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

Let $\mathrm{\Gamma }(\mathcal{H})$ 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 $\mathrm{\Gamma }(\mathcal{H})$. 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 $\rho$ admits a factorization $\rho = Z_{1}^{\dagger}Z_{1}$,where$Z_{1}$isanelementof$$\mathcal{E}$2($\mathcal{H}$)$andhastheform$Z{1} = \sqrt{c}\Gamma(\sqrt{\Lambda})\exp{\sum_{r=1}^{n} \lambda_ra_r+\sum_{r,s=1}^{n} \alpha_{rs}a_{r}a_{s}}$ on the dense linear manifold generated by all exponential vectors, $\mathrm{\Lambda }$ being a positive operator in $\mathcal{H}$, $a_r,1\le r\le n$ are the annihilation operators corresponding to the $n$ different modes in $\mathrm{\Gamma }(\mathcal{H})$, ${\lambda }_r\in \mathbb{C}$ and $[{\alpha }_{rs}]$ 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 $\mathrm{\Gamma }({\mathbb{C}}^n)$ by the estimation of its ${\mathcal{E}}_2({\mathbb{C}}^n)$-parameters using $O(n^2)$ measurements with a finite number of outcomes.