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Gaussian dynamics in the double Siegel disk

Giacomo Pantaleoni, Nicolas C. Menicucci

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.

Gaussian dynamics in the double Siegel disk

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.
Paper Structure (30 sections, 13 theorems, 150 equations, 9 figures)

This paper contains 30 sections, 13 theorems, 150 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Basic definitions and conventions
  3. Pure Gaussian Dynamics in Symmetric Spaces
  4. Kinematics
  5. Dynamics
  6. Non-unitary dynamics in the Siegel upper half-plane
  7. The Siegel Disk
  8. Identifying the double disk representative
  9. The Double Siegel Disk
  10. Notation
  11. Single-Kraus Gaussian operations in $\Delta_{2n}$
  12. The state space
  13. Gaussian unitaries on mixed states
  14. Normal-mode decomposition in the disk
  15. Gaussian operations and channels in the Siegel Disk
  16. ...and 15 more sections

Key Result

theorem 1

If $S_{r}\in\mathrm{Sp}_{2n}$, then $\phi_{S_{r}}(\Sigma_n)\subseteq\Sigma_n$. Equivalently, $\phi_{S_{r}}$ maps pure Gaussian states to pure Gaussian states in the upper half-plane representation. See Ref. folland_harmonic_1989.

Figures (9)

  • Figure 1: Upper half-plane intuition ($n=1$). The Siegel upper half-plane is $\Sigma_n=\{Z=Z^{T},\ \operatorname{Im}Z>0\}$; for one mode it reduces to $\{z\in\mathbb{C}\mid \operatorname{Im}z>0\}$. The vacuum corresponds to $Z_{0}=i$ (i.e. $iI_n$ in general). A Gaussian unitary $S_r\in\mathrm{Sp}_{2n}$ acts by a Möbius-type fractional transformation $Z\mapsto\phi_{S_r}(Z)$, which preserves $\operatorname{Im}Z>0$.
  • Figure 2: Schematic inclusion $\mathrm{Sp}_{2n}\subset\mathrm{Sp}_{2n}^+\subset\mathrm{Sp}_{2n}(\mathbb{C})$. The complex symplectic group $\mathrm{Sp}_{2n}(\mathbb{C})$ consists of $2n\times2n$ complex matrices satisfying $S^{T}\sigma_{y}S=\sigma_{y}$. The Siegel-domain preserving semigroup is $\mathrm{Sp}_{2n}^+=\{T\in\mathrm{Sp}_{2n}(\mathbb{C})\mid T^{\dag}\sigma_{y}T\ge\sigma_{y}\}$; its induced fractional transformations preserve the Siegel upper half-plane $\Sigma_n=\{Z=Z^{T},\ \operatorname{Im}Z>0\}$. The boundary $\partial\mathrm{Sp}_{2n}^+=\mathrm{Sp}_{2n}$ (where the inequality saturates) recovers Gaussian unitaries.
  • Figure 3: Oscillator-semigroup dictionary (upper half-plane). Elements of the normalized oscillator semigroup $\Omega^{0}$ are Gaussian Kraus operators (Gauss-kernel operators) parametrized by the doubled Siegel domain $\Sigma_{2n}=\Sigma_{2n}$. Modulo the standard $\pm$ ambiguity, $\Omega^{0}$ projects to the matrix semigroup $\mathrm{Sp}_{2n}^+\subset\mathrm{Sp}_{2n}(\mathbb{C})$. The notation $G\curvearrowright X$ denotes an action: each $T\in G$ determines an endomorphism $\phi_T\colon X\to X$, and evaluating it gives the update $Z\mapsto \phi_T(Z)$. In our case $X=\Sigma_n$ and $\phi_T$ is the linear fractional map induced by $T$. This is the pure-state precursor of the channel-level picture developed later.
  • Figure 4: Disk intuition ($n=1$). The Siegel disk is $\Delta_n=\{K=K^{T},\ I-K^{\dag}K>0\}$; for one mode it reduces to the unit disk $\{k\in\mathbb{C}\mid |k|<1\}$. The vacuum corresponds to $K_{0}=0$. A Gaussian unitary $S\in\mathrm{Sp}_{2n}^{\Gamma}$ acts by a Möbius-type fractional transformation $K\mapsto\phi_{S}(K)$, which preserves $\Delta_n$ and is related to the upper half-plane action by the Cayley transform.
  • Figure 5: Disk-preserving semigroup. The Siegel-disk preserving semigroup is $\mathrm{Sp}_{2n}^{+\Gamma}=\{T\in\mathrm{Sp}_{2n}(\mathbb{C})\mid T^{\dag}\sigma_{z}T\ge\sigma_{z}\}$ (with $T^{T}\sigma_{y}T=\sigma_{y}$ automatically since $\mathrm{Sp}_{2n}^{+\Gamma}\subset\mathrm{Sp}_{2n}(\mathbb{C})$). Its induced fractional transformations map the Siegel disk $\Delta_n=\{K=K^{T},\ I-K^{\dag}K>0\}$ into itself, describing single-Kraus Gaussian operations on pure states in disk coordinates. The boundary $\partial\mathrm{Sp}_{2n}^{+\Gamma}=\mathrm{Sp}_{2n}^{\Gamma}$ (equality $S^{\dag}\sigma_{z}S=\sigma_{z}$) recovers Gaussian unitaries in the ABC/disk realization.
  • ...and 4 more figures

Theorems & Definitions (54)

  • definition 1: Real symplectic group
  • definition 2: Lifted Pauli matrices
  • definition 3: Gaussian state
  • definition 4: Metaplectic representation and group
  • definition 5: Cayley matrix
  • definition 6: Adjoint-by-Cayley (ABC)
  • definition 7: Siegel upper half-plane
  • definition 8: Fock-Bargmann wavefunction
  • definition 9: Siegel disk
  • definition 10: Linear fractional transformation
  • ...and 44 more