The Symmetry Group of Gaussian States in $L^2 (\mathbb{R}^n)$

TL;DR

This paper characterizes the symmetry group of Gaussian states in quantum harmonic analysis, describing the structure of covariance matrices and the form of unitary operators that preserve Gaussian states.

Contribution

It provides a detailed description of the symmetry group of Gaussian states, including the structure of covariance matrices and the form of unitary transformations that leave the set invariant.

Findings

K_n is a closed convex set with extreme points characterized by symplectic matrices.

Every element of K_n can be expressed as an average of two symplectic transformations.

The symmetry group consists of unitary operators combining Weyl operators and symplectic transformations.

Abstract

This is a continuation of the expository article \cite{krp} with some new remarks. Let $S_n$ denote the set of all Gaussian states in the complex Hilbert space $L^2 (\mathbb{R}^n),$ $K_n$ the convex set of all momentum and position covariance matrices of order $2n$ in Gaussian states and let $\mathcal{G}_n$ be the group of all unitary operators in $L^2 (\mathbb{R}^n)$ conjugations by which leave $S_n$ invariant. Here we prove the following results. $K_n$ is a closed convex set for which a matrix $S$ is an extreme point if and only if $S=\frac{1}{2} L^{T} L$ for some $L$ in the symplectic group $Sp (2n, \mathbb{R}).$ Every element in $K_n$ is of the form $\frac{1}{2} (L^{T} L + M^{T} M)$ for some $L,M$ in $Sp (2n, \mathbb{R}).$ Every Gaussian state in $L^2 (\mathbb{R}^n)$ can be purified to a Gaussian state in $L^2 (\mathbb{R}^{2n}).$ Any element $U$ in the group $\mathcal{G}_n$ is of the form $U = \lambda W ({\bm {\alpha}}) \Gamma (L)$ where $\lambda$ is a complex scalar of modulus unity, ${\bm {\alpha}} \in \mathbb{C}^n,$ $L \in Sp (2n, \mathbb{R}),$ $W({\bm {\alpha}})$ is the Weyl operator corresponding to ${\bm {\alpha}} $ and $\Gamma (L)$ is a unitary operator which implements the Bogolioubov automorphism of the Lie algebra generated by the canonical momentum and position observables induced by the symplectic linear transformation $L.$