2011/01/26 by Parthasarathy, K. R.
#42A82 #60B15 #81R30 #81S25 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1101.5041
This is a continuation of the expository article \citekrp with some new remarks. Let Sn denote the set of all Gaussian states in the complex Hilbert space L2 (ℝn), Kn the convex set of all momentum and position covariance matrices of order 2n in Gaussian states and let Gn be the group of all unitary operators in L2 (ℝn) conjugations by which leave Sn invariant. Here we prove the following results. Kn is a closed convex set for which a matrix S is an extreme point if and only if S=(1)/(2) LT L for some L in the symplectic group Sp (2n, ℝ). Every element in Kn is of the form (1)/(2) (LT L + MT M) for some L,M in Sp (2n, ℝ). Every Gaussian state in L2 (ℝn) can be purified to a Gaussian state in L2 (ℝ2n). Any element U in the group Gn is of the form U = λW (\bm α) Γ(L) where λ is a complex scalar of modulus unity, \bm α ∈ ℂn, L ∈ Sp (2n, ℝ), W(\bm α) is the Weyl operator corresponding to \bm α and Γ(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.