A Common Parametrization for Finite Mode Gaussian States, their Symmetries and associated Contractions with some Applications

Let Γ\(mathcal{H}\) be the boson Fock space over a finite dimensional Hilbert space mathcal{H}. It is shown that every gaussian symmetry admits a Klauder-Bargmann integral representation in terms of coherent states. Furthermore, gaussian symmetries, gaussian states and second quantization contractions, all of these operators belong to a weakly closed, selfadjoint semigroup mathcal{E}₂\(mathcal{H}\) of bounded operators in Γ\(mathcal{H}\). This yields, a new parametrization of gaussian states, which is a very 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^daggerZ₁, where Z₁ is an element of mathcal{E}₂\(mathcal{H}\) and has the form Z₁ = sqrt{c}Γ\(sqrtΛ\)exp{sum_r=1ⁿ λ_rar+sum_r,s=1ⁿ α_rsa_ra_s} on the dense linear manifold generated by all exponential vectors, Λ being a positive operator in mathcal{H}, a_r, 1leq r leq n are the annihilation operators corresponding to the n different modes in Γ\(mathcal{H}\), λ_rin mathbb{C} and \[α_rs\] is a symmetric matrix in M_n\(mathbb{C}\); (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 mathcal{E}₂\(mathcal{H}\)-parameters; (iii) an easy test for the entanglement of pure gaussian states and a class of examples of pure n-mode gaussian states which are completely entangled; (iv) tomography of an unknown gaussian state in Γ\(mathbb{C}ⁿ\) by the estimation of its mathcal{E}₂\(mathbb{C}ⁿ\)-parameters using O\(n²\) measurements with a finite number of outcomes.