A common parametrization for finite mode Gaussian states, their symmetries, and associated contractions with some applications

Let $Γ(H)$ be the boson Fock space over a finite dimensional Hilbert space $H$⁠. It is shown that every Gaussian symmetry admits a Klauder–Bargmann integral representation in terms of coherent states. Furthermore, Gaussian states, Gaussian symmetries, and second quantization contractions belong to a weakly closed self-adjoint semigroup $E2(H)$ of bounded operators in $Γ(H)$⁠. This yields a common parametrization for these operators. It is shown that the new parametrization for Gaussian states is a fruitful alternative to the customary parametrization by position–momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries: (i) every Gaussian state ρ admits a factorization $ρ=Z1†Z1$⁠, where Z₁ is an element of $E2(H)$ and has the form $Z1=cΓ(P)exp∑r=1nλrar+∑r,s=1nαrsaras$ on the dense linear manifold generated by all exponential vectors, where c is a positive scalar, Γ(P) is the second quantization of a positive contractive operator P in $H$⁠, a_r, 1 ≤ r ≤ n, are the annihilation operators corresponding to the n different modes in $Γ(H)$⁠, $λr∈C$⁠, and [α_rs] is a symmetric matrix in $Mn(C)$⁠; (ii) an explicit particle basis expansion of an arbitrary mean zero pure Gaussian state vector along with a density matrix formula for a general Gaussian state in terms of its $E2(H)$-parameters; (iii) a class of examples of pure n-mode Gaussian states that are completely entangled; (iv) tomography of an unknown Gaussian state in $Γ(Cn)$ by the estimation of its $E2(Cn)$ parameters using O(n²) measurements with a finite number of outcomes.