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

This is a continuation of the expository article by Parthasarathy (Commun Stoch Anal 4:143–160, 2010) 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 \(\frac{1} {4}({L}^{T}L + {M}^{T}M)\) 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 \(U = \lambda W(\boldsymbol \alpha )\varGamma (L)\) where λ is a complex scalar of modulus unity, \(\boldsymbol \alpha \in {\mathbb{C}}^{n},\) \(L \in Sp(2n, \mathbb{R}),\) \(W(\boldsymbol \alpha )\) is the Weyl operator corresponding to \(\boldsymbol \alpha \) and \(\varGamma (L)\) 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.