Gaussian dynamics in the double Siegel disk

• URL: http://arxiv.org/abs/2603.05824v2
• Date: Mon, 09 Mar 2026 04:15:59 GMT
• Title: Gaussian dynamics in the double Siegel disk
• Abstract summary: We show that deterministic multimode Gaussian channels admit a symmetric-space description.<n>This gives a bridge between covariance-matrix channel theory and the adjacency-matrix or symmetric-space picture.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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