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

Let $\Gamma(\mathcal{H})$ be the boson Fock space over a finite dimensional Hilbert $\mathcal{H}$. It is shown that every gaussian symmetry admits Klauder-Bargmann integral representation in terms of coherent states. Furthermore, symmetries, states and second quantization contractions, all these operators belong to weakly closed, selfadjoint semigroup $\mathcal{E}_2(\mathcal{H})$ bounded $\Gamma(\mathcal{H})$. This yields, new parametrization states, which very fruitful alternative customary by position-momentum mean vectors covariance matrices. leads rich harvest corollaries: (i) state $\rho$ factorization $\rho = Z_{1}^{\dagger}Z_{1}$, where $Z_{1}$ an element has form $Z_{1} \sqrt{c}\Gamma(\sqrt{\Lambda})\exp{\sum_{r=1}^{n} \lambda_ra_r+\sum_{r,s=1}^{n} \alpha_{rs}a_{r}a_{s}}$ on dense linear manifold generated exponential vectors, $\Lambda$ being positive operator $\mathcal{H}$, $a_{r}, 1\leq r \leq n$ are annihilation corresponding $n$ different modes $\Gamma(\mathcal{H})$, $\lambda_r\in \mathbb{C}$ $[\alpha_{rs}]$ symmetric matrix $M_n(\mathbb{C})$; (ii) explicit particle basis expansion arbitrary zero pure vector along with density formula for general its $\mathcal{E}_2(\mathcal{H})$-parameters; (iii) easy test entanglement class examples $n$-mode completely entangled; (iv) tomography unknown $\Gamma(\mathbb{C}^n)$ estimation $\mathcal{E}_2(\mathbb{C}^n)$-parameters using $O(n^2)$ measurements number outcomes.