset characterization, bridging covariance-matrix theory with symmetric-space representations.

Contribution

It develops a symmetric-space model for Gaussian channels, linking covariance matrices with Möbius transformations and providing explicit physical state parametrization.

Findings

Gaussian channels are described by linear-fractional transformations.

The framework includes a physical subset corresponding to valid Gaussian states.

A composition law for channels is established via matrix multiplication.

Abstract

We show that deterministic multimode Gaussian channels admit a symmetric-space description. Passing from the n-mode Siegel disk to a doubled version of that space lets general Gaussian dynamics act by a linear-fractional (Mobius) transformation on a single matrix parameter. This doubled disk naturally parametrizes Gaussian kernels in the Fock-Bargmann representation, and contains an explicit physical subset corresponding to valid mixed Gaussian states. Starting from the standard X,Y parametrization of a deterministic Gaussian channel, we construct a normalized oscillator-semigroup element whose fractional action reproduces the channel update on that subset; Gaussian unitaries appear as the symplectic, isometric special case. This gives a bridge between covariance-matrix channel theory and the adjacency-matrix or symmetric-space picture, preserves a simple composition law given by matrix multiplication of the acting blocks, and suggests a direct route to graphical update rules beyond pure states.